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arxiv |
A STABILITY THEOREM FOR BIGRADED PERSISTENCE BARCODES
Anthony Bahri
Ivan Limonchenko
Taras Panov
Jongbaek Song
Donald Stanley
A STABILITY THEOREM FOR BIGRADED PERSISTENCE BARCODES
We define the bigraded persistent homology modules and the bigraded barcodes of a finite pseudo-metric space X using the ordinary and double homology of the moment-angle complex associated with the Vietoris-Rips filtration of X. We prove the stability theorem for the bigraded persistent double homology modules and barcodes.2020 Mathematics Subject Classification. Primary 57S12, 57Z25; Secondary 13F55, 55U10.
Introduction
Given a finite pseudo-metric space (X, d X ), the Vietoris-Rips filtration is a sequence {R(X, t)} t≥0 of filtered flag simplicial complexes associated with X. The simplicial homology of R(X, t) is used to define the most basic persistence modules in data science, the persistent homology of X.
In toric topology, a finer homological invariant of a simplicial complex K is considered, the bigraded homology of the moment-angle complex Z K associated with K. The moment-angle complex Z K is a toric space patched from products of discs and circles parametrised by simplices in a simplicial complex K. It has a bigraded cell decomposition and the corresponding bigraded homology groups H −i,2j (Z K ) contain the simplicial homology groups H k (K) as a direct summand. Algebraically, the bigraded homology modules H −i,2j (Z K ) are the bigraded components of the Tor-modules of the Stanley-Reisner ring k[K] and can be expressed via the Hochster decomposition as the sum of the reduced simplicial homology groups of all full subcomplexes K I of K.
The bigraded homology of the moment-angle complexes Z R(X,t) associated with the Vietoris-Rips filtration {R(X, t)} t≥0 can be used to define bigraded persistent homology modules and bigraded barcodes of a point cloud (data set) X, as observed in [LPSS]. Similar ideas have been pursued in [LFLX, LX]. Simple examples show that bigraded persistent homology can distinguish between points clouds that are indistinguishable by the ordinary persistent homology (see Example 3.4).
However, the approach to define bigraded persistence via the homology of momentangle complexes has two fundamental drawbacks. First, calculating simplicial homology groups of all full subcomplexes (or samples) in R(X, t) is much resourceconsuming from the computational viewpoint. Second, bigraded persistent homology lacks the stability property, which is important for applications of persistent homology in data science [CEH].
Bigraded double homology of moment-angle complexes was introduced and studied in [LPSS]. Besides theoretical interest in toric topology and polyhedral products theory, bigraded double homology fixes both drawbacks of ordinary bigraded homology mentioned above, therefore opening a way for a more efficient application in data analysis.
Double homology HH * (Z K ) is the homology of the chain complex CH * (Z K ) = (H * (Z K ), ∂ ) obtained by endowing the bigraded homology of Z K with the second differential ∂ . The bigraded double homology modules are smaller than the bigraded homology modules, and therefore might be more accessible from a computational viewpoint. More importantly, persistent homology modules defined from the bigraded double homology of the Vietoris-Rips filtration have the stability property, which roughly says that the bigraded barcode is robust to changes in the input data.
In Section 2 we review persistence modules, standard persistent homology and barcodes.
In Section 3 we define the bigraded persistent homology and persistent double homology modules and the corresponding bigraded barcodes.
The Stability Theorem for persistent homology says that the Gromov-Hausdorff distance between two finite pseudo-metric spaces (data sets) is bounded from below by the interleaving distance between their persistent homology modules. The interleaving distance between persistent homology modules can be identified, via the Isometry Theorem, with the ∞-Wasserstein (bottleneck) distance between the corresponding barcodes.
In Section 4 we prove the Stability Theorem for bigraded persistent double homology (Theorem 4.22). This is done in two stages. First, we prove that bigraded persistent homology satisfies the stability property with respect to a modified Gromov-Hausdorff distance, in which the infimum is taken over bijective correspondences only (Theorem 4.20). Second, the stability for bigraded persistent double homology is established using the invariance of double homology with respect to the doubling operation on simplicial complexes.
Acknowledgements. Research of Limonchenko and
• φ s1,s2 = φ s1,s3 whenever s 1 ≤ s 2 ≤ s 3 in R ≥0 .
In a more general version of persistence modules, the domain category R ≥0 can be replaced by any small category and the codomain k-mod by any Grothendieck category. See for example [BSS].
Example 2.1. An interval is a subset I ⊂ R ≥0 such that if r ∈ I and t ∈ I and r < s < t, then s ∈ I (such an interval may be closed or open from each side, and may have infinite length). Given an interval I, we define
k I s := k if s ∈ I; 0 otherwise.
Then, the family {k I s } s∈R ≥0 together with the family {φ s1,s2 : k I s1 → k I s2 } s1≤s2 of morphisms, where φ s1,s2 is the identity if s 1 , s 2 ∈ I and the zero map otherwise, forms a persistence module. We denote it by k(I) and refer to as the interval module corresponding to I.
The direct sum M ⊕ N of two persistence modules M = {M s } s∈R ≥0 and N = {N s } s∈R ≥0 is defined pointwise as the family of k-modules {M s ⊕ N s } s∈R ≥0 . The following decomposition theorem for a persistence module is originally due to Zomorodian and Carlsson [ZC,Theorem 2.1] for the case where the domain category is Z ≥0 and k is a field. It is generalized to the case where the domain category is R ≥0 or R by Crawley-Boevey [Cr,Theorem 1.1]. We also refer to [BC,Theorem 1.2] for a simpler proof. The above result gives us a discrete invariant of a persistence module M, namely the multiset B(M) of (2.2), which is called the barcode of M.
2.2. The Vietoris-Rips filtration and persistent homology. A typical example which fits into Theorem 2.2 is the persistent homology of filtered simplicial complexes.
A finite pseudo-metric space is a nonempty finite set X together with a function
d X : X × X → R ≥0 satisfying d X (x, x) = 0, d X (x, y) = d X (y, x) and d X (x, z) ≤ d X (x, y) + d X (y, z)
for every x, y, z ∈ X. Note that two distinct points may have distance 0 in X. We can think of a finite pseudo-metric space (X, d X ) as a point cloud.
The Vietoris-Rips filtration {R(X, t)} t≥0 associated with (X, d X ) consists of the Vietoris-Rips simplicial complexes R(X, t). The latter is defined as the clique complex of the graph whose vertex set is X and two vertices x and y are connected by an edge if d X (x, y) ≤ t. We have a simplicial inclusion
(2.3) R(X, t 1 ) → R(X, t 2 )
whenever t 1 ≤ t 2 . See Figure 1 for an example of Vietoris-Rips complexes. The n-dimensional persistent homology module
X = R(X, 0) → · · · → R(X, t 1 ) → · · · → R(X, t 2 ) → · · · → R(X, t 3 ) → · · ·PH n (X) : R ≥0 → k-mod, t → H n (R(X, t)),
maps t ∈ R ≥0 to the reduced simplicial homology group H n (R(X, t)) with coefficients in k and maps a morphism t 1 ≤ t 2 to the homomorphism H n (R(X, t 1 )) → H n (R(X, t 2 )) induced by (2.3). We also define the graded persistent homology module
(2.4) PH(X) = n≥0 PH n (X).
Applying Theorem 2.2 to the graded persistence module (2.4), we obtain the barcode of PH(X), which we denote simply by B(X). In this case, B(X) can be described more explicitly as follows.
A homology class α ∈ H n (R(X, t)) is said to
(1) be born at r if (i) α ∈ im H n (R(X, r)) → H n (R(X, t)) ; (ii) α / ∈ im H n (R(X, p)) → H n (R(X, t)) for p < r, (2) die at s if (i) α ∈ ker H n (R(X, t)) → H n (R(X, s)) ; (ii) α / ∈ ker H n (R(X, t)) → H n (R(X, q)) for q < s. We set s = ∞ if α / ∈ ker H n (R(X, t)) → H n (R(X, q)) for any q.
If α ∈ H n (R(X, t)) is born at r and dies at s, then [r, s) is called the persistence interval of α. Then, the barcode B(X) is the collection of persistence intervals of generators of the homology groups H * (R(X, t)). For each t ∈ R ≥0 , the dimension of H n (R(X, t)) is equal to the number of n-dimensional persistence intervals containing t.
Since (X, d X ) is a finite pseudo-metric space, it follows that R(X, t) is contractible when t is large enough. Therefore, the corresponding barcode B(X) is a finite collection of half-open persistence intervals with finite lengths. If we use the unreduced simplicial homology in the definition of persistent homology, then the barcode would contain a single infinite interval [0, ∞) in dimension 0. Figure 2 displays the unreduced barcode corresponding to the Vietoris-Rips filtration in Figure 1.
(i 1 , . . . , i k ) ⊂ [m] that is contained in K as a simplex. A one-element simplex {i} ∈ K is a vertex.
We also assume that ∅ ∈ K and, unless explicitly stated otherwise, that K contains all one-element subsets {i} ∈ [m] (that is, K is a simplicial complex without ghost vertices).
Let D 2 be the unit disc in R 2 , and let S 1 be its boundary circle. For each simplex I ∈ K, consider the topological space
(D 2 , S 1 ) I := {(z 1 , . . . , z m ) ∈ (D 2 ) m : |z j | = 1 if j / ∈ I} ⊂ (D 2 ) m .
Note that (D 2 , S 1 ) I is a natural subspace of (D 2 , S 1 ) J whenever I ⊂ J. The moment-angle complex corresponding to K is
Z K := I∈K (D 2 , S 1 ) I ⊂ (D 2 ) m .
We refer to [BP,Chapter 4] for more details and examples. Let k be a coefficient ring, which we assume to be a principal ideal domain. The face ring (the Stanley-Reisner ring) of a simplicial complex K is
k[K] := k[v 1 , . . . , v m ]/I K , where I K is the ideal generated by square-free monomials i∈I v i for which I ⊂ [m]
is not a simplex of K.
The following theorem summarizes several presentations of the cohomology ring H * (Z K ).
Theorem 3.1 ( [BP,Section 4.5]). There are isomorphisms of bigraded commutative algebras
H * (Z K ) ∼ = Tor k[v1,...,vm] k, k[K] ∼ = H Λ[u 1 , . . . , u m ] ⊗ k[K], d (3.1) ∼ = I⊂[m] H * (K I ). (3.2)
Here, (3.1) is cohomology of the bigraded algebra with bidegrees bideg u i = (−1, 2), bideg v i = (0, 2) and differential d =
m i=1 v i ∂ ∂ui of bidegree (1, 0) (the Koszul complex). In (3.2), each component H * (K I ) denotes the reduced simplicial coho- mology of the full subcomplex K I ⊂ K (the restriction of K to I ⊂ [m]). The last isomorphism is the sum of isomorphisms H p (Z K ) ∼ = I⊂[m] H p−|I|−1 (K I ),
and the ring structure is given by the maps
H p−|I|−1 (K I ) ⊗ H q−|J|−1 (K J ) → H p+q−|I|−|J|−1 (K I∪J )
which are induced by the canonical simplicial maps K I∪J → K I * K J for I ∩ J = ∅ and zero otherwise.
Isomorphism (3.2) is often referred to as the Hochster decomposition, as it comes from Hochster's theorem describing Tor k[v1,...,vm] (k, k[K]) as a sum of cohomology of full subcomplexes K I .
The bigraded components of cohomology of Z K are given by
H −i,2j (Z K ) ∼ = H −i,2j Λ[u 1 , . . . , u m ] ⊗ k[K], d ∼ = J⊂[m] : |J|=j H j−i−1 (K J ), so that H p (Z K ) = −i+2j=p H −i,2j (Z K ) ∼ = J⊂[m] H p−|J|−1 (K J ),
There is a similar description of bigraded homology of Z K :
H p (Z K ) = −i+2j=p H −i,2j (Z K ) ∼ = J⊂[m] H p−|J|−1 (K J ).
The construction of moment-angle complex and its bigraded homology is functorial with respect to inclusion of simplicial complexes:
Proposition 3.2. An inclusion of subcomplex K ⊂ L induces an inclusion Z K ⊂ Z L and a homomorphism H −i,2j (Z K ) → H −i,2j (Z L ) of bigraded homology modules.
When k is a field, the bigraded Betti numbers of K (with coefficients in k) are defined by
β −i,2j (K) := dim H −i,2j (Z K ) = J⊂[m] : |J|=j dim H j−i−1 (K J ). In particular, when j = m, we obtain β −i,2m (K) = dim H m−i−1 (K). Moreover, dim H j−i−1 (K J ) agrees with dim H j−i−1 (K J ) obviously.
Dimensional considerations imply that possible locations of nonzero bigraded Betti numbers form a trapezoid in the table, see Figure 3. We refer to [BP,Section 4.5] for more details.
3.2. Bigraded persistence and barcodes. Definition 3.3. Let (X, d X ) be a finite pseudo-metric space and (X, d X ) and {R(X, t)} t≥0 its associated Vietoris-Rips filtration. We define the bigraded persistent homology module of bidegree (−i, 2j) as
PHZ −i,2j (X) : R ≥0 → k-mod, t → H −i,2j (Z R(X,t) ). 2j −i 0 −1 0 2 4
1 * * * * * * * * * * * * * * * * * * * * * * * * * * 2m t2) ), see Proposition 3.2. We also define the bigraded persistent homology module
H0(K) Hn−1(K) · · · −(m − 1) −(m − n) Figure 3.It maps t ∈ R ≥0 to the bigraded homology module H −i,2j (Z R(X,t) ) and maps a morphism t 1 ≤ t 2 to the homomorphism H −i,2j (Z R(X,t1) ) → H −i,2j (Z R(X,(3.3) PHZ(X) = 0≤i≤m−1; 0≤j≤m PHZ −i,2j (X).
For any subset J ⊂ X, the Vietoris-Rips complex R(J, t) is the full subcomplex R(X, t) J . Hence, the Hochster decomposition (3.2) yields the decomposition of the persistence modules:
(3.4) PHZ(X) = J⊂X PH(J).
The bigraded barcode BB (X) is the collection of persistence intervals of generators of the bigraded homology groups H −i,2j (Z R(X,t) ). We note that bigraded persistence intervals are defined in the same way as for ordinary persistent homology, see
Subsection 2.2. For each t ∈ R ≥0 , the dimension of H −i,2j (Z R(X,t) ) is equal to the number of persistence intervals of bidegree (−i, 2j) containing t.
The bigraded barcode of X is drawn as a diagram in 3-dimensional space, which contains the original barcode of X in its top level. See Figure 4. Here is a simple example of two sequences of point clouds with the same barcodes, but different bigraded barcodes.
Example 3.4. Let X 1 and X 2 consist of three points (0, 0), (2, 0), (0, 4) and (0, 0), (2, 0), (1, √ 15) in R 3 , respectively. The Vietoris-Rips filtration {R(X 1 , t)} t≥0 at t = 0, 2, 4, 2 √ 5 is shown in Figure 5, top, and the Vietoris-Rips filtration {R(X 2 , t)} t≥0 at t = 0, 2, 4 is shown in Figure 5, bottom. The corresponding bigraded barcodes are shown in Figure 6. We notice that the two bars in the top levels of the bigraded barcodes are identical. These two bars represent the ordinary (single graded) barcodes of X 1 and X 2 , respectively. The two data sets X 1 and X 2 are distinguished by their bigraded barcodes. Double cohomology of Z K is defined similarly:
HH * (Z K ) = H(H * (Z K ), d ), d : H −i,2j (Z K ) → H −i+1,2j−2 (Z K )
Double cohomology HH * (Z K ) can also be defined as the first double cohomology of the bicomplex Λ[u 1 , . . . ,
u m ] ⊗ k[K], d, d with d = m i=1 v i ∂ ∂ui , d = m i=1
∂ ∂ui and dd = −d d:
HH * (Z K ) ∼ = H H Λ[u 1 , . . . , u m ] ⊗ k[K], d , d ,
see [LPSS,Theorem 4.3].
If K = ∆ m−1 , the full simplex on [m], then double homology is trivial:
HH * (Z K ) = k (0,0)
(the latter means that HH 0,0 (Z K ) ∼ = k and HH −i,2j (Z K ) = 0 if (−i, 2j) = (0, 0)). An important property of double homology is that attaching a simplex to K along a face (in particular, adding a disjoint simplex) destroys most of HH * (Z K ), as described next:
Theorem 3.5 ([LPSS, Theorem 6.7]). Let K = K ∪ I ∆ n be a simplicial complex obtained from a nonempty simplicial complex K by gluing an n-simplex, n ≥ 0, along a proper, possibly empty, face I ∈ K. Then either K is a simplex, or HH * (Z K ) = k (0,0) ⊕ k (−1,4) .
Definition 3.6. Let (X, d X ) be a finite metric space. A point x ∈ X is an outlier if for any y = x holds the inequality
d X (x, y) ≥ max y =x d X (y, y ).
Proposition 3.7. Suppose a finite metric space (X, d X ) has an outlier. Let D be the diameter of (X, d X ), and let R(X, t) be the Vietoris-Rips complex associated with (X, d X ). Then,
HH * (Z R(X,t) ) = k (0,0) t ≥ D; k (0,0) ⊕ k (−1,4) t < D
with all maps HH * (Z R(X,t) ) → HH * (Z R(X,t ) ) being the identity if D ≤ t ≤ t , and the projection if t < D ≤ t .
Proof. The definition of an outlier x implies that the Vietoris-Rips complex R(X, t) is obtained from the Vietoris-Rips complex R(X \ x, t) by attaching a simplex (with vertices x and {y : d(x, y) ≤ t}) along its facet (with vertices {y : d(x, y) ≤ t}). For t ≥ D, the complex R(X, t) is a simplex itself. Then the result follows from Theorem 3.5.
We define the bigraded persistent double homology and the corresponding barcodes by analogy with Definition 3.3:
Definition 3.8. Let (X, d X ) be a finite pseudo-metric space and {R(X, t)} t≥0 its associated Vietoris-Rips filtration. The bigraded persistent double homology module of bidegree (−i, 2j) is
PHHZ −i,2j (X) : R ≥0 → k-mod, t → HH −i,2j (Z R(X,t) ).
It maps t ∈ R ≥0 to the bigraded module HH −i,2j (Z R(X,t) ) and maps a morphism t 1 ≤ t 2 to the homomorphism HH −i,2j (Z R(X,t1) ) → HH −i,2j (Z R(X,t2) ) induced by the inclusion Z R(X,t1) → Z R(X,t2) (see [LPSS,Proposition 4.8]). We also define the bigraded persistent homology module (3.6) PHHZ(X) = 0≤i≤m−1; 0≤j≤m
PHHZ −i,2j (X).
One can view the bigraded persistent homology module (3.3) as a functor to differential bigraded k-modules,
PHZ(X) : R ≥0 → dg(k-mod), t → (H * , * (Z R(X,t) ), ∂ ).
Then we have
(3.7) PHHZ(X) = H • PHZ(X),
where H : dg(k-mod) → k-mod is the homology functor. This will be convenient when we compare interleaving distances. We denote by BB(X) the barcode corresponding to the bigraded persistence module PHHZ(X).
Example 3.9. Suppose (X, d X ) has an outlier. Then BB(X) consists of an infinite bar [0, ∞) at bidegree (0, 0) and a bar [0, D) at bidegree (−1, 4), by Proposition 3.7.
Stability
In this section, we establish stability properties of the bigraded persistence modules PHZ(X) and PHHZ(X), see Theorems 4.20 and 4.22.
4.1. Distance functions. We first summarize several distance functions defined on the categories of finite pseudo-metric spaces and persistence modules. The Gromov-Hausdorff distance between two finite pseudo-metric spaces X and Y is
d GH (X, Y ) := inf Z,f,g d H (f (X), g(Y )),
where the infimum is taken over all isometric embeddings f : X → Z and g : Y → Z into a pseudo-metric space Z.
Using the notion of a correspondence between two sets, one can give an alternative definition of the Gromov-Hausdorff distance, which is often convenient for computational purposes.
Definition 4.2. Given two sets X and Y and a subset C of X × Y , define D C (x) as the number of y ∈ Y for which (x, y) ∈ C. Similarly, define D C (y) as the number of x ∈ X for which (x, y) ∈ C. A correspondence between two sets X and Y is a subset C of X × Y such that both D C (x) and D C (y) are nonzero.
Note that the condition D C (x) ≥ 1 for all x ∈ C is equivalent to saying that C is a (multi-valued) mapping with domain X and range Y . Furthermore, the condition D C (y) ≥ 1 is equivalent to saying that C is surjective.
(4.1) d GH (X, Y ) = 1 2 min C max (x1,y1),(x2,y2)∈C |d X (x 1 , x 2 ) − d Y (y 1 , y 2 )|.
We have the following modification of the Gromov-Hausdorff distance (4.1), to be used below.
Definition 4.4. For two finite pseudo-metric spaces (X, d X ) and (Y, d Y ) of the same cardinality, define d GH (X, Y ) by formula (4.1) in which the infimum is taken over bijective correspondences C only.
Obviously, d GH (X, Y ) ≥ d GH (X, Y ) for any pair of pseudo-metric spaces of the same cardinality. The next example shows that for two pseudo-metric spaces X and Y of the same cardinality, the difference between d GH (X, Y ) and d GH (X, Y ) can be arbitrary large. Example 4.7. The interleaving distance between two interval modules k ([a, b)) and k ([a , b )) (see Example 2.1) is given by
d IL k ([a, b)) , k ([a , b )) = min max b − a 2 , b − a 2 , max{|a − a|, |b − b|} .
This formula has the following meaning. If the intervals are close to each other (the closure of each interval contains the midpoint of the other), then the distance is the maximum of the distances between their endpoints, i. e. max{|a − a|, |b − b|}, which is the standard l ∞ -distance between the two intervals. If the intervals are far apart, then the distance is max b−a 2 , b −a 2 , where b−a 2 is the l ∞ -distance between [a, b) and a 'zero length interval' at the midpoint of [a, b).
The zero persistence module can be thought of as the interval module k(∅) of an empty interval. In this case, the interleaving distance is given by
d IL k ([a, b)) , k(∅) = b − a 2 .
Below, we record the following properties of d IL for future use.
Proposition 4.8 (see [CS,Proposition 5.5]). Let M 1 , M 2 , N 1 and N 2 be persistence modules. Then
d IL (M 1 ⊕ M 2 , N 1 ⊕ N 2 ) ≤ max d IL (M 1 , N 1 ), d IL (M 2 , N 2 ) .
More generally, the interleaving distance can be defined for persistence modules M : R ≥0 → c taking values in any Grothendieck category c instead of k-mod, see [BSS]. The following result is clear from the definition.
Proposition 4.9. Given two functors M and N from R ≥0 to c, and a functor F : c → d, we have:
d IL (F • M, F • N ) ≤ d IL (M, N ).
Next, we introduce the bottleneck distance between multisets of intervals (or barcodes). It is defined in the standard way as the infimum over matchings between them, in which some of the intervals can be matched to zero length intervals at their midpoints.
Let B and B be finite multisets of intervals of the form [a, b) with a ∈ R ≥0 and b ∈ R ≥0 ∪ {∞}. Define the multiset B = B ∪ ∅ |B | , obtained by adding to B the multiset containing the empty interval ∅ with cardinality |B |. Similarly, define B = B ∪ ∅ |B| . Note that B and B have the same cardinality. We define the distance function π : B × B → R ≥0 ∪ {∞} by the formulae:
π ([a, b), [a , b )) = max{|a − a|, |b − b|}, π([a, ∞), [a , ∞)) = |a − a|, π([a, b), ∅) = b − a 2 , π(∅, [a , b )) = b − a 2 , π(∅, ∅) = 0, π([a, ∞), [a , b )) = π([a, b), [a , ∞)) = π([a, ∞), ∅) = π(∅, [a , ∞)) = ∞
for finite a, b, a , b . It can be shown [BS,Definition 4.11] that
W ∞ (B, B ) = min θ∈D(B,B ) max I∈B d IL k(I), k(θ(I)) ,
where the interleaving distance between interval modules can be expressed via the l ∞ -distance as in Example 4.7. The minimum of max I∈B π(I, θ(I)) is achieved at θ 1 , which reflects the fact that the two intervals overlap sufficiently. We have W ∞ (B, B ) = π([0, 5), [1, 7)) = 7 − 5 = 2. 2. Let B = {[0, 5)} and B = {[3, 9)}. These two intervals are 'far apart', so the minimum of max I∈B π(I, θ(I)) is achieved when both [0, 5) and [3,9) are matched to an empty interval. We have W ∞ (B, B ) = π(∅, [3, 9)) = 3.
In the literature, the following result is often referred to as the isometry theorem, see [Le,Theorem 3.4]. We also refer to [BS,Theorem 4.16] and [CS,Theorem 5.14] for different versions of this result: The stability theorem asserts that the persistent homology barcodes are stable under perturbations of the data sets in the Gromov-Hausdorff metric. It is a key result justifying the use of persistent homology in data science. We refer to [CC,Theorem 3.1] and [CDO,Lemma 4.3,Theorem 5.2] for the proof:
Theorem 4.13. Let (X, d X ) and (Y, d Y ) be two finite pseudo-metric spaces, and let B(X) and B(Y ) be the barcodes corresponding to the persistence modules PH(X) and PH(Y ). Then,
d IL (PH(X), PH(Y )) ≤ 2d GH (X, Y ),
and when working over a field,
W ∞ (B(X), B(Y )) ≤ 2d GH (X, Y ).
4.2. The doubling operation. The following construction plays a key role in proving the stability property of persistent double homology.
Definition 4.14. Given a pseudo-metric space (X, d X ) and a point x ∈ X, we refer to the pseudo-metric space X = X {x } with d X (x, x ) = 0 as the doubling of X at x. The doubling at a vertex is a particular case of the following operation introduced in [AP]. Let K be a simplicial complex on the set [m], and let K 1 , . . . , K m be simplicial complexes. The substitution of K 1 , . . . , K m into K is the simplicial complex K(K 1 , . . . , K m ) = {I j1 · · · I j k : I j l ∈ K j l , l = 1, . . . , k and {j 1 , . . . , j k } ∈ K}. Observe that if X is the doubling of a pseudo-metric space X at a point, then the Vietoris-Rips complex R(X , t) is the doubling of R(X, t) at a vertex, i. e. R(X , t) = R(X, t) .
Proposition 4.15. Suppose X is obtained from X by an arbitrary number of doubling operations performed at arbitrary points on the way. Then d GH (X, X) = 0.
Proof. Using the triangle inequality, we reduce the claim to the case when X is obtained from X by a single doubling operation, i. e. X = X = X {x 0 } is a doubling at x 0 ∈ X.
We use (4.1) and consider the following correspondence C between X and X :
C := {(x, x) ∈ X × X | x ∈ X} {(x 0 , x 0 )}.
Note that for any s ∈ X with s = x 0 , the number D C (s) of x ∈ X for which (s, x) ∈ C is 1. Similarly, D C (x 0 ) = 2 and D C (t) = 1 for all t ∈ X . See Definition 4.2. The triangle inequality together with d X (x 0 , x 0 ) = 0 implies d X (x 1 , x 2 ) = d X (y 1 , y 2 ) for any (x 1 , y 1 ), (x 2 , y 2 ) ∈ C, hence d GH (X, X ) = 0.
Proposition 4.16. Given two finite pseudo-metric spaces X and Y , there exist two finite pseudo-metric spaces X and Y such that | X| = | Y | and
d GH (X, Y ) = d GH ( X, Y ),
where d GH is the modified Gromov-Hausdorff distance, see Definition 4.4.
Proof. For two finite pseudo-metric spaces S and T of the same cardinality, we have
(4.2) d GH (S, T ) ≤ d GH (S, T ),
because any bijection is a correspondence between S and T . Now let X and Y be pseudo-metric spaces obtained by iterated doublings of X and Y , respectively, so that X and Y have the same cardinality. The triangle inequality together with Proposition 4.15 implies
d GH (X, Y ) ≤ d GH (X, X) + d GH ( X, Y ) + d GH ( Y , Y ) = d GH ( X, Y ) ≤ d GH ( X, Y ),
where second equality follows from Proposition 4.15 and the last inequality follows from (4.2).
To complete the proof, we construct iterated doublings X and Y such that d GH ( X, Y ) ≤ d GH (X, Y ). Let C be a correspondence which realizes d GH (X, Y ). If C is a bijection (in particular, |X| = |Y |), then there is nothing to prove. So we assume that C is not a bijection.
First, consider all the vertices x ∈ X such that D C (x) > 1 and double each of them D C (x) − 1 times. We get a new pseudo-metric space X and a new correspondence C between X and Y which matches each double of x with a single point of Y . If C is a bijection, we set Y = Y and we have
d GH (X, Y ) = 1 2 max (x,y),(x ,y )∈C |d X (x, x ) − d Y (y, y )| = 1 2 max (x,y),(x ,y )∈C |d X (x, x ) − d Y (y, y )| ≥ d GH ( X, Y ), (4.3)
where the second equality follows because the set {|d X (x, x )−d Y (y, y )|} (x,y),(x ,y )∈C coincides with {|d X (x, x ) − d Y (y, y )} (x,y),(x ,y )∈C and the last inequality follows from Definition 4.4. If C is not a bijection, then it is a single-valued surjective mapping from X to Y , which is not yet injective, so we perform the next step. Consider all the
x 2 x 1 y 1 y 2 y 3 C x 1,1 x 1,2 x 2,1 x 2,2 x 2,3 y 1 y 2 y 3 C x 1,1 x 1,2
x 2,1 x 2,2 x 2,3 y 1,1 y 1,2 y 2 y 3,1 y 3,2 C Figure 9. From a correspondence to a bijection via iterated doublings.
vertices y ∈ Y such that D C (y) > 1 and double each of them D C (y) − 1 times. We get a new pseudo-metric space Y and a new correspondence C between X and Y which matches each double of y with a single point of X. Now, C is an injective mapping from X to Y , therefore it is a bijection. See Figure 9 for an example of this procedure. Now the same argument as in (4.3) establishes the claim.
The proof of the last proposition also gives the following proposition. This could be helpful in proving other stability results.
Proposition 4.17. Any correspondence between finite sets can be written as a composition of doublings and a single bijection.
Next we show that the doubling of simplicial complexes preserves the double homology of the corresponding moment-angle complexes. We claim that the chain map ϕ defined above is a weak equivalence. Since ϕ is surjective, it suffices to show that ker ϕ is acyclic. Observe that ker ϕ ∼ = The associated spectral sequence has E 0 n = (F n /F n+1 , ∂ ), where the differential ∂ induced by ∂ , decomposes as a direct sum i / ∈L, |L|=n
H * (K L {i } ) → H * (K L {i,i } ) .
Each map in the parentheses above is an isomorphism, because K L {i } ⊂ K L {i,i } is a homotopy equivalence. Hence, the E 1 -page of the spectral sequence is zero, so ker ϕ is acyclic.
Proposition 4.19. Suppose X is a pseudo-metric space obtained from X by an arbitrary number of doubling operations performed at arbitrary points. Then the persistent double homology modules PHHZ(X) and PHHZ( X) are isomorphic.
The inequality in Theorem 4.20 does not hold if we replace d GH by d GH on the right hand side. Indeed, let Y = X {x } be the doubling of X at a point x ∈ X, see Definition 4.14. Then d GH (X, Y ) = 0 by Proposition 4.15. On the other hand, d IL (PHZ(X), PHZ(Y )) = 0 even when X consists of two points with nonzero distance. The reason is that at t = 0 we have R(X, 0) = K is two disjoint points, while R(X , 0) = K is a union of a point and a segment, so H * (Z K ) = H * (Z K ). 4.4. Stability for PHHZ. The main result of this subsection (Theorem 4.22) shows that the interleaving distance between persistence modules PHHZ(X) and PHHZ(Y ) defined in (3.6) for two finite pseudo-metric spaces (X, d X ) and (Y, d Y ) is bounded above by the usual Gromov-Hausdorff distance between X and Y . It also tells us that the persistent double homology module PHHZ(X) is more stable than the persistence module PHZ(X) defined by the ordinary homology of moment-angle complexes.
Theorem 4.22. Let X and Y be finite pseudo-metric spaces. Then d IL (PHHZ(X), PHHZ(Y )) ≤ 2d GH (X, Y ).
Proof. Let X and Y be finite pseudo-metric spaces of the same cardinality obtained as iterated doublings of X and Y by the procedure discussed in the proof of Proposition 4.16. Then, we have
d IL (PHHZ(X), PHHZ(Y )) = d IL (PHHZ( X), PHHZ( Y )) ≤ d IL (PHZ( X), PHZ( Y )) ≤ 2d GH ( X, Y ) = 2d GH (X, Y ).
Here the first identity follows from Proposition 4.19. The second inequality follows from decomposition (3.7) and Proposition 4.9. The third inequality is Theorem 4.20 and the last identity is Proposition 4.16.
When using field coefficients we also obtain stability for the bigraded barcodes using Theorem 4.12:
Theorem 2 . 2 .
22Let M = {M s } s∈R ≥0 be a persistence module as in (2.1). If k is a field and all M s are finite dimensional k-vector spaces, then multiset B(M) of intervals in R ≥0 , where k(I) is the interval module defined in Example 2.1.
Figure 1 .
1A point cloud and the corresponding Vietoris-Rips filtration.
Figure 2 .
2The barcode corresponding to the Vietoris-Rips complex inFigure 1. The image is generated by Ripser for Python.3. Moment-angle complexes and bigraded Betti numbers3.1. Cohomology of moment angle complexes. Let K be a simplicial complex on the set [m] = {1, 2, . . . , m}. We refer to a subset I =
Figure 4 .
4A bigraded barcode.
Figure 5 .Figure 6 .
56Two sequences of Vietoris-Rips complexes.3.3. Double cohomology of moment angle complexes. In[LPSS], the homology of the moment-angle complex H * (Z K ) = I⊂[m] H * (K I ) was endowed with the second differential ∂ . The homology of the resulting chain complex CH * (Z K ) = (H * (Z K ), ∂ ) was called the double homology of Z K . We use this construction to define a new bigraded persistence module PHHZ(X).Given j ∈ [m] \ I, consider the homomorphismφ p;I,j : H p (K I ) → H p (K I {j} ) induced by the inclusion K I → K I∪{j} . Then define ∂ p = (−1) p+1 I⊂[m], j∈[m]\I ε(j, I) φ p;I,j , where ε(j, I) = (−1) #{i∈I : i<j} . The bigraded barcode for 3 points in R 2 . The homomorphism ∂ p : I⊂[m] H p (K I ) → I⊂[m] H p (K I ) satisfies (∂ p ) 2 = 0. We therefore have a chain complex (3.5) CH * (Z K ) := (H * (Z K ), ∂ ), ∂ : H −i,2j (Z K ) → H −i−1,2j+2 (Z K ), and bigraded double homology of Z K : HH * (Z K ) = H(H * (Z K ), ∂ ).
Definition 4. 1 .
1The Hausdorff distance between two nonempty subsets A and B in a finite pseudo-metric space (Z, d) is d H(A, B)
Proposition 4.3 ([BBI, Theorem 7.3.25]). For two finite pseudo-metric spaces (X, d X ) and (Y, d Y ),
Example 4 . 5 .Figure 7 .
457Fix n > 2. Let X consist of four vertices of a rectangle in R 2 with two edges of length 1 and diagonals of length n. Let Y consist of four vertices of a tetrahedron in R 3 with the base an equilateral triangle with edges of length 1 and the other three edges of length n. SeeFigure 7. It is easy to see thatd GH (X, Y ) = 1 2 (n − 1), while d GH (X, Y ) = Pseudo-metric spaces X and Y with d GH (X, Y ) = d GH (X, Y ).Next, we introduce a distance function for persistence modules. Given a persistence module M as in (2.1) and an arbitrary ∈ R ≥0 , we defineM : R ≥0 → k-mod by M (s) = M(s + ) and M (φ s1,s2 ) = φ s1+ ,s2+ . There is a natural transformation η M : M → M defined by the family of morphisms {M s → M s+ | s ∈ R ≥0 }. Given two persistence modules M and N , we say that M and N are -interleaved if there exist natural transformations β : M → N and γ : N → M such that β • γ = η 2 N and γ • β = η 2 M , where β : M → N 2 and γ : N → M 2 denote the -shifts of β and γ, respectively.
Definition 4 . 6 .
46The interleaving distance between two persistence modules M and N is d IL (M, N ) := inf{ ∈ R ≥0 | M and N are -interleaved}.
Definition 4 . 10 .
410Denote by D B, B the set of all bijections θ : B → B . Then the ∞-Wasserstein distance, also called the bottleneck distance, is defined as follows: W ∞ (B, B ) = min θ∈D(B,B ) max I∈B π(I, θ(I)).
Example 4.11. 1 .
1Let B = {[0, 5)} and B = {[1, 7)}. Then B = {[0, 5), ∅} and B = {[1, 7), ∅}. There are two bijections between B and B : θ 1 : [0, 5) → [1, 7), ∅ → ∅, and θ 2 : [0, 5) → ∅, ∅ → [1, 7).
Theorem 4 . 12 .
412Let M and N be persistence modules satisfying the hypothesis of Theorem 2.2. Then, d IL (M, N ) = W ∞ (B(M), B(N )), where B(M) and B(N ) are the barcodes corresponding to M and N , respectively.
8 Figure 8 .
88Given a simplicial complex K on [m] and a vertex {i} of K, the doubling of K at {i} is the minimal simplicial complex K on the set [m] {i } which contains K and all subsets I {i }, where i ∈ I ∈ K. There is a deformation retraction K → K sending {i } to {i} and identity on other vertices. See Figure Doubling at a vertex.
The doubling of K at {i} is the substitution complex K(1, . . . , ∆[i, i ], . . . , m), where ∆[i, i ] denotes the 1-simplex on i and i .
Proposition 4 . 18 .
418Let K be obtained from K by doubling a vertex. Then,HH * (Z K ) ∼ = HH * (Z K ).Proof. Let K be obtained by doubling of K at i ∈ [m]. We recall the chain complex (3.5) and define the chain mapϕ : CH * (Z K ) = I⊂[m] {i } H * (K I ), ∂ → CH * (Z K ) = J⊂[m]H * (K J ), ∂ by the property that for α ∈ H * (K I ),ϕ(α) = 0 if i ∈ I; α if i / ∈ I.
H
* (K L {i } ). Let T n := {L ⊂ [m] : |L \ {i}| ≥ n} and define a decreasing filtration ker ϕ = F 0 ⊃ F 1 ⊃ · · · ⊃ F n ⊃ · · · with F n = L∈Tn H * (K L {i } ), F n /F n+1 = L⊂[m] : |L\{i}|=n H * (K L {i } ).
Panov is funded within the framework of the HSE University Basic Research Program. Limonchenko is also a Young Russian Mathematics award winner and would like to thank its sponsors and jury. Song was supported by a KIAS Individual Grant (SP076101) at Korea Institute for Advanced Study. Stanley is supported by NSERC. The authors are grateful to Daniela Egas Santander for helpful conversations at the Fields Institute. This work began at the Fields Institute during the Thematic Program on Toric Topology and Polyhedral Products. 2. Preliminaries 2.1. Persistence modules and barcodes. Consider the set R ≥0 of nonnegative real numbers, which we regard as a poset category with respect to the standard inequality ≤. A persistence module is a (covariant) functor (2.1) M : R ≥0 → k-mod where k-mod denotes the category of modules over a principal ideal domain k. A persistence module can be thought of as a family of k-modules {M s } s∈R ≥0 together with morphisms {φ s1,s2 : M s1 → M s2 } s1≤s2 such that φ s,s is the identity on M s and φ s2,s3
Table of bigraded Betti numbers.
W ∞ (BB (X), BB (Y )) ≤ 2d GH (X, Y ).
Corollary 4.23. Let BB(X) and BB(Y ) be the bigraded barcodes corresponding to the persistence modules PHHZ(X) and PHHZ(Y ), respectively. Then, we have W ∞ (BB(X), BB(Y )) ≤ 2d GH (X, Y ).Proof. This follows from Proposition 4.18 and obvious functorial properties of the doubling construction. 4.3. Stability for PHZ. First, we establish a stability result for the bigraded persistence module PHZ(X) of a finite pseudo-metric space (X, d X ), see Definition 3.3.Theorem 4.20. For two finite pseudo-metric spaces (X, d X ) and (Y, d Y ) of the same cardinality, we havewhere d GH is the modified Gromov-Hausdorff distance, see Definition 4.4.Proof. Let θ : X → Y be the bijection that realizes d GH (X, Y ). Then, for arbitrary subset J ⊂ X, we haveLet PH(J) and PH(θ(J)) be the ordinary persistent homology modules (2.4) corresponding to the subspaces J and θ(J), respectively. Then,where the first inequality follows from the stability of persistent homology (Theorem 4.13), the second follows by Definition 4.4 and the third is (4.4). Hence,where the first identity follows from 3.4, the second to last inequality follows from Proposition 4.8 and the last inequality follows from (4.5).When using field coefficients, we also get the stability result for the bigraded barcodes using Theorem 4.12:Corollary 4.21. Suppose (X, d X ) and (Y, d Y ) are two finite pseudo-metric spaces of the same cardinality. Let BB (X) and BB (Y ) be the bigraded barcodes corresponding to the persistence modules PHZ(X) and PHZ(Y ). Then,
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arxiv |
An inexact augmented Lagrangian method for nonsmooth optimization on Riemannian manifold *
29 Nov 2019
Kang-Kang Deng
Zheng Peng
An inexact augmented Lagrangian method for nonsmooth optimization on Riemannian manifold *
29 Nov 2019Manifold optimizationNonsmooth optimizationAugmented Lagrangian methodMoreau envelope Mathematics Subject Classification: 90C30, 90C26
We consider a nonsmooth optimization problem on Riemannian manifold, whose objective function is the sum of a differentiable component and a nonsmooth convex function.The problem is reformulated to a separable form. We propose a manifold inexact augmented Lagrangian method (MIALM) for the considered problem. By utilizing the Moreau envelope, we get a smoothing subproblem at each iteration of the proposed method. Theoretically, the convergence to critical point of the proposed method is established under suitable assumptions.In particular, if an approximate global minimizer of the iteration subproblem is obtained at each iteration, we prove that the sequence generated by the proposed method converges to a global minimizer of the origin problem. Numerical experiments show that, the MIALM is a competitive method compared to some existing methods.
Introduction
Riemannian manifold optimization is a class of constrained optimization problems, in which the constraint set is a subset of Riemannian manifold M. It has recently aroused considerable research interests due to the wide applications in different fields such as computer vision, signal processing, etc [3]. In these applications, manifold M could be Stiefel manifold, Grassmann manifold, or symmetric positive definite manifold. Analogy to classical optimization methods in Euclidean space, some Riemannian optimization methods have been explored, e.g., gradient-type methods [3,10,37], Newton-type methods [28,36,8] and trust region methods [1,9,27].
In this paper, we consider a nonsmooth nonconvex Riemannian manifold optimization problem as follows min X∈R n×r F (X) := f (X) + g(AX)
s.t. X ∈ M, (1.1) where f : M → R is a smooth but possibly nonconvex function, g is convex but nonsmooth, and M is a Riemannian submanifold embedded in Euclidean space E. Many convex or nonconvex problems in machine learning applications have the form of problem (1.1), e.g., sparse principle component analysis [40], sparse canonical correlation analysis [34], robust low-rank matrix completion [13,26] and multi-antenna channel communications [39,19], etc.
Absil and Hosseini [2] presented many examples of manifold optimization with nonsmooth objective. We list three representative examples in the following. Problem (1.1) is reformulated to a separable form in this paper, and then a manifold inexact augmented Lagrangian method (MIALM) is proposed for the resulting separable optimization problem. The iteration subproblem of the MIALM is formulated to a smooth optimization problem by utilizing the Moreau envelope, it could be solved by some classical Riemannian optimization methods such as Riemannian gradient/Newton/Quasi-Newton method. This algorithmic framework is adapted from [31,32,17] for classical nonsmooth composite problem in Euclidean space, which has drawn significant research attentions. The convergence to critical point of the proposed MIALM method is established under some mild assumptions. In particular, under the assumption that an approximate global minimizer of the iteration subproblem could be obtained, the convergence to global minimizer of the original problem is proved. Numerical experiments show that, the MIALM is competitive compared to some existing methods.
The rest of this paper is organized as follows. Some related works on nonsmooth manifold optimization problem are summarized in Section 2, and some preliminaries on manifold are given in Section 3. In Section 4, a manifold inexact augmented Lagrangian method is proposed and the iteration subproblem solver is presented. The convergence of the proposed method is established in Section 5. Numerical results on compressed modes problems in physics and sparse PCA are reported in Section 6. Finally, Section 7 concludes this paper by some remarks.
Related works
We summarize some related works for nonsmooth optimization problem on manifold in this section. The existing results mainly focused on two classes of nonsmooth manifold optimization problem: nonsmooth optimization problem with locally Lipschitz objective function, and structured optimization problem having the form of problem (1.1).
Grohs and Hosseini [21] proposed the ǫ-subgradient algorithm for minimizing a locally Lipschitz function on Riemannian manifold. By utilizing ǫ-subgradient-oriented descent directions and the generalized Wolfe line-search on Riemannian manifold, Hosseini, Huang and Yousefpour [24] presented a nonsmooth Riemannian line search algorithm and established the convergence to a stationary point. Grohs [20] presented a nonsmooth trust region algorithm for minimizing locally Lipschitz objective function on Riemannian manifold. The iteration complexity of these subgradient algorithms was also investigated in [5] and [18]. In [25] and [12], the authors proposed the Riemannian gradient sampling algorithms. At each iteration of these Riemannian gradient sampling methods, the subdifferential of the objective function is approximated by the convex hull of transported gradients of nearby points, and the nearby points are randomly generated in the tangent space of the current iterate.
Some proximal point algorithms on Riemannian manifold were investigated in the recent.
Bento, Ferreira and Melo [5] analyzed the iteration complexity of a proximal point algorithm on Hadamard manifold having non-positive sectional curvature. Bento, et al [16] gave the full convergence for any bounded sequence generated by the proximal point method, without assumption on the sign of the sectional curvature on manifold. The Kurdyka-Lojasiewicz inequality on Riemannian manifold is a powerful tool for convergence analysis of optimization methods on manifold. Bento, et al [6] analyzed the full convergence of a steepest descent method and a proximal point method via Kurdyka-Lojasiewicz inequality. Seyedehsomayeh [23] pro-posed a subgradient-oriented descent method and proved that, if the objective function has the Kurdyka-Lojasiewicz property, the sequence generated by the subgradient-oriented descent method converges to a singular critical point.
By a separable reformulation of problem (1.1), the variable involving Riemannian manifold constraint and that one involving nonsmooth term could be handled separately. To do so, it results in two tractable subproblems. Based on this idea, Lai, et al [30] proposed a splitting of orthogonality constraints (SOC) method for a special case of problem (1.1), in which f ≡ 0
and A = I, M is a Stiefel manifold. That is min X g(X), s.t. X ∈ M. (2.1)
To solve problem (2.1), the SOC method considered the following separable reformulation:
min X,Y g(Y ), s.t. X ∈ M, X = Y. (2.2)
The associated partial augmented Lagrangian function is
L β := g(Y ) − Λ, X − Y + β 2 X − Y 2 F (2.3)
where Λ is the Lagrangian multiplier, and β is a penalty parameter. The SOC method updates
iterate via X k+1 = arg min X∈M β 2 X − Y k − 1 β Λ k 2 F , Y k+1 = arg min g(Y ) + β 2 X k+1 − Y − 1 β Λ k 2 F , Λ k+1 = Λ k − β(X k+1 − Y k+1 ). (2.4)
The X-subproblem is "easy" via projection on M, and the Y -subproblem is often structured in real applications.
Chen, et al [15] proposed a proximal alternating minimization augmented Lagrangian (PA-MAL) method of multipliers for problem (1.1) with A = I and M = St n . Specifically, the PAMAL method first reformulates the problem to: Then it considers the augmented Lagrangian method of multipliers framework aiming to obtain the solution for the jointed variable (X, Y, Q) at each iteration. The iterate is produced by
min X,Y,Q f (Y ) + h(Q), (X k+1 , Y k+1 , Q k+1 ) = arg min X,Y,Q L β (X, Y, Q; Λ k 1 , Λ k 2 ), Λ k+1 1 = Λ k 1 − β(X k+1 − Y k+1 ), Λ k+1 2 = Λ k 2 − β(X k+1 − Q k+1 ), (2.6)
where L β is the augmented Lagrangian function associated to (2.5). The subproblem on the jointed variable (X, Y, Q) is intractable, hence the authors proposed a proximal alternating minimization method to handle it. Hong, et al [22] considered a more general form where M is the generalized orthogonal constraint, and proposed a PAMAL-type algorithm in which a proximal alternating linearized minimization method was used for iteration subproblem.
Kovnatsky, et al [29] proposed a manifold ADMM (MADMM) for a general manifold optimization problem as follows
min X,Y f (X) + g(Y ) s.t. AX = Y, X ∈ M.
(2.7)
The associated partial augmented Lagrangian function is
L β (X, Y ; Λ) := f (X) + g(Y ) − Λ, AX − Y + β 2 AX − Y 2 F .
The MADMM has the iterate as follows
X k+1 = arg min X∈M L β (X, Y k , Λ k ) Y k+1 = arg min Y L β (X k+1 , Y, Λ k ) Λ k+1 =Λ k − β(AX k+1 − Y k+1 ) (2.8)
More recently, Chen, et al [14] proposed a manifold proximal gradient method (ManPG) for
problem (1.1) with A = I, i.e. min X f (X) + g(X), s.t. X ∈ M (2.9)
At the k-th iteration, the search direction D k of ManPG is obtained by
min D D, gradf (X k ) + β 2 D 2 F + g(X k + D), s.t. D ∈ T X k M,(2.10)
where D ∈ T X k M can be represented by a linear system A k (D) = 0 . The subproblem (2.10)
is solved by applying the semi-smooth Newton method to the KKT system. The next iterate X k+1 is then obtained by
X k+1 = R X k (α k D k ).Definition 3.1 (Riemannian Gradient). Riemannian gradient, denoted by gradf (x) ∈ T x M,
is the unique tangent vector satisfying
gradf (x), ξ x = df (x)[ξ], ∀ ξ ∈ T x M. (3.1) If M is an embedded manifold of E, the Riemannian metric between u, v ∈ T x M could be introduced by an inner product in E, i.e. u, v x = u, v , where the later is classical inner product in E.
In the sense, we have
gradf (x) = Proj TxM (∇f (x)) (3.2)
where ∇f (x) is the gradient in E, Proj TxM is a projection on tangent space T x M.
(x) of T x M into itself, defined by Hessf (x)[ξ x ] = ∇ ξx gradf (x) (3.3) for ∀ ξ x ∈ T x M, where ∇ is the Riemannian connection on M.
Definition 3.3 (Retraction). A retraction on manifold M is a smooth mapping R : T M → M
which has the following properties:
let R x : T x M → M be the restriction of R to T x M, then • R x (0 x ) = x, where 0 x is zero element of T x M • dR x (0 x ) = id TxM ,
where id TxM is the identity mapping on T x M Definition 3.4 (Vector Transport). The vector transport T is a smooth mapping with
T M ⊕ T M → T M : (η x , ξ x ) → T ηx (ξ x ) ∈ T M, ∀ x ∈ M,(3.4)
where T satisfies that
• T 0x ξ x = ξ x holds for ∀ ξ x ∈ T x M; • T ηx (aξ x + bζ x ) = aT ηx (ξ x ) + bT ηx (ζ x ).
Definition 3.5 (The Clarke subdifferential on Riemannian manifold). For a locally Lipschitz
continuous function f on M, the Riemannian generalized directional derivative of f at x ∈ M in direction v ∈ T x M is given by f • (x; v) = lim y→x sup t↓0 f • ϕ −1 (ϕ(y) + tDϕ(y)[v]) − f • ϕ −1 (ϕ(y)) t , (3.5)
where (ϕ, U ) is coordinate chart at x. The generalized gradient or the Clarke subdifferential of
f at x ∈ M is ∂f (x) = {ξ ∈ T x M : ξ, v x ≤ f • (x; v), ∀v ∈ T x M}. (3.6)
Consider a Riemannian manifold minimization problem
min x f (x) s.t. c i (x) = 0, i = 1, · · · , m,
x ∈ M.
(3.7)
Let Ω := {x ∈ M : c i (x) = 0, i = 1 · · · , m}. Given x * ∈ Ω, assume that the Linear Independent Constraint Qualification (LICQ) holds at x * , then normal cone N Ω (x * ) is [35]:
N Ω (x * ) = m i=1 λ i gradc i (x * ) λ ∈ R m (3.8)
For the first-oder optimality condition of problem (3.7), we have
Lemma 3.1 ([38],Proposition 2.7). If x * ∈ Ω, and ∂f (x * ) ∩ (−N Ω (x * )) = ∅, (3.9)
then x * is a stationary solution of problem (3.7).
Proximal operator and retraction-smooth
For a proper, convex and low semicontinuous function g : E → R, the proximal operator with parameter µ ≥ 0, denoted by prox µg , is defined by
prox µg (v) := arg min x {g(x) + 1 2µ x − v 2 }. (3.10)
The associated Moreau envelope is a function M : E → R given by
M µg (v) : = min x {g(x) + 1 2µ x − v 2 } = g(prox µg (v)) + 1 2µ prox µg (v) − v 2 . (3.11)
The Moreau envelope is a continuously differentiable function, even when g is not.
Lemma 3.2 (Theorem 6.60 in [4]). Let g : E → R be a proper closed and convex function, and
µ ≥ 0. Then M µg is 1 µ -smooth in E, and for ∀ v ∈ E one has ∇M µg (v) = 1 µ (v − prox µg (v)). (3.12)
Lemma 3.2 states that, the Moreau envelope is continuously differentiable in Euclidean space E. Next we will show the relationship between Retraction smoothness in submanifold of Euclidean space and smoothness in Euclidean space.
Definition 3.6 (Retraction-Smooth). A function f : M → R is said to be retraction ℓ-smooth if, for ∀ x, y ∈ M it holds that f (y) ≤ f (x) + gradf (x), ξ x + ℓ 2 ξ 2 x ,(3.13)
where ξ ∈ T x M and R x (ξ) = y.
Let M be a Riemannian submanifold of E. The following lemma states that, if f : R n → R has Lipschitz continuous gradient, then f is also retraction smooth on M.
f (R x k (η)) ≤ f (x k ) + η, gradf (x k ) + ℓ g 2 η 2 (3.14) holds at ∀ η ∈ T x k M.
Lemma 3.3 was proved in [10]. For the sake of completeness, we give a proof as follows.
Proof. By Lipschitz continuity, ∇f is Lipschitz along any line segment in E jointing x and y.
Hence, there exists ℓ > 0 such that
f (y) ≤ f (x) + ∇f (x), y − x + ℓ 2 y − x 2 , ∀x, y ∈ M. (3.15) It also holds if y = R x (η), ∀η ∈ T x M. Since gradf (x) is a orthogonal projection of ∇f (x) on T x M, we have ∇f (x), R x (η) − x = ∇f (x), η + R x (η) − x − η = gradf (x), η + ∇f (x), R x (η) − x − η . (3.16)
It is easy to deduce from (3.15) and (3.16) that
f (R x (η)) ≤ f (x) + gradf (x), η + ℓ 2 R x (η) − x 2 + ∇f (x) R x (η) − x − η . Since ∇f (x) is continuous on compact set M, there exists G > 0 such that ∇f (x) ≤ G, ∀ x ∈ M.
By Definition 3.3 and the compactness of manifold, there exists α, β ≥ 0 such that, for all
x ∈ M and all η ∈ T x M, we have
R x (η) − x ≤ α η 2 , and R x (η) − x − η ≤ β η 2 . Hence f (R x (η)) ≤ f (x) + gradf (x), η + ℓ 2 α 2 + Gβ η 2 .
Let ℓ g = ℓ 2 α 2 + Gβ , we have (3.14) and complete the proof.
The proposed method 4.1 Problem reformulation
For regularity, we need the following assumptions on problem (1.1).
Assumption 4.1.
A. M is a compact Riemannian submanifold embedded in Euclidean space E;
B. f is smooth but not necessary convex, g is a nonsmooth convex function on E, A ∈ R d×n and ∂g(Y ) is uniformly bounded for ∀Y ∈ R d×r , where ∂g(Y ) is subdifferential of g at Y .
By introducing auxiliary variable Y = AX, problem (1.1) can be reformulated to
min X,Y f (X) + g(Y ) s.t. AX = Y, X ∈ M.
(4.1)
The partial Lagrangian function associated to problem (4.1) is
L(X, Y ; Z) := f (X) + g(Y ) − Z, AX − Y (4.2)
By Lemma 3.1, the KKT system of problem (4.1) is as follows:
0 ∈ Proj T X * M (∇f (X * ) − A T Z * ), 0 ∈ ∂g(Y * ) + Z * , AX * = Y * .
Manifold inexact augmented Lagrangian method
The augmented Lagrangian associated with (4.1) is
L ρ (X, Y ; Z) = L(X, Y ; Z) + ρ 2 AX − Y 2 F = f (X) + g(Y ) − Z, AX − Y + ρ 2 AX − Y 2 F . (4.4)
For a given (X k , Y k , Z k ), the next iterate generated by our manifold inexact augmented Lagrangian method (MIALM) is given by
(X k+1 , Y k+1 ) = arg min X∈M,Y L ρ (X, Y ; Z k ), Z k+1 = Z k − ρ(AX k+1 − Y k+1 ).
(4.5)
The (X, Y )-subproblem is intractable due to the nonsmoothess and joint minimization. Hence, an efficient Riemannian optimization method should be proposed for this subproblem in MIALM (4.5). Notice that, for fixed ρ > 0 and Z we aim to solve
min X∈M,Y ∈R d×r Ψ(X, Y ) := L ρ (X, Y ; Z) (4.6) Let ψ Z (X) := inf Y Ψ(X, Y ) = f (X) + g(Prox g/ρ (AX − µZ)) + ρ 2 AX − 1 ρ Z − Prox µg (AX − 1 ρ Z) 2 F − 1 2ρ Z 2 F . (4.7)
The new iterate (X,Ȳ ) could be produced sequentially bȳ
X = arg min X∈M ψ Z (X),Ȳ = Prox g/ρ (AX − 1 ρ Z). (4.8)
In the sense, the MIALM iterate is rewritten to
X k+1 = arg min X∈M ψ Z k (X), Y k+1 = Prox g/ρ (AX k+1 − 1 ρ Z k ), Z k+1 = Z k − ρ(AX k+1 − Y k+1 ).
(4.9) By (3.12), we have
∇ψ Z (X) = ∇f (X) + ρA T AX − 1 ρ Z − Prox µg (AX − 1 ρ Z) = ∇f (X) + ρA T Prox ρg * (AX − 1 ρ Z) (4.10)
where g * is the conjugate operator of g and defined by g
* (x) = sup v { x, v − g(v)}. By Lemma 3.3, ψ Z (·)
is retraction smooth over Riemannian manifold M, and its Riemannian gradient is gradψ Z (X) = Proj TX M (∇ψ Z (X)).
Thus, at the k-th iteration, the X-subproblem is identical to find X k+1 such that gradψ Z k (X k+1 ) = 0.
Algorithm 1 below summarizes the proposed manifold inexact augmented Lagrangian method in details.
Remark 4.1.
1) The proposed method is an ALM-type method. The complexity of Xsubproblem is as same as that of MADMM. However, our method obtains a joint optimal solution which guarantees the convergence, while the MADMM does not.
2) All efficient Riemannian optimization methods are applicable for the X-subproblem, e.g., Riemannian gradient method, Riemannian Newton method, etc.
3) The proposed method is utilizable for smooth Riemannian optimization problem under set-constrained, in which g(X) = δ Ω (X), the indictor function of constraint set Ω.
Riemannian optimization subproblem
The main computational cost of Algorithm 1 is to solve the X-subproblem. It is a smooth optimization problem on Riemannian manifold. The X-subproblem could be restated as follows min X ψ(X), s.t. X ∈ M. where ψ = ψZ given by (4.7). It is a retraction smooth function, so problem (4.6) can be solved by some Riemannian gradient methods including Riemannian gradient descent (RGD),
Riemannian conjugate gradient (RCG) and Riemannian trust region (RTR) method, etc. In this paper, we adopt a RGD method to problem (4.15), see Algorithm 2 for details.
Algorithm 1 Manifold inexact augmented Lagrangian method for problem (1.1)
1: Input: Let Z min < Z max , X 0 ∈ M,Z 0 ∈ R d×r . Given ǫ min ≥ 0, ǫ 0 > 0, ρ 0 > 1, σ > 1, 0 < τ < 1.
2: for k = 0, 1, · · · do 3:
Produce the next iterate (X k+1 , Y k+1 ): get X k+1 by solving problem min X∈M ψZk (X) (4.11)
inexactly with a tolerance ǫ k where {ǫ k } k∈N ↓ 0; let
Y k+1 = Prox g/ρ k (AX k+1 −Z k ).
(4.12)
4:
Update Lagrangian multiplier Z k+1 by
Z k+1 =Z k − ρ k (AX k+1 − Y k+1 ) (4.13) 5: Project Z k+1 onto {Z : Z min ≤ Z ≤ Z max } to getZ k+1 .
6:
Update penalty parameter by
ρ k+1 = ρ k , if AX k+1 − Y k+1 ∞ ≤ τ AX k − Y k ∞ σρ k , otherwiseL ρ (W ; Z) = θ(W ) + d i=1 r j=1 Z ij [h(W )] ij + ρ 2 d i=1 r j=1 [h(W )] 2 ij (5.2)
The KKT conditions of problem (5.1) are given by Pick η k = −gradψ(X k ) and a step size α k , compute grad[h(W * )] ij |i = 1, · · · , d; j = 1, · · · , r are linearly independent in T W * N .
0 ∈ ∂θ(W * ) + d i=1 r j=1 Z * ij grad[h(W * )] ij , h(W * ) = 0, W * ∈ N ,(5.X k+1 = R X k (α k η k ).
We will analyze the convergence of Algorithm 1 in the following two cases:
1) The iterate (X k+1 , Y k+1 ) is an ǫ k -stationary point of iteration subproblem, i.e.,
gradψZk (X k+1 ) ≤ ǫ k . (5.4)
2) The iterate (X k+1 , Y k+1 ) is an ǫ k -global minimizer of iteration subproblem, i.e.,
L ρ k (W k+1 ;Z k ) ≤ L ρ k (W ;Z k ) + ǫ k , ∀ W ∈ N . (5.5)
Remark 5.1. In the case 1), (5.4) is indeed to find W k+1 such that
δ k ∈ ∂L ρ k (W k+1 ;Z k ), δ k ≤ ǫ k .Y k+1 = Prox g/ρ (AX k+1 − 1 ρZ k ), there exists ν k ∈ ∂g(Y k+1 ) such that 0 = ν k − ρ k (AX k+1 − 1 ρZ k − Y k+1 ).
Again by Assumption 4.1, ∂g(Y k+1 ) is bounded, and hence ν k is also bounded. It is obvious
thatZ k ∈ [Z min , Z max ] is bounded. Since sequence {ρ k } k∈N is nondescreasing, we have ρ k ≥ ρ 0 (∀k ∈ N). Hence {Y k } k∈N is bounded. In summary, sequence {W k } k∈N is bounded.
Next, we will show that W * is a feasible point of (5.1). By the updating rule of W in
Algorithm 1, we have W k ∈ N .
If {ρ k } k∈N is bounded, by the updating rule of ρ k , there exists a k 0 ∈ N such that
h(W k ) ∞ ≤ τ h(W k−1 ) ∞ , ∀k ≥ k 0 ,
where τ ∈ (0, 1). Hence, h(W * ) = 0.
If {ρ k } is unbounded, by Remark 5.1 we have
δ k ∈ ∂L ρ k (W k+1 ;Z k ), δ k ≤ ǫ k .
where ǫ k ↓ 0 as k → ∞. Thus there exists U k ∈ ∂θ(W k ) such that
U k + d i=1 r j=1 Z k ij + ρ k [h(W k )] ij grad[h(W k )] ij = δ k . (5.6)
Dividing both sides of (5.6) by ρ k , we have
d i=1 r j=1 Z k ij /ρ k + [h(W k )] ij grad[h(W k )] ij = (δ k − U k )/ρ k (5.7)
where {Z k } is bounded, and δ k ↓ 0. Since θ(W ) = f (X) + g(Y ), where g is a convex function on E, and
∂θ(W ) = gradf (X) ∂g(Y ) ,
where ∂g(Y ) is a subdifferential (set) in usual sense. Invoked by Proposition B.24(b) in [7], the set k∈K ∂g(Y k ) is bounded because that {Y k } k∈K is a bounded set. In addition, f (X)
is a retraction smooth function, hence the Riemannian gradient sequence {gradf (X k )} k∈K is bounded. Thus, we have that k∈K ∂θ(W k ) is bounded. This means that {U k } is bounded.
Taking limits as k ∈ K going to infinity on both sides of (5.6), and using the continuity and differentiability of h, we have,
d i=1 r j=1 ([h(W * )] ij ) grad[h(W * )] ij = 0 (5.8)
Note that LICQ holds at W * , we conclude that [h(W * )] ij = 0 for all i, j.
Since {U k } k∈K is bounded, there exists a subsequence K 1 ⊂ K such that lim k→∞,k∈K1
U k = U * .
Recall that lim k→∞,k∈K1
W k = W * . We get U * ∈ ∂θ(W * )
by the closedness property of the limiting subdifferential. Together with Z k+1
ij =Z k +ρ k [h(W k )] ij
for all i, j, one can get from Algorithm 1 that, for all k ∈ K 1 ,
U k + d i=1 r j=1 Z k+1 ij grad[h(W k )] ij = δ k (5.9)
where δ k satisfying δ k ≤ ǫ k , and U k ∈ ∂θ(W k ).
We claim that {Z k } is bounded. Otherwise, assume {Z k } is unbounded, we have
U k Z k+1 ∞ + d i=1 r j=1 Z k+1 ij Z k+1 ∞ grad[h(W k )] ij = δ k Z k+1 ∞ Since Z k+1 Z k+1 ∞ ∈ [−1, 1] is bounded, there exists a subsequence K 2 ⊂ K 1 such that lim k→∞,k∈K2 Z k+1 Z k+1 ∞ = Z,
whereZ is a nonzero matrix. Taking the limit on k ∈ K 2 going to infinity, we obtain Since {U k } is bounded and {δ k } ↓ 0, there exists a subsequence K 3 ⊂ K 2 such that lim k→∞,k∈K3 U k = U * and lim k→∞,k∈K3 Z k = Z * . By the continuity of mapping grad h, and taking limits on k ∈ K 3 going to infinity on both sides of (5.9), we have
U * + d i=1 r j=1 Z * ij grad[h(W * )] ij = 0. (5.11)
Lemma 5.1. Suppose that W ∈ N = M × R d×r , and M is a stiefel manifold denoted by St(n, r)). Then the LICQ always holds at ∀ W ∈ N .
Proof. Let e i ∈ R d be a m-dimensional coordinate vector in which the i-th entry is 1 and 0 for others, andē j ∈ R r be a r-dimensional coordinate vector. Then
∇[h(W )] ij = A T e iē T j −e iē
T j , i = 1, · · · , d; j = 1, · · · , r.
A basis of the normal cone of St(n, r) at X, denoted by N X St(n, r), is given by
X(ē iē T j +ē jē T i ) : i = 1, · · · , r, j = 1, · · · , r .
It is easy to show that, for ∀ W ∈ N , all the vectors in the set
A T e iē T j −e iē T j , i = 1, · · · , d; j = 1, · · · , r. X(ē iē T j +ē jē T i ) 0 , i = 1, · · · , r; j = 1, · · · , r. are linearly independent. Hence, if there exists Z such that d i=1 r j=1 Z ij ∇[h(W )] ij ∈ N W N ,(5.12)
we have Z = 0. Since N is a submanifold of Euclidean space, it derives immediately that
d i=1 r j=1 Z ij grad[h(W )] ij = 0,
holds if and only if Z = 0. Which implies LICQ holds at W and completes the proof.
Next, we consider the case that a ǫ k -global minimizer of the iteration subproblem could be obtained at each iteration of Algorithm 1.
Theorem 5.2. Assume that {W k } k∈N is a sequence generated by Algorithm 1, Assumption 4.1 holds, and (5.5) is satisfied at each iteration of Algorithm 1. Let W * be a limit point of
{W k } k∈N . Then we have d i=1 r j=1 [h(W * )] 2 ij ≤ d i=1 r j=1 [h(W )] 2 ij , ∀ W ∈ N . (5.13)
Proof. Consider the following two cases: {ρ k } bounded and ρ k → ∞.
If {ρ k } is bounded, then there exists k 0 such that ρ k = ρ k0 for all k ≥ k 0 . Hence
d i=1 r j=1 [h(W k+1 )] 2 ij ≤ τ d i=1 r j=1
[h(W k )] 2 ij , i = 1, · · · , m; j = 1, · · · , r.
Which implies that h(W k ) → 0 as k → ∞. We have h(W * ) = 0, and (5.13) holds.
Then to the case ρ k → ∞. Since W * is a limit point of {W k }, there exists a subsequence K ⊂ N such that lim k→∞, k∈K
W k = W * .
Assume by contradiction there exists W ∈ N such that
d i=1 r j=1 [h(W * )] 2 ij ≥ d i=1 r j=1 [h(W )] 2 ij .
By the boundedness of {Z k } and ρ k → ∞, there exist c > 0 and k 0 ∈ N such that, for all k ∈ K and k ≥ k 0 we have
d i=1 r j=1 ([h(W k+1 )] ij + 1 ρ kZ k ij ) 2 ≥ d i=1 r j=1 ([h(W )] ij + 1 ρ kZ k ij ) 2 + c.
Therefore
θ(W k+1 ) + ρ k 2 d i=1 r j=1 ([h(W k+1 )] ij + 1 ρ kZ k ij ) 2 ≥ θ(W ) + ρ k 2 d i=1 r j=1 ([h(W )] ij + 1 ρ kZ k ij ) 2 + ρ k c 2 + θ(W k+1 ) − θ(W ).
Since lim k→∞,k∈K
W k = W * , and {ǫ k } is bounded, there exists k 1 > k 0 such that, for all k ∈ K, k ≥ k 1 we have ρ k c 2 + θ(W k+1 ) − θ(W ) > ǫ k .
Therefore,
θ(W k+1 ) + ρ k 2 d i=1 r j=1 ([h(W k+1 )] ij + 1 ρ kZ k ij ) 2 ≥ θ(W ) + ρ k 2 d i=1 r j=1 ([h(W )] ij + 1 ρ kZ k ij ) 2 + ǫ k .
This contradicts (5.5). We have (5.13) and complete the proof.
Theorem 5.3. In Algorithm 1, let ǫ min = 0 and W * be a limit point of sequence {W k } k∈N .
If iterate W k+1 is a ǫ k -global minimizer satisfying (5.5), then W * is a global minimizer of problem (4.1). Meanwhile, X * is a global minimizer of problem (1.1).
Proof. By (5.5), we have
θ(W k+1 ) + ρ k 2 d i=1 r j=1 ([h(W k+1 )] ij + 1 ρ kZ k ij ) 2 ≤ θ(W ) + ρ k 2 d i=1 r j=1 ([h(W )] ij + 1 ρ kZ k ij ) 2 + ǫ k for all W ∈ N . Since h(W ) = 0, we get θ(W k+1 ) + ρ k 2 d i=1 r j=1 ([h(W k+1 )] ij + 1 ρ kZ k ij ) 2 ≤ θ(W ) + ρ k 2 d i=1 r j=1 ( 1 ρ kZ k ij ) 2 + ǫ k .
Which implies that
θ(W k+1 ) ≤ θ(W ) + ρ k 2 d i=1 r j=1 ( 1 ρ kZ k ij ) 2 + ǫ k . (5.14)
If ρ k → ∞, by taking limits on both sides of (5.14) as k ∈ K, k → ∞, and using lim k→∞,k∈K
ǫ k = 0, we get θ(W * ) ≤ θ(W ), ∀ W ∈ N .
In case of that {ρ k } is bounded, there exists k 0 ∈ N such that ρ k = ρ k0 for all k > k 0 . By
(5.5) we have θ(W k+1 ) + ρ k0 2 d i=1 r j=1 ([h(W k+1 )] ij + 1 ρ k0Z k ij ) 2 ≤ θ(W ) + ρ k0 2 d i=1 r j=1 ([h(W )] ij + 1 ρ k0Z k ij ) 2 + ǫ k for W ∈ N . Since h(W ) = 0, we get θ(W k+1 ) + ρ k0 2 d i=1 r j=1 ([h(W k+1 )] ij + 1 ρ k0Z k ij ) 2 ≤ θ(W ) + ρ k0 2 d i=1 r j=1 ( 1 ρ k0Z k ij ) 2 + ǫ k (5.15) for all k ≥ k 0 . Let K 1 ⊂ K and lim k→∞,k∈K1Z k = Z * .
Taking limits on both sides of (5.15) as k → ∞, k ∈ K 1 , and noting that h(W * ) = 0, we get
θ(W * ) + ρ k0 2 d i=1 r j=1 ( 1 ρ k0 Z * ij ) 2 ≤ θ(W ) + ρ k0 2 d i=1 r j=1 ( 1 ρ k0 Z * ij ) 2 .
Hence θ(W * ) ≤ θ(W ), ∀ W ∈ N , and the proof is completed.
Experiments
Numerical experiments for testing the performance of the proposed MIALM method, with compared to some existing methods including SOC [30], PAMAL [22], MADMM [29] and
ManPG [14], are presented in the current section. All the methods are used to solve the compressed modes and sparse PCA problem. In the MIALM and MADMM, the Riemannian manifold optimization subproblem is handled by "Manopt", a Matlab toolbox for optimization on manifolds [11]. In the SOC, PAMAL and ManPG methods, the code provided by Chen [14] are used (all codes are available in online). All experiments are run on a personal computer with 4.0GHz Intel Core i7 CPU and 16 GB RAM.
Compressed modes in Physics
In physics, the compressed modes problem (CMs) seeks spatially localized solutions of the independent-particle Schrödinger equation: tr(X T HX) + µ X 1 ,
Hφ(x) = λφ(x), x ∈ Ω,(6.s.t. X T X = I d ,(6.2)
where H is the discretized Schrödinger operator, µ is a regularization parameter. The interesting readers are referred to [33] for more details. For problem (6.2), both SOC and PAMAL consider the identical form as follows:
min Ψ,Q,P ∈R n×r tr(X T HX) + µ Q 1 ,
s.t. Q = X, P = X, P T P = I r . In our experiments, the domain Ω := [0, 50] is discretized with n equally spaced nodes. The parameters of our MIALM are set to : τ = 0.99, σ = 1.05, ρ 0 = λ max (H)/2, Z min = −100 · 1 d×r , Z max = 100 · 1 d×r , Z 0 = 0 d×r and ǫ k = max(10 −5 , 0.9 k ), where k ∈ N is the iteration counter. We terminated MIALM if X k − Y k 2 F ≤ 10 −9 or k ≥ 500. The qr retraction is used in inner iteration of the MIALM, and a Barzilai-Borwein stepsize is used to accelerate it. The inner iteration is terminated if gradΨZk (X) X ≤ ǫ k or the iteration number exceeds 20. The final objective value obtained by the MIALM method is denoted by F M .
For the MADMM, the penalty parameter is set to ρ = λ max (H)/2. We terminated MADMM if X k − Y k 2 F ≤ 10 −9 or F (X k ) ≤ F M + 10 −7 , or k ≥ 500. The inner iteration of the MADMM terminates if the norm of Riemannian gradient of X-subproblem is less than 10 −5 or the inner iteration number exceeds 20. For the SOC, PAMAL and ManPG, the parameters are set as same as in [14], except that the penalty parameter ρ = 2λ max (H) in SOC and PAMAL. The ManPG terminates if stopping criterion described in [14] is met or F (X k ) ≤ F M + 10 −7 . For easy comparisons, Table 1 lists the objective function value, sparsity of solution and cpu time.
One can find from Table 1 that, our MILAM method outperforms to the other methods.
, · · · , b m } where b i ∈ R n×1 . The sparse PCA problem is min X∈R n×r m i=1 b i − XX T b i 2 2 + µ X 1 ,
s.t. X T X = I r ,
(6.5)
where µ is a regularization parameter. Let B = [b 1 , · · · , b m ] T ∈ R m×n , problem (6.5) has the form:
min X∈R n×r − tr(X T B T BX) + µ X 1 ,
s.t. X T X = I r .
(6.6)
In our experiments, the random data matrix B ∈ R m×n is generated by MATLAB function randn(m, n), in which the entries of B follow the standard Gaussian distribution. We shift the columns of B such that they have mean 0, and finally the column-vectors are normalized.
The parameters of our MIALM are set as same as that of used for the CMs problem, except that the stopping criterion is modified to X k − Y k 2 F ≤ 10 −8 and the penalty parameter ρ 0 = λ 2 max (B T B)/2. Similarly, the parameters of the MADMM are also set as the same as that used for the CMs problem, except that the penalty parameter ρ 0 = λ 2 max (B T B)/2. For the SOC, PAMAL and ManPG methods, the stopping criterion and parameter settings provided in [14] are copied. The interesting readers are referred to [14] for details. Table 2 lists performance of all methods on the sparse PCA problem for comparisons.
Conclusions
We proposed a manifold inexact augmented Lagrangian method for nonsmooth composite minimization problem on Riemannian manifold. At each iteration of the proposed method, we only need to solve a smooth Riemannian manifold minimization subproblem based on the Moreau envelope. The convergence of the proposed method is established under some mild assumptions. Numerical experiments show that, the proposed method is competitive compared to some existing state-of-the-art methods.
T A T AX + λ X 1 , s.t. X T X = I r .
n×r tr(Ψ T ∆Ψ) + µ Ψ 1 , s.t. Ψ T Ψ = I r .
P
Ω (X − M ) 1 , s.t. X ∈ M r := {X| rank (X) = r}.
s.t. X = Y, X = Q, X ∈ M.
Let M be a smooth manifold, and E be the Euclidean space. The tangent space of M atx ∈ M is denoted by T x M. A Riemannian manifold (M, ·, · ) is a smooth manifold equipped with inner product ·, · x on each point x ∈ M. Let (U, ϕ) be a chart, where U is an open set with x ∈ U ⊂ M and ϕ is a homeomorphism between U and open set ϕ(U ) ⊂ E. Given a Riemannian manifold M, the chart exists at each point x ∈ M.
Definition 3. 2 (
2Riemannian Hessian). Given a smooth function f : M → R, the Riemannian Hessian of f at x in M is linear mapping Hessf
Lemma 3. 3 .
3[Lemma 4 in[10]] Let E be a Euclidean space (for example, E = R n ) and M be a compact Riemannian submanifold of E. If f : E → R has Lipschitz continuous gradient in the convex hull of M, then there exists a positive constant ℓ g such that
Proposition 4. 1 .
1Suppose in problem (4.1) that f is smooth with Lipschitz continuous gradient and g is convex and locally Lipschitz continuous. Then, (X * , Y * ) satisfies the KKT conditions if there exists a Lagrange multiplier Z * such that
convenience of notation, we rewrite problem (4.1) to a standard constraint optimization problem on manifold. Let W = [X; Y ] ∈ R (n+d)×r , and N = M × R d×r be a product manifold. Then, problem (4.1) can be rewritten to min W θ(W ), s.t. h(W ) = 0, W ∈ N . (5.1) where θ(W ) = f (X)+g(Y ), and h(W ) = [A, −I]W ∈ R d×r . The partial augmented Lagrangian function associated to problem (5.1) is
3) 1 :
1Given: X 0 ∈ M, tolerance ǫ > 0. Let η 0 = −gradψ(X 0 ) .
where ∂θ(W * ) is Riemannian subdifferential of θ at W * . The KKT system (5.3) is identical to(4.3) because of that M is a Riemannian submanifold embedded in Euclidean space. Inspired by Zhang, Yang and Song [35], the constraint qualifications of problem (5.1) is given by: Definition 5.1 (LICQ). Linear independence constraint qualifications (LICQ) are said to hold at W * ∈ N for problem (5.1) if
the LICQ condition at W * .
Ψ,Q∈R n×r tr(X T HX) + µ Q 1 , s.t. Q = X, X T X = I r .
Theorem 5.1. Suppose {W k } k∈N is a sequence generated by Algorithm 1, Assumption 4.1 and (5.4) hold.Then, sequence {W k } k∈N has at least one cluster point. Furthermore, if W * is a cluster point, and LICQ holds at W * , then W * is a KKT point of problem (5.1). Proof. To prove the first part of Theorem 5.1, we need to show that sequence {W k } k∈N is bounded. By Assumption 4.1, M is a compact manifold, hence {X k } is bounded. By
Table 1 :
1Comparisons of MIALM and ManPG, MADMM, PAMAL, SOC on CMs problem Given a data set {b 1µ
MIALM
ManPG
MADMM
time
F M
sp
time
F
sp
time
F
sp
r = 2, n = 128
0.1 0.021 0.943 0.835 0.036 0.943 0.836 0.112 0.943 0.836
0.2 0.016 1.639 0.881 0.024 1.639 0.882 0.024 1.639 0.882
0.3 0.020 2.265 0.901 0.029 2.265 0.900 0.167 2.265 0.903
µ
PAMAL
SOC
time
F
sp
time
F
sp
0.1 0.049 0.943 0.837 0.024 0.943 0.837
0.2 0.038 1.639 0.882 0.017 1.639 0.882
0.3 0.088 2.265 0.901 0.026 2.265 0.901
r
MIALM
ManPG
MADMM
time
F M
sp
time
F
sp
time
F
sp
µ = 0.2, n = 256
2 0.021 2.167 0.892 0.071 2.167 0.892 0.153 2.167 0.892
4 0.063 4.334 0.887 0.233 4.334 0.886 0.311 4.338 0.884
6 0.345 6.500 0.889 0.722 6.500 0.884 0.531 6.509 0.881
r
PAMAL
SOC
time
F
sp
time
F
sp
2 0.127 2.167 0.892 0.057 2.167 0.892
4 0.709 4.334 0.888 0.273 4.334 0.888
6 3.036 6.500 0.887 0.980 6.500 0.887
n
MIALM
ManPG
MADMM
time
F M
sp
time
F
sp
time
F
sp
µ = 0.6, r = 2
200 0.018 2.265 0.901 0.028 2.265 0.901 0.167 2.265 0.903
300 0.017 2.996 0.910 0.051 2.996 0.910 0.128 3.005 0.909
500 0.026 3.956 0.920 0.132 3.956 0.920 0.282 4.048 0.916
n
PAMAL
SOC
time
F
sp
time
F
sp
200 0.045 2.265 0.902 0.028 2.265 0.901
300 0.085 2.996 0.910 0.041 2.996 0.910
500 0.253 3.956 0.920 0.137 3.956 0.920
Table 2 :
2Comparisons of MIALM and ManPG, MADMM, PAMAL, SOC on SPCA (m = 50)µ
MIALM
ManPG
MADMM
time
F M
sp
time
F
sp
time
F
sp
r = 2, n = 200
0.5 0.038 -6.839 0.461 0.035 -6.819 0.458 0.193 -6.767 0.454
0.6 0.038 -5.304 0.543 0.042 -5.248 0.545 0.201 -5.147 0.539
0.8 0.043 -2.439 0.722 0.047 -2.369 0.732 0.199 -2.285 0.732
µ
PAMAL
SOC
time
F
sp
time
F
sp
0.5 1.919 -6.847 0.460 0.251 -6.826 0.458
0.6 2.123 -5.267 0.545 0.302 -5.262 0.544
0.8 2.247 -2.387 0.733 0.281 -2.371 0.732
r
MIALM
ManPG
MADMM
time
F M
sp
time
F
sp
time
F
sp
µ = 0.6, n = 200
2 0.040 -5.308
0.548 0.039 -5.290
0.547 0.199 -5.209
0.538
3 0.047 -7.563
0.562 0.058 -7.530
0.561 0.223 -7.369
0.552
5 0.091 -11.625 0.594 0.117 -11.571 0.591 0.291 -11.304 0.582
r
PAMAL
SOC
time
F
sp
time
F
sp
2 0.040 -5.308
0.548 0.251 -5.329
0.544
3 3.322 -7.597
0.562 0.442 -7.552
0.561
5 6.828 -11.687 0.592 0.674 -11.727 0.588
n
MIALM
ManPG
MADMM
time
F M
sp
time
F
sp
time
F
sp
µ = 0.6, r = 2
200 0.039 -5.323
0.539 0.040 -5.283
0.541 0.203 -5.166
0.538
300 0.048 -8.128
0.473 0.043 -8.112
0.473 0.227 -7.955
0.467
500 0.085 -14.139 0.399 0.054 -14.134 0.399 0.303 -13.698 0.385
n
PAMAL
SOC
time
F
sp
time
F
sp
200 2.187 -5.282
0.545 0.288 -5.290
0.542
300 3.037 -8.106
0.477 0.443 -8.108
0.474
500 9.618 -14.106 0.400 1.283 -14.109 0.398
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| {'fraction_non_alphanumeric': 0.09467830638071678, 'fraction_numerical': 0.05644716128636559, 'mean_word_length': 3.2582049273091322, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 11, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 34, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We consider a nonsmooth optimization problem on Riemannian manifold, whose objective function is the sum of a differentiable component and a nonsmooth convex function.The problem is reformulated to a separable form. We propose a manifold inexact augmented Lagrangian method (MIALM) for the considered problem. By utilizing the Moreau envelope, we get a smoothing subproblem at each iteration of the proposed method. Theoretically, the convergence to critical point of the proposed method is established under suitable assumptions.In particular, if an approximate global minimizer of the iteration subproblem is obtained at each iteration, we prove that the sequence generated by the proposed method converges to a global minimizer of the origin problem. Numerical experiments show that, the MIALM is a competitive method compared to some existing methods.', 'arxivid': '1911.09900', 'author': ['Kang-Kang Deng ', 'Zheng Peng '], 'authoraffiliation': [], 'corpusid': 208247938, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 20110, 'n_tokens_neox': 16946, 'n_words': 8956, 'pdfsha': 'fafb277b42aa5a8299aa966e0391b650a2eebcc8', 'pdfurls': ['https://arxiv.org/pdf/1911.09900v2.pdf'], 'title': ['An inexact augmented Lagrangian method for nonsmooth optimization on Riemannian manifold *', 'An inexact augmented Lagrangian method for nonsmooth optimization on Riemannian manifold *'], 'venue': []} |
arxiv |
Parametrically controlling solitary wave dynamics in modified Kortweg-de Vries equation
8 Dec 2005
Kallol Pradhan
Physics Department
University of Wisconsin-Madison
1150 University AvenueWI53706 -1390MadisonU.S.A
Prasanta K Panigrahi
Physical Research Laboratory
380 009Navrangpura, AhmedabadIndia
Parametrically controlling solitary wave dynamics in modified Kortweg-de Vries equation
8 Dec 2005(Dated: March 30, 2022)
We demonstrate the control of solitary wave dynamics of modified Kortweg-de Vries (MKdV) equation through the temporal variations of the distributed coefficients. This is explicated through exact cnoidal wave and localized soliton solutions of the MKdV equation with variable coefficients. The solitons can be accelerated and their propagation can be manipulated by suitable variations of the above parameters. In sharp contrast with nonlinear Schrödinger equation, the soliton amplitude and widths are time independent. PACS numbers: 03.75.Lm,05.45.Yv,03.75.-b Modified Kortweg-de Vries (MKdV) equation manifests in diverse areas of physics [1, 2, 3, 4, 5, 6]. For example, it appears in the context of, electromagnetic waves in size-quantized films, van Alfvén waves in collisionless plasma [7], phonons in anharmonic lattice [8], interfacial waves in two layer liquid with gradually varying depth [9], transmission lines in Schottky barrier [10], ion acoustic solitons[11,12,13], elastic media[14], and traffic flow problems[15,16]. It is an integrable dynamical system with an infinite number of conserved quantities; the solutions of this equation are well studied[17,18].
Recently non-linear equations with variable coefficients have attracted considerable attention in the literature. Nonlinear Schrödinger equation (NLSE) with variable non-linearity and dispersion is relevant to both optical fibers and Bose-Einstein condensates [19,20,21,22]. Nonlinear Schrödinger equation with source, having distributed coefficients like variable dispersion, variable Kerr nonlinearity and gain or loss, is applicable to asymmetric twin-core optical fibers [23,24]. It has been shown that, solitons can be compressed and their dynamics effectively controlled through these variable parameters. The Kortweg-de Vries (KdV) equation with variable coefficients [25] has been studied recently in the context of ocean waves, where the spatio-temporal variability of the coefficients are due to the changes in the water depth and other physical conditions. The fact that, MKdV equation is relevant to hydrodynamics and a variety of physical phenomena, it is natural to expect the possibility of temporal variations in the equation parameters occurring in the same. Furthermore, for propagating solitons, the first integral of the MKdV equation yields NLSE with a source, making it imperative to investigate the effect of the temporal variation of the distributed parameters on the solitary wave solutions of this dynamical system.
The goal of the present paper is to study the effect of * [email protected] the variable coefficients on the solution space of MKdV equation, both for positive and negative cases. We find that the effect of distributed coefficients on the soliton dynamics of MKdV equation is quite different than that of NLSE. In case of NLSE, the amplitude and width are affected by the time dependence of the distributed coefficients. This leads to compression of solitary waves in NLSE. Through explicit construction, it is shown that, solitary waves in this system can be effectively controlled through the equation parameters. The solitons can be accelerated and manipulated by suitable variations of the above parameters. However the width and amplitudes are not amenable for manipulation and control.
We consider the modified KdV equation with variable coefficients in the form
u t + α(t)u x − β(t)u 2 u x + γ(t)u xxx = 0,(1)
where γ(t), α(t) and β(t) are time dependent variables. Although, the first derivative term in the field variable can be removed by a suitable change of the coordinate frame, the same has been kept here explicitly to contrast its effect with the nonlinear and dispersion terms. We consider the ansatz solution of the form,
u = A 1 (t)g[ω(x, t)] + A 0 (t),(2)where, ω(x, t) = f (t)x − h(t).
The variable coefficient MKdV equation can be mapped to Jacobi elliptic equation:
g ′′ = P g + 2Qg 3 ,(3)
having a conserved quantity,
(g ′ ) 2 = P g 2 + Qg 4 .(4)
Here prime indicates differentiation with respect to the argument ω, and P and Q are constants. Substituting the ansatz in Eq.(1) and using the relations (3) and (4), we collect the coefficients of g α and g α g ′ (where α=0,1,2) to find the consistency conditions:
g 0 : ∂ t A 0 = 0,(5)g : ∂ t A 1 = 0,(6)g ′ : A 1 ∂ t ω+A 1 α(t)∂ x ω+A 1 P γ(t)[∂ x ω] 3 −A 2 0 A 1 β(t)∂ x ω = 0,(7)gg ′ : −2A 0 A 2 1 β(t)∂ x ω = 0,(8)g 2 g ′ : −A 3 1 β(t)∂ x ω + 6A 1 Qγ(t)[∂ x ω] 3 = 0,(9)
For obtaining solutions to the above set of equations, we require, A 0 = 0, and A 1 =constant. From Eq. (9), further simplification yields,
f 2 (t) = A 2 1 β(t) 6Qγ(t) ,(10)
and from Eq. (7) we get,
∂ t h(t) = x∂ t f (t) + α(t)f (t) + P γ(t)f 3 (t).(11)
Condition (11) requires that, f (t) should be a constant so that the term containing x vanishes. This implies β(t)/γ(t) = κ, where κ is a constant. We then have the relations,
f = A 2 1 κ 6Q ,(12)
and,
h(t) = A 2 1 κ 6Q [α(t) + γ(t)P A 2 1 κ 6Q ]dt.(13)
Hence, the exact travelling wave solution can be written in the form,
u = A 1 g A 2 1 κ 6Q x − [α(t) + γ(t)P A 2 1 κ 6Q ]dt .(14)
It is worth noting that, A 1 is unconstrained and controls the width of the solution. Unlike the case of NLSE the amplitude and width are independent of time. Since, the solution involves κ, positive and negative MKdV equations have different type of solutions. g can be any of the twelve Jacobi elliptic functions, with the modulus parameter m 2 (0 ≤ m 2 ≤ 1) [26,27]. The following are some identities of the Jacobi elliptic functions, which are used: cn 2 (w, m) + sn 2 (w, m) = 1, dn 2 (w, m) + m 2 sn 2 (w, m) = 1, sn ′ (w, m) = cn(w, m)·dn(w, m), m). For m = 1, cn(w, 1) = dn(w, 1) = sech(w) and sn(w, 1) = tanh(w).
cn ′ (w, m) = −sn(w, m)·dn(w, m), dn ′ (w, m) = −m 2 sn(w, m)·cn(w,
Below we analyze some explicit solutions and corresponding parameter ranges. For the sake of specificity we consider β(t) > 0. Case I: With g = cn(ω(x, t)) one finds, the cnoidal wave solution as, Fig. (1) depicts the temporal evolution of the above bell shaped localized solution. For illustrative purpose we have considered two different cases where, α(t) = 0 and α(t) = −3t 3 cos(t 3 ). Fig. (2) depicts the same solution when γ(t) and α(t) have polynomial time dependence. One clearly sees that the temporal variations of γ and α can effectively modulate and control the propagation of the solitons. Case II: g = sn(w, m)
u = A 1 cn A 2 1 κ −6m 2 x − [α(t) + γ(t)(2m 2 − 1)A 2 1 κ −6m 2 ]dt ,(15)u = A 1 sech −A 2 1 κ 6 x − [α(t) − γ(t)A 2 1 κ 6 ]dt .(16)
We now study the cases where g = sn(ω(x, t)), for which the solution corresponds to negative MKdV equation:
u = A 1 sn A 2 1 κ 6m 2 x − [α(t) − γ(t)(m 2 + 1)A 2 1 κ 6m 2 ]dt .(17)
Here γ(t) > 0, Q = m 2 > 0 and P = −(m 2 + 1) < 0. For m 2 = 1, we have the kink type solitary wave solution Fig. (3) depicts the kink solution in the presence of time dependent dispersion. The soliton motion can be controlled through the external parameters. Case III: g = dn(w, m)
u = A 1 tanh A 2 1 κ 6 x − [α(t) − γ(t)A 2 1 κ 3 ]dt .(18)
We get exact solitary wave solution of positive MKdV equation, in the case when g = dn(w(x, t)):
u = A 1 dn −A 2 1 κ 6 x − [α(t) − γ(t)(2 − m 2 )A 2 1 κ 6 ]dt .(19)
Here, γ(t) < 0 and P = (2 − m 2 ) > 0. For, m 2 = 1, we get bell shaped solitary wave solution as,
u = A 1 dn −A 2 1 κ 6 x − [α(t) − γ(t)A 2 1 κ 6 ]dt .(20)
In conclusion, MKdV equation with time varying coefficients has solitary waves solutions, provided the tem-poral variations of the coefficients are of the form given in the text. The temporal variation of these parameters allow effective control of the solitary wave profile. These continuous waves and localized solutions can be made to accelerate. The amplitude and widths are not modulated by the distributed coefficients. The induction of time dependent u x term allows us to control the motion of the solitons more efficiently.
where, γ(t) < 0, P = (2m 2 − 1) > 0, Q = −m 2 < 0 and m 2 > 1/2. This corresponds to the positive MKdV equation. In the case, when m 2 = 1/2, P = 0, and γ(t) does not affect the solution. For, m 2 = 1 case, we have the exact solitary wave solution of the form,
FIG. 1 :
1Propagating localized solitary wave solution of MKdV equation with g = cn(x, t), where γ = cos(t), κ = −24, A1 = 1, P = 1, Q = −1 and m = 1; left α(t) = 0 and right α(t) = −3t 3 cos(t 3 ). FIG. 2: Propagating localized solitary wave solutions for g = cn(x, t), where γ(t) = 3t 2 , κ = −24,A1 = 1, P = 1, Q = −1 and m = 1; left α(t) = 0 and right α(t) = −t 9 . FIG. 3: Kink type solitary wave solution of the MKdV equation, where γ(t) = cos(t), α(t) = 5t 4 , κ = −48, A1 = 1, P = 1, Q = −1 and m = 1.
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arxiv |
Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models
arXiv:hep-th/0105164v1 17 May 2001 May 2001
V I Inozemtsev
Yukawa Institute for Theoretical Physics
Kyoto University
606-8502KyotoJapan
R Sasaki
Yukawa Institute for Theoretical Physics
Kyoto University
606-8502KyotoJapan
Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models
arXiv:hep-th/0105164v1 17 May 2001 May 2001
For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a "site", we associate a vector space V µ whose element is called a "spin". Its dynamical variables are the canonical coordinates {q j , p j } of a particle in R r , (r = rank of ∆), and spin exchange operators {P ρ } (ρ ∈ ∆) which exchange the spins at the sites µ and s ρ (µ). Here s ρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For ∆ = A r and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.
Introduction
The essential part of our knowledge of quantum many-body systems is concerned with integrable models in one dimension. Among them, the Calogero-Moser models [1,2,3,4] with long-range interactions are most popular during last decade. Their links to the models of solid-state physics [5,6,7,8,9,10,11,13] have been found, and they are based on the possibility to introduce also the spin exchange interaction in a translation-invariant form.
However, the CM models can be formulated n classical and quantum mechanics for any root system [14,15,16,17,18,19], and one can guess that introduction of spin exchange can be done at least for some root systems too. There were several attempts [8,9,10,12] in this direction, but they were far from being universal in a way for introducing spin into the CM models.
In this paper, we consider the possibility of unifying all the previous approaches to spin Calogero-Moser models and related models of spin exchange interactions obtained by "freezing" the canonical variables at the equilibrium points of the corresponding classical CM systems. This can be done by constructing universal Lax representations for degenerate forms of the CM potentials. There are also some indications that the corresponding models with most general elliptic potentials are also integrable [6,13], but the construction of Lax pair in this case does not lead directly to integrability.
The organization of the paper is as follows. In Section 2, the universal Lax operators for the CM models with degenerate potentials [17,18] is briefly recapitulated. The way of introducing spin exchange in the framework of the above formalism is proposed in Section 3 so as to prove the integrability of the spin CM models for all root systems. The existence of conserved quantities is guaranteed by the "sum to zero" condition for the second Lax operator. Section 4 is devoted to the models with spin exchange operators only. The corresponding Lax operators lead in the trigonometric case and A r root system to Haldane-Shastry model [5]. The Polychronakos model [8,12] corresponds in this approach to the rational case with a confining q 2 potential. The final section is devoted to summary and comments.
Universal Lax Operator for Calogero-Moser Model with Degenerate Potential
In this section we briefly recapitulate the essence of Calogero-Moser models based on any root system ∆ (applicable to the exceptional and non-crystallographic root system) and
the associated universal Lax pair formalism along with appropriate notation [16,17,18,19] and background [14,15] for the main body of this paper. Those who are familiar with the universal Lax pair formulation may skip this section and return when necessity arises. A Calogero-Moser model is a Hamiltonian system associated with a root system ∆ of rank r, which is a set of vectors in R r with its standard inner product, invariant under reflections in the hyperplane perpendicular to each vector in ∆. In other words,
s α (β) ∈ ∆, ∀α, β ∈ ∆, s α (β) = β − (α ∨ · β)α, α ∨ ≡ 2α/|α| 2 . (2.1)
The set of reflections {s α , α ∈ ∆} generates a group G ∆ , a finite reflection group, known as the Coxeter (Weyl) group. The set of roots ∆ is decomposed into a disjoint sum of the positive roots ∆ + and negative roots ∆ − . The dynamical variables of the Calogero-Moser model are the coordinates {q j } and their canonically conjugate momenta {p j },which will be denoted by vectors in R r with the standard inner product: q = (q 1 , . . . , q r ), p = (p 1 , . . . , p r ),
p 2 = p · p = r j=1 p 2 j . (2.2)
The Hamiltonian for classical Calogero-Moser model with a degenerate potential reads:
H C = 1 2 p 2 + 1 2 ρ∈∆ + g 2 |ρ| |ρ| 2 V (ρ · q),(2.3)
in which the potential function V is listed in the following Table 1: Here we have omitted the scale factor for the trigonometric (hyperbolic) potential, for simplicity. The associated universal Lax pair operators read
V (u) x(u) y(u) rational 1/u 2 1/u -1/u 2 trigonometric 1/ sin 2 u cot u -1/ sin 2 u hyperbolic 1/ sinh 2 u coth u -1/ sinh 2 uL = p ·Ĥ + X, X = i ρ∈∆ + g |ρ| (ρ ·Ĥ) x(ρ · q)ŝ ρ ,(2.
4)
M = i 2 ρ∈∆ + g |ρ| |ρ| 2 y(ρ · q)ŝ ρ ,(2.5)
in which the functions x(u) and y(u) are listed in the Table 1. These functions are related by
y(u) ≡ dx(u)/du, V (u) = −y(u) = x 2 (u) + constant. (2.6)
The real positive coupling constants g |ρ| are defined on orbits of the corresponding reflection group, i.e. they are identical for roots in the same orbit. That is, for the simple Lie algebra cases one coupling constant g |ρ| = g for all roots in simply-laced models and two independent coupling constants, g |ρ| = g L for long roots and g |ρ| = g S for short roots in non-simply laced models. The operatorsĤ j andŝ ρ obey the following commutation relations
[Ĥ j ,Ĥ k ] = 0, (2.7) [Ĥ j ,ŝ α ] = α j (α ∨ ·Ĥ)ŝ α , (2.8) s αŝβŝα =ŝ sα(β) ,ŝ 2 α = 1,ŝ −α =ŝ α . (2.9)
In terms of these commutation relations it is easy to show that the canonical equations of motion can be represented in an operator form:
q j = p j ,ṗ j = − ∂H C ∂q j ⇐⇒ d dt L = [L, M ].
(2.10)
Let us choose an irreducible D-dimensional representation of the reflection (Weyl) group G ∆ to be denoted by R. It is a collection of R r vectors, to be called a "site", which form a single Weyl orbit:
R = {µ (1) , . . . , µ (D) |µ (k) ∈ R r }. (2.11)
That is any site of R can be obtained from any other site by the action of the reflection (Weyl) group. Thus the (length) 2 of the vectors µ (k) are equal:
(µ (j) ) 2 = (µ (k) ) 2 , ∀µ (j) , µ (k) ∈ R.
(2.12)
Then L and M are D × D matrices whose elements are given by
(Ĥ j ) µν = µ j δ µν , (ŝ ρ ) µν = δ µ,sρ(ν) = δ ν,sρ(µ) . (2.13)
The essence of the Lax pair is the following set of identities among the functions {x(ρ · q)} and {y(ρ · q)} expressed in matrix forms:
[X, M ] = −Ĥ · ∂V ∂q , V = 1 2 ρ∈∆ + g 2 |ρ| |ρ| 2 V (ρ · q),(2.
14) (2.15) in which the right hand side of (2.14) is a diagonal matrix. The matrix M has a special property (see (2.36) of [18]): Since the elements of the matrices X and M are numbers and V S × I commutes with X we have from (2.14)
[ p ·Ĥ, M ] = i[− 1 2 ∂ 2 ∂q 2 , X],µ∈R M µν = ν∈R M µν = −iV S , V S = 1 2 ρ∈∆ + g |ρ| |ρ| 2 V (ρ · q), (2.16) in which V S is independent of µ and ν. Note that V S is different from V in (2.14),[X, M] = −Ĥ · ∂V ∂q , (2.19)
which is the content of the usual Lax pair.
Spin Calogero-Moser Model with Degenerate Potential
Now let us define a spin Calogero-Moser model associated with a root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ , that is the set of "sites". A dynamical state of the model is a wavefunction ψ(q) times a vector ψ S which takes value in the D multiple of a vector space V;
ψ S ∈ D ⊗V. (3.1)
Each V is associated with site µ. In other words ψ S can be represented by its component
spin ψ (µ) S at the site µ, or ψ (j)
S at site j for short:
ψ S = |ψ (1) S , . . . , ψ (D) S > . (3.2)
Let us introduce a spin exchange operatorP ρ associated with each root ρ ∈ ∆:
P ρ : ψ S →P ρ ψ S , (P ρ ψ S ) (µ) = ψ (sρ(µ)) S , ∀µ ∈ R. (3.3)
Obviously {P ρ } (ρ ∈ ∆) satisfy the same commutation relations as {ŝ ρ }:
P αPβPα =P sα(β) ,P 2 α = 1,P −α =P α , (3.4)
andŝ α ,Ĥ j andP β commute since they act on different spaces
[ŝ α ,P β ] = 0 = [Ĥ j ,P β ]. (3.5)
Likewise the quantum operators {q j } and {p k } commute withP ρ :
[q j ,P ρ ] = 0 = [p k ,P ρ ], j, k = 1, . . . , r, ∀ρ ∈ ∆. (3.6)
By multiplyingP ρ to the functions x(ρ · q) and y(ρ · q) in X and M, we define new matrices X S and M S :
X S = i ρ∈∆ + g |ρ| (ρ ·Ĥ) x(ρ · q)P ρŝρ , (3.7) M S = i 2 ρ∈∆ + g |ρ| |ρ| 2 y(ρ · q)P ρŝρ ,(3.8)
whose elements are no longer numbers but operators now. As in the previous section we define a new matrix M S ,
M S = M S + iA × I,(3.9)
which satisfies sum up to zero condition, too
µ∈R (M S ) µν = ν∈R (M S ) µν = 0. (3.10)
The operator A now depends on the spin exchange operators {P ρ }:
A = 1 2 ρ∈∆ + g |ρ| |ρ| 2 V (ρ · q)P ρ . (3.11)
Since the commutation relations of {Ĥ j ,ŝ ρ } and {Ĥ j ,ŝ ρ ≡P ρŝρ } are identical we have the following main result
[X S , M S ] = −Ĥ · ∂V ∂q , (3.12)
in which the right hand side does not contain operators {P ρ }. This is because they cancel out by the relationP 2 ρ = 1. The right hand side can be replaced by the obvious identity in quantum theory (3.14) in which the second commutator in the left hand side no longer vanishes. By adding (3.14) to
−Ĥ · ∂V ∂q = i[H C , p ·Ĥ]. (3.13) If we rewrite M S in terms of M S , we obtain [X S , M S − iA] = i[H C , p ·Ĥ],[p ·Ĥ, M S − iA] = i[ p 2 2 , X S ],(3.15)
we arrive at the desired equation
[p ·Ĥ + X S , M S ] = i[H S , p ·Ĥ + X S ], (3.16) H S ≡ H C − A = 1 2 p 2 + 1 2 ρ∈∆ + |ρ| 2 g |ρ| (g |ρ| −P ρ ) V (ρ · q), (3.17)
which is a universal Lax equation for the spin Calogero-Moser model
i[H S , L S ] = [L S , M S ], L S = p ·Ĥ + X S (3.18)
defined by the Hamiltonian H S (3.17). That is, this applies to any spin Calogero-Moser models based on any root system ∆ and any irreducible representation R of the reflection (Weyl) group G ∆ and for any degenerate potentials. From this follows
i[H S , L k S ] = [L k S , M S ], or i[H S , (L k S ) µν ] = κ∈R (L k S ) µκ (M S ) κν − (M S ) µκ (L k S ) κν . (3.19)
Thanks to the sum up to zero condition of M S (3.10) we obtain the conserved quantity as the Total sum (Ts) of L k S instead of the diagonal sum (Tr):
[H S , Ts(L k S )] = 0, Ts(L k S ) ≡ µ,ν∈R (L k S ) µν , k = 2, . . . , .(3.20)
This type of conserved quantities was known for the A r spin Calogero-Moser models for the vector representation [7,9]. Note that ( 4. For the A r model with the vector representation the number of "sites" is r + 1 which is equal to the degrees of freedom of the associated particle motion, if the A r root system is embedded into R r+1 as is done customarily. This is a rather exceptional situation.
In all the other irreducible representations S r+1 and for all the other root systems (except for the trivial representation), the number of sites, or the dimensions of R, is bigger than r, the rank of ∆. For example, the vector representation of D r or the set of short roots for B r consists of 2r vectors, which in a conventional parametrisation of the roots take the form R = {±e j , j = 1, . . . , r|e j ∈ R r , e j · e k = δ jk }. Our spin Calogero-Moser models require all these 2r sites. There are some references in which spin Calogero-Moser models for B r , C r , D r or BC r are discussed [10,11,12]. In all these papers, the number of sites is equal to the rank of the root systems. These are different from the present spin Calogero-Moser models.
The present formulation of the spin Calogero-Moser models together with the Lax pair
formulation does not require any specific structure of the "spin" space V attached to each site.
6. It is well-known that for the spin 1/2 case in the A r model with the vector representation, the spin exchange operators {P ρ } can be expressed in terms of the local Pauli spin matrix at each site asP e j −e k = (1 + σ j · σ k )/2. For the vector representation of D r or R being the set of short roots for B r mentioned above, we havê 8. The conserved quantities {Ts(L k S )} are essentially the same as those obtained in terms of the Dunkl [20] operators, and/or the exchange operator formalism [8]. The same remark applies to the conserved quantities of the spin exchange models to be discussed in the following section. For the quantum CM models without spin, the equivalence of the Lax pair formalism and Dunkl operator formalism was proven in [19]. 9. The Yangian symmetry [21,22] for the spin CM model and spin exchange model based on any root system is an interesting challenge. 10. The commutativity of the conserved quantities obtained from the above Lax pair formulation will be discussed elsewhere.
P e j = [(1 + σ j · σ −j )/2],P e j −e k = [(1 + σ j · σ k )/2][(1 + σ −j · σ −k )/2], P e j +e k = [(1 + σ j · σ −k )/2][(1 + σ −j · σ k )/2].
Rational Spin Calogero Model
In this subsection we will define rational spin Calogero-Moser model with quadratic confining potential to be called rational spin Calogero model, for brevity. The Hamiltonian is given by
H RS = 1 2 p 2 + 1 2 ω 2 q 2 + 1 2 ρ∈∆ + |ρ| 2 g |ρ| (g |ρ| −P ρ ) (ρ · q) 2 . (3.23)
The construction of the Lax pair follows the same pattern as the case without the spin degrees of freedom. Since the added potential 1 2 ω 2 q 2 commutes with X S , the canonical equations of motion to be obtained from H RS are equivalent tȯ
L S = i[H RS , L S ] = [L S , M S ] − ω 2 Q, Q ≡ q ·Ĥ, (3.24)
in which L S and M S are the Lax pair for the rational (1/(ρ · q) 2 ) potential only. Let us define
L ± S = L S ± iωQ, (3.25) whose time evolution readL ± S = [L ± S , M S ] ± iωL ± . (3.26)
Here we have used well-known relations [17,18]
Q = p ·Ĥ = L S − X S , [Q, M S ] = −X S . (3.27)
If we define
L S = L + S L − S ,(3.
Spin Exchange Model
The spin exchange model is defined for a root system ∆ and an irreducible representation S at site j for short:
ψ S = |ψ (1) S , . . . , ψ (D) S > .
In fact, the spin exchange model is obtained from the corresponding spin Calogero-Moser model by "freezing" the particle degrees of freedom:
p = 0, q = q 0 ,(4.2)
in which q 0 is an equilibrium position of the classical Calogero-Moser potential
∂V ∂q q=q 0 = 0, V = 1 2 ρ∈∆ + g 2 |ρ| |ρ| 2 V (ρ · q). (4.3)
Since the rational potential without the quadratic confining potential or the hyperbolic potential do not have any equilibrium points, this automatically selects the trigonometric potential. The rational potential with the quadratic confining potential case will be discussed in the next subsection separately. The equilibrium position q 0 for the trigonometric potential is determined uniquely in each Weyl alcove. In other words, if q 0 is an equilibrium point so is s α (q 0 ) which defines an equally integrable model. Let us fix q 0 and define X E and M E in terms of the Lax pair operators of the corresponding spin Calogero-Moser model at q = q 0 : and as before M E has a special property:
X E = X S | q=q 0 , M E = M S | q=q 0 .µ∈R ( M E ) µν = ν∈R ( M E ) µν = −iA E , A E = 1 2 ρ∈∆ + g |ρ| |ρ| 2 V (ρ · q 0 )P ρ ,
As in the previous section we define a new matrix M E ,
M E = M E + iA E × I,
which satisfies sum up to zero condition, too
µ∈R (M E ) µν = ν∈R (M E ) µν = 0. (4.6)
By rewriting (4.5) in terms of M E we arrive at the Lax representation of the spin exchange model:
i[H E , X E ] = [X E , M E ], (4.7)
in which the Hamiltonian H E of the spin exchange model is
H E = 1 2 ρ∈∆ + g |ρ| |ρ| 2 V (ρ · q 0 )(1 −P ρ ) = −A E + const. (4.8)
The added constant simply shifts the ground state energy. The Lax pair supplies the conserved quantities as the Total sum of X k E :
[H E , Ts(X k E )] = 0, Ts(X k E ) ≡ µ,ν∈R (X k E ) µν , k = 3, . . . , . (4.9)
It is interesting to note that the first two members Ts(X 1 E ) and Ts(X 2 E ) are trivial, in contrast to the spin Calogero-Moser case.
Some remarks are in order.
As in the spin Calogero-Moser model, the form of the spin exchange model Hamiltonian
H E (4.8) depends on the root system ∆ only, although its actual operator contents depend on the chosen representation R. The infinitely many models corresponding to various irreducible representations, sharing the same set of conserved quantities, can be considered to constitute an integrable hierarchy belonging to the root system ∆.
If one considers a series of representations with increasing dimensionality (i.e. more spins), the thermodynamic limit could be achieved within models belonging to a fixed root system ∆. This is a novel situation, since in the Haldane-Shastry model the rank r grows indefinitely in the thermodynamic limit. 3. It should be emphasised that the "coordinates" q or rather q 0 are just a set of numbers rather than dynamical variables. Thus, in contrast to the conventional approach [5,8], the notion of 'position exchange operator' is not used in our approach. 4. For the A r model, q 0 can be chosen to be "equidistant": q 0 = π(1, 2, . . . , r, r + 1)/(r + 1), (4.10) thanks to the well-known trigonometric identity r+1 k =j cos [π(j − k)/(r + 1)] sin 3 [π(j − k)/(r + 1)] = 0.
The Haldane-Shastry model [5], i.e. the A r spin exchange model for the vector representation, has been understood quite well because of this simplifying feature.
5. The equidistance of q 0 for A r seems rather fortuitous. As remarked above, any transposition of the above q 0 (4.10) provides an equally integrable spin exchange model, but the equidistance property is lost. As for D r (r ≥ 4), we have not been able to find equidistant q 0 . For BC r model, equidistant q 0 can be achieved for certain ratios of the coupling constants. For the following parametrisation of the potential [4,16],
V = r j<k g 2 M sin 2 (q j − q k ) + g 2 M sin 2 (q j + q k ) + r j=1 g S (g S + 2g L ) 2 sin 2 (q j ) + r j=1 2g 2 L sin 2 (2q j ) , (4.11)
one obtains equidistant equilibrium positions: q 0 = π(1, 3, . . . , 2r − 1)/4r, for g L /g M = 1/2, g S = 0, (4.12) q 0 = π(1, 2, . . . , r)/2(r + 1), for g L /g M = 3/2, g S = 0, (4.13) q 0 = π(1, 2, . . . , r)/(2r + 1), for g L /g M = 1/2, g S /g M = 1. (4.14)
These cases were discussed in some detail by Bernard-Pasquier-Serban [11].
6. Note that the present derivation of the spin exchange model and its Lax pair does not adopt the strong coupling limit.
7. For most general elliptic potentials, the Lax pair can be constructed in a usual manner [14]. But the second Lax operator does not satisfy the "sum to zero" condition, hence the integrability of these models is not yet established.
Rational Spin Exchange Model
The above formulation fails to give integrable spin exchange model with rational potential.
This can be remedied by adding a harmonic confining potential [8,12] which creates equilibrium points in each Weyl chamber. Here we derive the Lax operator formalism for these models. Let us start with the Lax pair for the rational Calogero-Moser models and for the time being keep the value of q unspecified. We have as in (2.14) [X, M] = −Ĥ · ∂V ∂q and after multiplyingP ρ to functions x(ρ · q) and y(ρ · q), we obtain ( If we define two new matrices X ± S X ± S = X S ± iωQ, (4.17) they satisfy simple commutation relations thanks to (4.15) and (4.16)
[X ± S , M S ] = ∓iωX ± S −Ĥ · ∂V RC ∂q , (4.18)
in which V RC is the potential of the classical rational Calogero-Moser model with harmonic confining potential V RC = 1 2 ω 2 q 2 + 1 2 ρ∈∆ + g 2 |ρ| |ρ| 2 (ρ · q) 2 . The conserved quantities are obtained as Total sum of (X + RE X − RE ) k :
A RE = 1 2 ρ∈∆ + g |ρ| |ρ| 2P ρ (ρ · q 0 ) 2 ,(4.H RE , Ts (X + RE X − RE ) k = 0, k = 1, . . . , .(4.27)
It is interesting to note that the above Hamiltonian H RE depends on the harmonic confining potential 1 2 ω 2 q 2 only through the value q 0 .
Summary and comments
We have shown that the integrability of spin Calogero-Moser model and the spin exchange model with degenerate potential and based on any root system is a direct consequence of the integrability of the corresponding classical Calogero-Moser system. For a given root system ∆ there are infinitely many integrable spin Calogero-Moser models and the spin exchange models corresponding to infinitely many irreducible representations R of the reflection group.
These define physically different models sharing the same exchange features.
After completion of the present work, we came across [23] which discusses the integrability of spin BC r model with harmonic confining potential, or "spin Inozemtsev model" [4] in terms of the Dunkl operator formalism [20].
which is quadratic in the coupling g |ρ| , whereas V S is linear. We can define a new matrix M, M = M + iV S × I, I : Identity operator,
S
3.15) is obtained from (2.15) by replacing X and M by X S and M S . = · · · = ψ (D) S the action of the spin exchange operators become that of the identity operator P ρ = 1, ∀ρ ∈ ∆. Then the Hamiltonian H S (3.17) reduces to that of the quantum Calogero-Moser models and the Lax operator L S and M S become identical to the universal quantum Lax pair operator derived by Bordner, Manton and Sasaki [18]. 2. The form of the spin Calogero-Moser Hamiltonian (3.17) depends on the root system ∆ only, although its actual operator contents depend on the chosen representation R. 3. For the A r model with the vector representation, the present spin Calogero-Moser coincides with the existing one. For the other root systems the present model is completely new, to the best of our knowledge (see the remarks in the following entry). It should be emphasised that even for the A r root system the present formulation of the spin Calogero-Moser models defines an infinitely many different models corresponding to the infinitely many irreducible representations of the symmetric group S r+1 , which is the Weyl group of A r .
words,P e j +e k exchanges the spins at site j and −k and simultaneously the spins at −j and k. Similar expressions exist for other representations and root systems for su(2), su(N) or other spins. 7. It is easy to verify, as in the Calogero-Moser models, that the Hamiltonian H S (3.17) is obtained as the lowest member of the conserved quantities derived from the Lax pair formulation: H S ∝ Ts(L 2 S ). (3.22)
28) its time evolution is Lax like:L S = i[H RS , L S ] = [L S , M S ]. (3.29) Thus we obtain conserved quantities Ts(L k ), k = 1, . . . . (3.30) The lowest conserved quantity Ts(L) gives the Hamiltonian H RS (3.23) Ts(L) ∝ H RS + ( terms which commute with all the spin exchange operators {P ρ }.
R
of the reflection (Weyl) group G ∆ . Its dynamical state is represented by a vector ψ S only which takes value in the D multiple of a vector space V; the spin Calogero-Moser model case each V is associated with site µ. In other words ψ S can be represented by its component spin ψ (µ) S at the site µ, or ψ (j)
of the matrices X E and M E are linear combinations of the spin exchange operatorsP ρ and the coefficients are just numbers. They satisfy a simple matrix identity [X E , M E ] = 0 (4.5)
2 .
2It should be remarked that the q 0 is the equilibrium point not of the function appearing in the Hamiltonian H E (4.8) which is linear in the coupling constants g |ρ| but that of the potential of the classical Calogero-Moser Hamiltonian H C (2.3) which is quadratic in the coupling constants. This difference is meaningful only for the models based on non-simply laced root systems.
matrix Q (3.19) satisfies the relation (3.27) [Q, M S ] = −X S . (4.16)
choose q 0 as an equilibrium point of V RC and defineX ± RE = X ± S | q=q 0 , M RE = M S | q=q 0 , RE X − RE , M RE ] = X + RE [X − RE , M RE ] + [X + RE , M RE ]X − RE = 0. (4.21)We define M RE byM RE = M RE + iA RE × I,(4.22)
Table 1 :
1Functions appearing in the Hamiltonian and Lax pair.
23 )
23so that M RE satisfies the sum to zero condition µ∈R (M RE ) µν = Then (4.21) can be rewritten as a Lax representation for the rational spin exchange modeli[H RE , X + RE X − RE ] = [X + RE X − RE , M RE ], (4.25)in which the rational spin exchange Hamiltonian H RE is defined byg |ρ| |ρ| 2 (ρ · q 0 ) 2 (1 −P ρ ) − A RE + const.(4.26)ν∈R
(M RE ) µν = 0.
(4.24)
H RE =
1
2 ρ∈∆ +
AcknowledgementsWe thank D. B. Fairlie for bringing[23]to our attention. V. I. I. is supported by JSPS long term fellowship. R. S. is partially supported by the Grant-in-aid from the Ministry of Education, Culture, Sports, Science and Technology, Japan, priority area (#707) "Supersymmetry and unified theory of elementary particles".
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New spin Calogero-Sutherland models related to B N -type Dunkl operators. F Finkel, D Gomez-Ullate, A Gonzalez-Lopez, M A Rodriguez, R Zhdanov, hep-th/0103190A Ntype Dunkl operators and new spin Calogero-Sutherland modelsF. Finkel, D. Gomez-Ullate, A. Gonzalez-Lopez, M.A. Rodriguez, R. Zhdanov, "A N - type Dunkl operators and new spin Calogero-Sutherland models", hep-th/0102039, February 2001; "New spin Calogero-Sutherland models related to B N -type Dunkl oper- ators", hep-th/0103190, March 2001.
| {'fraction_non_alphanumeric': 0.08324342740332746, 'fraction_numerical': 0.03881097041621714, 'mean_word_length': 3.549012010848508, 'pattern_counts': {'":': 1, '<': 1, '<?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': 21, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'For any root system ∆ and an irreducible representation R of the reflection (Weyl) group G ∆ generated by ∆, a spin Calogero-Moser model can be defined for each of the potentials: rational, hyperbolic, trigonometric and elliptic. For each member µ of R, to be called a "site", we associate a vector space V µ whose element is called a "spin". Its dynamical variables are the canonical coordinates {q j , p j } of a particle in R r , (r = rank of ∆), and spin exchange operators {P ρ } (ρ ∈ ∆) which exchange the spins at the sites µ and s ρ (µ). Here s ρ is the reflection generated by ρ. For each ∆ and R a spin exchange model can be defined. The Hamiltonian of a spin exchange model is a linear combination of the spin exchange operators only. It is obtained by "freezing" the canonical variables at the equilibrium point of the corresponding classical Calogero-Moser model. For ∆ = A r and R = vector representation it reduces to the well-known Haldane-Shastry model. Universal Lax pair operators for both spin Calogero-Moser models and spin exchange models are presented which enable us to construct as many conserved quantities as the number of sites for degenerate potentials.', 'arxivid': 'hep-th/0105164', 'author': ['V I Inozemtsev \nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n', 'R Sasaki \nYukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan\n'], 'authoraffiliation': ['Yukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan', 'Yukawa Institute for Theoretical Physics\nKyoto University\n606-8502KyotoJapan'], 'corpusid': 17116772, 'doi': '10.1088/0305-4470/34/37/314', 'github_urls': [], 'n_tokens_mistral': 12455, 'n_tokens_neox': 10981, 'n_words': 6215, 'pdfsha': 'f55d3125c59c7d46c8409f4d8a805ad207223f3a', 'pdfurls': ['https://export.arxiv.org/pdf/hep-th/0105164v1.pdf'], 'title': ['Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models', 'Universal Lax pairs for Spin Calogero-Moser Models and Spin Exchange Models'], 'venue': []} |
arxiv |
Energy Stored by Radiating Systems
Kurt Schab
Lukas Jelinek
Miloslav Capek
Casimir Ehrenborg
Doruk Tayli
Guy A E Vandenbosch
Mats Gustafsson
Energy Stored by Radiating Systems
1Index Terms Electromagnetic theoryantenna theoryPoynting's theoremQ-factorenergy storage
Though commonly used to calculate Q-factor and fractional bandwidth, the energy stored by radiating systems (antennas) is a subtle and challenging concept that has perplexed researchers for over half a century. Here, the obstacles in defining and calculating stored energy in general electromagnetic systems are presented from first principles as well as using demonstrative examples from electrostatics, circuits, and radiating systems. Along the way, the concept of unobservable energy is introduced to formalize such challenges. Existing methods of defining stored energy in radiating systems are then reviewed in a framework based on technical commonalities rather than chronological order. Equivalences between some methods under common assumptions are highlighted, along with the strengths, weaknesses, and unique applications of certain techniques. Numerical examples are provided to compare the relative margin between methods on several radiating structures.
I. INTRODUCTION
For many in the field of electromagnetics, stored energy is best known by its appearance in the definition of a time-harmonic system's Q-factor (quality factor, antenna Q, radiation Q) [1], [2],
Q = 2πW sto W diss ,(1)
from which an estimate of fractional bandwidth is available. In the above expression, W sto and W diss denote the cycle-mean stored and dissipated energies within the system, respectively. The dissipated energy is typically well defined and can be easily calculated, while in many cases the definition of stored energy is ambiguous. This issue is particularly troublesome in distributed and radiating systems, where there exists no consistent, physicallyintuitive method of delineating the overlap between energy which is stored and that which is propagating. Analogous problems can be encountered in lumped circuits, where specific networks can be arbitrarily inserted to increase the total energy without altering the impedance characteristics as seen from a port. The first of two goals of this paper is to elucidate the challenges involved in defining stored energy within a general electromagnetic system. To do so, we draw upon examples of lumped circuits and radiating systems which exhibit the general issue of "unobservable energy states". Although this concept is somewhat abstract, it provides a consistent framework for understanding what makes defining stored energy in certain systems so difficult. Because of the powerful relationship between fractional bandwidth and stored energy, many researchers have worked to rigorously define stored energy in an attempt to obtain bounds on the broadband behavior of systems. Of particular practical and historical importance is the study of stored energy in radiating systems, i.e., antennas. Work in this area dates back over half a century and has given rise to many unique (and sometimes controversial) interpretations and claims. One regime where most methods agree is in the quasi-static limit, i.e., for small antennas. However, for problems involving larger antennas or antennas next to larger objects (e.g., ground planes or human bodies), most methods disagree and there is no consensus on a definition of stored energy. In some cases, the similarities and differences between these existing approaches are clear, though in other instances the technical and philosophical connections between works from different eras are more subtle. The second goal of this paper is to provide a clear summary of the many previously published approaches to defining stored energy, with emphasis on works studying distributed and radiating systems. We aim to provide not a chronological history of this topic, but rather an organized guide to the major themes and concepts used in previous works.
The paper is organized as follows. In Section II, we present a general definition for stored energy within an electromagnetic system using the concept of unobservable energy states. In Section III, existing approaches to defining and calculating stored energy within radiating systems are summarized. Where applicable, the similarities and differences between these methods are highlighted, along with their strengths, weaknesses, and relation to the formal definition of stored energy given in Section II. Analytical and numerical examples are presented in Section IV, giving both quantitative and qualitative insight into the relative results obtained by the methods outlined in Section III. The paper concludes with a discussion of applications of certain methods in Section V and general conclusions in Section VI. Further details are provided on the classical definition of stored energy in Box 1, unobservable states in Boxes 2 and 3, and electrostatic energy in Box 4.
II. DEFINITION AND PHYSICAL RATIONALE OF STORED EM ENERGY
The total energy of a dynamic system, see Box 1, represents a well-known and fundamental characteristic describing the energy stored in all of its degrees of freedom. By contrast, the observable part of total energy is a more subtle quantity typically defined in such a way that its value has a direct correspondence with the input / output relation of the system as seen by a fixed observer [3]. In lossless systems, these two quantities are equal due to the Foster's reactance theorem [6,. In general dissipative systems, however, they lose their relation due to the presence of states not observable from outside the system, see Boxes 2 and 3.
The energy supplied to a radiating system is converted into several different forms. Consider a radiator made of non-dispersive isotropic medium with permittivity ε, permeability µ and conductivity σ, which is placed in otherwise free space (effects induced by frequency dispersion are discussed in Appendix A). The radiator is enclosed within a volume V with bounding surface S, see Figure 1. Here we use, E and H to represent the time-domain electric and magnetic fields, respectively, while J source denotes an impressed current distribution. Assuming the initial conditions E (r, t → −∞) = 0, H (r, t → −∞) = 0, Poynting's theorem can be written as [10], [11] W supp (t 0 ) = W EM (t 0 ) + W heat (t 0 ) + W rad (t 0 ) ,
where the supplied energy is
W supp (t 0 ) = − t0 −∞ V E · J source dV dt,(8)
the energy lost in heat is
W heat (t 0 ) = t0 −∞ V σ |E| 2 dV dt,(9)
and the net energy escaping the volume through the bounding surface S is
W rad (t 0 ) = t0 −∞ S (E × H) ·n dS dt.(10)
These terms account for energy supplied to and lost from the system, letting us define the remaining term in Poynting's theorem as the total electromagnetic energy stored within the volume V at time t = t 0 ,
W EM (t 0 ) = 1 2 V ε |E| 2 + µ |H| 2 dV.(11)
Box 1. Stored energy in circuits and systems
Many dynamic systems in nature can be modeled as
∂ ∂t Wu + Pu = Bv in with u out = B T u,(2)
where v in and u out denote the input and output signals, u the system's internal states, and W, P, and B are matrices describing the system [3]. To construct an energy balance of such a system over an interval [t 1 , t 2 ] we multiply with the states u and integrate to get
u T Wu 2 t2 t1 + t2 t1 u T Pu dt = t2 t1 u T out v in dt,(3)
in which T denotes matrix transpose. The left-hand side can be identified as the difference in stored energy and dissipation of energy during the interval and the right-hand side is the supplied energy, cf. the definition in Section II. The definition and interpretation of the stored energy depend on the properties of the matrices W, P, and B. Systems representable by (2) can contain states that are unobservable to an observer seeing only the input and output signals. These states can contain unobservable energy [3]. The time-average stored energy (3) for time-harmonic signals u(t) = Re{Ue jωt } is U H WU/4, where we note that the system matrix W can be determined by frequency differentiation of the matrix Z obtained from (2), i.e.,
Z = P + jωW with W = ∂ Im{Z} ∂ω .(4)
By (2), it is implicit that P and W are frequency-independent in this classical system model. Probably one of the most familiar systems which follows the form (3) is a lumped circuit. Here, the input and output states are the voltages V and currents I, respectively. These are related through either the explicit summation of all circuit components or their impedance matrix [4]
Z = R + jωL + 1 jω C i ,(5)
where R describes the resistive components of the circuit and matrices L and C i represent the reactive elements. The impedance matrix relates the current to the voltage as ZI = V. To reach the stored energy form in (3) we differentiate the impedance matrix with respect to ω and multiply with the current I and its hermitian conjugate I H from the right and left, respectively. This expresses the time-average stored energy, average of the first term in (3) for a time-harmonic signal, as the quadratic form [4] W sto = 1 4
I H LI + 1 4ω 2 I H C i I,(6)
where the classical expressions for the stored energy in inductors and capacitors are recognized [5].
All aforementioned quantities depend upon a choice of volume V and its bounding surface S. A specific choice of the surface S lying in the radiation zone 1 [10] leads to (11) representing the total electromagnetic energy and (10) the total radiated energy. This division, however, depends on surface S due to time retardation. The energy defined in (11) encompasses all electromagnetic energy localized in the chosen volume V containing the system. Nevertheless, for an observer situated at the input port of the system, the entirety of energy W EM is not necessarily observable, see Box 2. Unobservable energy states by definition cannot affect physical measurements at the location of the observer. For this observer a more sensible definition of the stored energy is,
W sto (t 0 ) = W EM (t 0 ) − W unobs (t 0 ) ,(12)
where W unobs (t 0 ) is the energy of all unobservable states. This definition suggests that the value of stored energy depends on the position of the observer. Throughout this paper it is assumed that the observer is positioned at the input port of the electromagnetic system and therefore perceives the minimum stored energy from all observers. Note, however, that the even the minimum value of energy W sto (t 0 ) is not necessarily recoverable [12], [13] by experiments performed at the location of the observer (recoverable energy W rec (t 0 ) is detailed later in Section III).
0.8 1 ω/ω 0 |Γ | Q = 20 Q = 5 (b) System Γ (a) C L R 1 2 3 4 (c) C R R 2 C R 3 4 R (d)
The unobservable states are defined as those states which cannot be identified by the observer. To provide an example, let us suppose a yet unknown system, schematically depicted in panel (a). This system is examined by an observer at its input port and quantified by its reflection coefficient Γ , [7]. From the information obtained at the port, we can attempt to construct the system within. The simplest circuit that fits the measured data, depicted in panel (b), is an RLC circuit, see panel (c). However, the resistor in the RLC circuit can be arbitrarily replaced by circuit elements of the Zöbel type [8], see panel (d), without affecting exterior results observed at the port. If we now assume to be able to access the internal structure of the constructed circuits, we can calculate the energy stored in the reactive elements. It then becomes apparent that the added Zöbel circuit does affect the stored energy without changing what is observed at the port. Thus, these two valid circuit realizations for the same measured reflection coefficient predict different values of stored energy. This illustrates that depending on the specific circuit realization, the stored energy, unlike the reflection coefficient, can potentially be altered by states unobservable to the outside observer. This is true for all quantities inferred from stored energy, including the Q-factor in (1). It is also important to appreciate that how much of a system's stored energy is observable explicitly depends on the observer. If, for example, the observation procedure would include both measurement of the the reflection coefficient Γ and measurement of heat produced by the circuit, the observer will be able to distinguish circuit (c) from circuit (d), since the time evolution of heat differs in them.
The stored energy is fully recoverable only in special cases, the most important being closed lossless systems satisfying W heat (t 0 ) + W rad (t 0 ) = 0. Examining the properties of aforementioned energy definitions, we arrive at the following inequality
0 ≤ W rec (t 0 ) ≤ W sto (t 0 ) ≤ W EM (t 0 ) ≤ W supp (t 0 ) .(13)
In the preceding discussion, all quantities are defined in the time domain. However, in many cases cycle mean values of the energies in (10), (11) and (12) in time-harmonic steady state are of interest, where time-harmonic quantities at angular frequency ω are defined as G(t) = Re{G(ω)e jωt } and cycle means are denoted as · . The conversion of all preceding energy terms into time-harmonic domain is straightforward, but induces an issue with potentially unbounded energy values. This happens when the volume V is chosen to consist of all space (denoted V ∞ ) with bounding surface S being a sphere at infinity (denoted S ∞ ). In such a case the time-averaged total electromagnetic energy
W EM = W EM = 1 4 V∞ ε|E (ω)| 2 + µ|H (ω)| 2 dV(14)
is infinite due to the infinite amount of radiation energy contained in propagating fields within the volume V ∞ . Subtracting this energy from the total energy W EM , i.e., to identify unobservable energy with radiation, is the aim of several approaches calculating the stored energy W sto = W sto (t 0 ) . These methods rely on the fact that Box 3. Unobservable energy, part 2
R R Z 0 = R l R (a) ∼ r → ∞ (b)
Unobservable energy can be encountered in many basic electromagnetic devices, such as a matched transmission line or a radiating antenna system, see panels (a) and (b) above. In both of these cases, traveling energy exists but is unobservable for an observer at the input port. Specifically, the total energy within the transmission line circuit in panel (a) can be arbitrarily altered through changes to the line length l with no effect on the impedance seen from the input port cf. with lumped circuit models for a transmission line [9]. Similarly, the energy stored within the radiating system in panel (b) depends on the definition of the spatial boundary at which energy "leaves" the system, though this boundary has no effect on the port impedance. For time-harmonic signals and a system boundary chosen at infinity, i.e., the far-field sphere, the system in panel (b) contains an infinite amount of traveling energy. Fig. 1: Sketch of an antenna region Ω, a smallest circumscribing sphere of radius a, an arbitrary volume V with its boundary surface S and the far-field sphere bounded by S ∞ . time-averaged radiated power
R 3 S ∞ V S ≡ ∂V n r = a O r → ∞ ΩP rad = S∞ P (ω) ·r dS = 1 2Z 0 S∞ |E (ω)| 2 dS = 1 2Z 0 S 2 |F (ω)| 2 dS(15)
in time-harmonic steady state is the same for all surfaces enclosing the sources. The quantities
F (ω) = lim r→∞ re jkr E(ω) and P (ω) = 1 2 Re{E (ω) × H * (ω)}(16)
used above denote the far field and the real part of the Poynting's vector, respectively. In the far right-hand-side of (15), surface S 2 denotes the unit sphere and k = ω/c 0 in (16) denotes the free-space wavenumber. When used to evaluate Q-factor, the cycle-mean stored energy W sto is normalized by the cycle-mean dissipated energy (see (1)).
In radiating systems without ohmic losses, the cycle-mean dissipation reduces to the radiated power P rad in (15). Note that in many cases, the Q-factor in (1) is assumed to be tuned such that the system as a whole is resonant. In general, a non-resonant system can be tuned by the addition of a specific reactance, which stores additional energy W tune . The tuned Q-factor can then be explicitly rewritten as
Q = 2π (W sto + W tune ) W diss .(17)
Box 4. Electrostatic energy expressed in fields, circuits, and charges
Electrostatic energy W e is thoroughly treated in many classical textbooks [10], [14], [9] with a clear consensus on its definition, see [9] for a discussion. The energy W e can be expressed in three equivalent ways as
W e = 1 2 R 3 ε 0 |E(r)| 2 dV = 1 2 Ω φ(r)ρ(r) dV = 1 2ε 0 Ω Ω ρ(r 1 )ρ(r 2 ) 4π|r 1 − r 2 | dV 1 dV 2 ,(18)
where E denotes electric field intensity, φ electric potential and ρ charge density supported in Ω ⊂ R 3 , see Figure 1. Below, we consider a perfect electric conductor (PEC) object Ω with the total charge ρ dV = q tot . From left to right, the terms in (18) represent energy expressed in:
• fields, where the electric energy density ε 0 |E| 2 /2 is integrated over all space, • circuits, where a constant potential φ = V on the PEC object is used to rewrite the energy W e = V q tot /2 = CV 2 /2 in terms of capacitance C, • charges, where a double integral over the source region is used. These representations offer alternative expressions and ways to evaluate the energy. Similar interpretations are observed for the electromagnetic energy discussed in Section III.
Since the stored energy in a pure reactance is well-defined, throughout this paper we discuss only the general stored energy W sto .
III. EXISTING METHODS
So far, we have discussed stored energy only in terms of the abstract definition in (12) involving the total and unobservable energies. For practical purposes, more specific expressions are required to evaluate a system's stored energy. This Section compares many methods developed to calculate the stored energy in electromagnetic systems. These methods vary in approach and generality, though most were motivated by the desire to calculate the Q-factor of radiating systems, as defined in (1).
The many attempts at defining and calculating stored energy in radiating systems can be classified and grouped in several ways, cf. the electrostatic case in Box 4. In this section, we briefly discuss these methods using the physical quantities required in each technique as a primary distinguishing feature. All discussed methods are listed in Table I, where they are grouped using this convention. Specifically, methodologies are grouped into those derived mainly from electromagnetic fields (blue color), those with energy values directly calculable from source current distributions (green color), and those which take a more abstract system-level approach (red and gray color). This particular division is by no means unique, and throughout this section mathematical equivalences and philosophical similarities between methods are discussed.
The data required for implementing each method are listed in the Requirements column, along with the region over which those data sets are required. These regions are denoted using R 3 to represent all space, Ω the support of sources, S ∞ the far-field sphere, and Port the port of the system. Three salient features are indicated for each method in the Properties column. These features are:
• coordinate independence, r ind : A check mark in this column indicates that energy expressions are coordinate independent, i.e., they are independent of an antenna's position within a coordinate system. • positive semi-definiteness, W sto ≥ 0: In Section II it was argued that the stored energy W sto should always be non-negative. A check mark in this column indicates that energies obtained by a given method obey this requirement. • applicability to current optimization, J -opt: A check mark in this column indicates that a given formulation of stored energy can be directly applied to source current optimization, useful in determining certain physical bounds. For the sake of simplicity, all the methods described in Section III are presented assuming radiators made only of PEC or assuming electric currents placed in a vacuum environment. All presented methods however allow generalization to non-dispersive inhomogeneous media of finite extent, although validations of such generalizations are scarce. Specific information regarding this procedure for each method is left to corresponding subsections.
Method
Properties Requirements Reference Similarly, certain methods may be applicable to systems containing dispersive media, though the accuracy and interpretation of results in these cases is still an open area of study.
r ind Wsto ≥ 0 J -opt Data Domain Region Field WP r E, H ω0 R 3 §III-A1 WP E, H ω0 R 3 §III-A2 WF E, H or Xin, F ω0 R 3 or Port, S∞ §III-A3 Current W X Z, I ω0 Ω §III-B1 Wreac J ω0 Ω §III-B2 W X Z, I ω0 Ω §III-B3 W td (t0) J t Ω, S∞ §III-B4 System W Z B in Zin, Iin ω Port §III-C1 Wrec(t0) Zin, Iin ω Port §III-C2 QFBW Zin ω Port §III-D1 Q Z Zin ω0 Port §III-D2
A. Stored energy expressed in terms of electromagnetic fields Methods derived from the fields E and H attempt to calculate stored energy (12) by subtracting unobservable energy from the total energy locally at the level of electromagnetic fields around the radiator, see Figure 2. These procedures commonly allow for the definition of a local stored energy density by identifying energy in radiating fields as unobservable energy. An advantage of these methods is that they require only field quantities, not the physical structure of the radiator. However, these methods are typically computationally demanding, rendering even simple optimization tasks prohibitively expensive. Other common issues are the unknown form of unobservable energy within the smallest sphere circumscribing a source region Ω (which can lead to over-subtraction [15]) and omission of other forms of unobservable energy such as non-radiating currents [16], see also Boxes 2 and 3. In all known cases, general dispersive materials cannot be treated with these methods. The inclusion of non-dispersive materials can be made [17], [18], [19] in all methods described in this subsection by changing ε 0 → ε and µ 0 → µ in the first two terms in (19), (20) and (21).
The published results are dominated by analytic evaluation of the stored energy for spherical modes in the exterior region of a sphere circumscribing the radiator [17], [18], [20]. The radiated power (15) expressed in the power flux and the far field are identical for this case and the classical expressions can be extended to arbitrary shapes in several ways. Here, we consider radiated energy expressed as the: power flux in the radial direction, magnitude of the power flux, and far-field amplitude, see first three rows in Table I. 1) Subtraction of the radial power flowr · P : Collin and Rothschild [17] suggested identification of radiated energy with the power flux in the radial direction to define the stored energy as
W Pr = 1 4 R 3 ε 0 |E| 2 + µ 0 |H| 2 − 4 √ ε 0 µ 0r · P dV.(19)
They used this expression to evaluate the stored energy in the exterior of a sphere using mode expansions and produced explicit results on the Chu [21] lower bound, see also [18] for a time-domain extension. The expression (19) is non-negative and does not subtract energy for standing waves, e.g., in the interior of a sphere for spherical mode expansions [17], [20]. The main drawbacks of (19) are the coordinate dependence and the need for numerical integration for general fields, see [22], [23] for spheroidal geometries and [24] for an FDTD approach.
2) Subtraction of the magnitude of the power flow |P |: The problem with coordinate dependence in (19) can be resolved by subtraction of the magnitude of the power flow |P |, i.e.,
W P = 1 4 R 3 ε 0 |E| 2 + µ 0 |H| 2 − 4 √ ε 0 µ 0 |P | dV.(20)
This expression for the stored energy was originally proposed in an equivalent form by Counter [25]. The expression is identical to (19) for fields expressed as a single spherical mode [25]. It is non-negative and less than or equal to (19) for general fields with a power flow in non-radial directions. The main drawback with (20) is the numerical evaluation of the energy density over R 3 .
3) Subtraction of the far-field amplitude |F | 2 : The energy of the radial component of the power flow, subtracted in the previous method (19), can be expressed in the far-field amplitude |F | 2 outside a circumscribing sphere. This leads to the formulation [26], [27], [19], [20], [28], [29], [30]
W F = 1 4 R 3 ε 0 |E| 2 + µ 0 |H| 2 − 2ε 0 |F | 2 |r| 2 dV = 1 4 ∂X in ∂ω |I 0 | 2 − Im 2Z 0 S 2 ∂F ∂ω · F * dS(21)
for the stored energy, where S 2 denotes the unit sphere and the frequency derivatives are evaluated for a frequency independent input current I 0 . Here, all radiated energy is subtracted and the expression makes no difference between standing and radiating waves, e.g., in the interior of the smallest circumscribing sphere. Hence, the energy W F differs from W Pr by kaP rad for spherical modes and implies a difference of the Chu bound by ka, i.e., Q Chu − ka.
Variations of (21) exist in the literature and, e.g., Rhodes [26] suggested to use subtraction (21) only in the exterior region, keeping the total electromagnetic energy in the interior region. A shielded power supply is also often excluded from the integration in (21), [19]. This is equivalent to setting the E and H to zero in the region of the power supply. The stored energy W F in (21) can be rewritten using the frequency-differentiated input reactance X in and far field F for antennas with a fixed feeding current I 0 using a reactance theorem [20], [26], [19]. This form of the stored energy is shown in the far right of (21) and simplifies the numerical evaluation from a volume integral to a surface integral. Moreover, it shows that the energy W F is coordinate dependent for non-symmetric radiation patterns [19], [28]. The reactance theorem is extended to complex media in [19], [31]. The formula (21) is also rewritten in the current density in [28], see Section III-B2. Several methods exist for calculating the energy stored by a source current distribution J placed in vacuum, see Figure 3. These methods can be used to evaluate stored energy from any system (including materials, feeds, and ports) which can be represented by an equivalent current distribution J . A powerful feature of this approach is an immense reduction of information needed to evaluate stored energy. Commonly, only current densities on finite surfaces are needed. These methods are also well suited for various tasks in antenna design [32], since the feeding which leads to the current density J need not to be known. This makes it possible to determine fundamental performance bounds on antennas with given support [33], [32], [34], [35] or to utilize modal decomposition methods [36].
Similarly to field approaches, the methods discussed in this subsection identify radiation energy as unobservable energy. Their use for evaluation of (12) for lumped circuits will thus always count the entire electromagnetic energy W EM regardless of the complexity of the circuit. The formulation of the methods for general dispersive materials is not well studied except for the state-space method of moments (MoM) approach in Section III-B3. In the case of non-dispersive materials, electric polarization can be included in the current density J .
1) Differentiated MoM reactance matrix X : Harrington and Mautz [37] proposed to use frequency differentiation of the MoM reactance matrix
W X = 1 4 I H ∂X ∂ω I = 1 4 I H X I(22)
to estimate the stored energy. The reactance matrix is determined from the impedance matrix Z = R + jX derived from the MoM approximation of the electric field integral equation (EFIE) [38]. The expression (22) is not derived in [37], but is merely motivated by the analogous expression of Foster's reactance theorem for lossless systems [39], see also (30). The stored energy for lumped circuit networks can be determined with the formula (22) by substituting the MoM impedance matrix with the lumped circuit impedance matrix, see (5) and [4]. For currents in free space, the expression (22) is identical to the MoM state-space approach in Section III-B3 and the MoM approximation of the stored energy expressions by Vandenbosch [40]. Hence, it also suffers from the matrix X being indefinite for large structures and potentially producing negative values for the stored energy [15]. The expression (22) is easily applied to temporally dispersive materials but is inaccurate for many cases [41], cf. the state-space MoM approach in Section III-B3.
2) Reactive energy: The expressions in the frequency domain introduced by Vandenbosch [40] start from the same classical idea as described by Collin and Rothschild [17]: the subtraction of the radiated energy density from the total energy density. However, the subtracted term is defined in a slightly different way on the basis of an energy balance equation involving the derivatives of Maxwell's laws. The resulting difference is analytically integrated over all space, yielding closed-form expressions for the reactive energy (both the electric and magnetic part) in terms of the currents flowing on the radiator. The new definition thus eliminates the coordinate dependency, resulting in the expression
W reac = Z 0 4ωk Ω Ω k 2 J 1 · J * 2 + ∇ 1 · J 1 ∇ 2 · J * 2 cos(kr 12 ) 4πr 12 − k k 2 J 1 · J * 2 − ∇ 1 · J 1 ∇ 2 · J * 2 sin(kr 12 ) 4π dV 1 dV 2 .(23)
This expression was later found to conform [28] to the coordinate independent part of energy W F given by (21). The same expression is found also from a line of reasoning starting in time domain [42], [43]. The expression is positive semi-definite for circuits and small radiators but indefinite for larger structures [15]. This method essentially can be seen as a "transformation" of the original field based definition (21), acting on all space, into a current based interpretation, acting only within the volume of the radiator. The MoM approximation of (23) is identical to (22) for the free-space case and hence (23) offers a rigorous motivation for (22). The first term in (23) is also similar to the time-domain formulation using the product of sources and potentials proposed by Carpenter in [44]. Moreover, Geyi presented an approximation of (23) for small antennas in [45]. This small regime formulation was also addressed in [46], [47]. The formulation based on (23) is generalized to electric and magnetic current densities in [48], [49].
3) State-space MoM model X : The state-space method is based on the classical approach to define stored energy in a dynamic system, see (3). The stored energy for a radiating system is more complex as the dynamics are not described by the simple system in (3). In [50], a state-space model
ZĨ = jωµL 1 −1 jωεC I U = B 0 V in(24)
is derived from the MoM impedance matrix Z = jωµL+C i /(jωε), where U is the voltage state and V = BV in = ZI is the excitation. The stored energy is constructed by differentiation of the state-space reactance matrix X = Im{ Z} with respect to the frequency, cf. (4). The resulting stored energy is identical to the X -formulation in Section III-B1 for PEC structures in free space and suffers from the same problem of being indefinite for larger structures. The advantage of the state-space approach is that the quadratic forms for the stored energy are derived for small structures in temporally dispersive and inhomogeneous materials.
4) Subtraction of the radiated power in time domain:
The subtraction of unobservable energy (12) in the form of radiation can advantageously be applied in time domain [51]. In this paradigm the system is brought into a given state (for example time-harmonic steady state) during time t < t 0 and then its excitation is switched off. The system is then let to pass a subsequent transient state in which all its energy is lost via radiation and heat. With the time-dependent current density J (t) existing in the system, which has been recorded during the entire time course, the stored energy can be calculated as
W td (t 0 ) = ∞ t0 P heat (J ) + P rad (J ) − P rad (J freeze ) dt,(25)
where P heat and P rad are the power lost and power radiated corresponding to the lost and radiated energy W heat and W rad defined by (9), (10), with bounding surface S far located in the far field. The current density J freeze (t) is defined as the current density at time t = t 0 artificially frozen for times t > t 0 , i.e., J freeze (t > t 0 ) = J (t 0 ). Cycle-mean stored energy in time-harmonic case is achieved by moving time t 0 within one period and averaging. Note that although the power terms in (25) are evaluated for time t > t 0 , the time retardation demands knowledge of the current density also in preceding times. This subtraction technique closely follows the stored energy definition (12) and its more detailed exposition [51] also shows that the method gives non-negative stored energy, is coordinate independent, and can subtract the radiation energy inside the smallest circumscribing sphere. Its major disadvantage is numerically expensive evaluation.
C. Approaches using system, port, or feed ε 0 a µ 0 a Z 0 Z in System-level approaches evaluate energy storage directly from quantities available in the input/output ports of the system, see Figure 4. Grounded in thermodynamic principles, energy balance calculations of this kind preceded local approaches in mechanics, however, they are not commonly seen in the domain of electromagnetic stored energy evaluation. The oldest application of system-level energy quantification in electromagnetics uses circuit synthesis [21], [4] and is also tightly related to the concept of recoverable energy [13]. The generality of these approaches is unprecedented as they are applicable to arbitrarily complex electromagnetic systems. Unfortunately, this generality comes at the price of losing all physical interpretation of the unobservable energy content. Additionally, application of these techniques require systems with well defined input ports. This latter restriction makes these techniques inappropriate for evaluating the Q-factors of currents without a well-defined port, such as those encountered in modal decompositions and current optimization.
1) Brune circuit synthesis: Chu's classical antenna bound was originally derived using the stored energy in lumped inductors and capacitors of a circuit model for the spherical modes [21]. Thal has extended this approach to hollow spheres [52] and arbitrarily shaped radiators [53]. The stored energy for arbitrarily shaped antennas can analogously be estimated from equivalent circuit networks synthesized solely from the input impedance [54], where Brune synthesis [55], [5] is used. Alternative synthesis methods [5] can be used but it is essential that the synthesized circuit is a reciprocal minimal representation [3]. Non-reciprocal methods such as the minimum-phase Darlington synthesis [4], [56] can be used to estimate the recoverable energy in Section III-C2.
It is hypothesized [54] that the Brune circuit synthesis procedure produces a circuit with minimal stored energy from all reciprocal realizations, and thus best estimates the stored energy W sto . By definition, this means the procedure only includes the observable part of the stored energy. Note that this is zero for the Zöbel network in Box. 2. The formulation can be used for arbitrary antennas and material models, but its application requires approximation of the input impedance Z in (ω) as a positive-real function. This approximation is computationally difficult for electrically large antennas that require high-order rational functions.
2) Recoverable energy: The recoverable energy W rec (t 0 ) is defined as the maximum energy which can be extracted from a system which has been driven for times t < t 0 by a known set of sources [12], [13]. In the most general sense, calculating W rec (t 0 ) involves finding the optimal "recovery source" [13] as a function of time t > t 0 . This recovery signal implicitly depends on the sources applied at times t < t 0 and the locations where recovery is allowed to occur. The optimal recovery source extracts maximum energy from the system and equivalently minimizes energy lost by the system during recovery. When both driving and recovery sources are confined to a single port as they are in many antenna systems, the task of finding the optimal recovery source is greatly simplified [57]. Given a port impedance Z c and a system reflection coefficient Γ (ω), the recovery source (in the form of an incident voltage u + in (t)) is obtained by solving
F −1 1 Z c 1 − |Γ (ω)| 2 * u + in (t) = 0(26)
for times t > t 0 , where * denotes convolution and F −1 {·} denotes the inverse Fourier transform. Applying this recovery source to the antenna port, the recoverable energy is given by
W rec (t 0 ) = − ∞ t0 u in (t)i in (t) dt,(27)
where u in and i in are the total port voltage and current corresponding to the optimal time course u + in (t) from (26). For time-harmonic excitation prior to time t 0 , the cycle-mean recoverable energy can be calculated directly in closed-form from a rational function fit of the system's input impedance [57]. The process of approximating an antenna's input impedance as a rational function, however, suffers from the same problems as Brune synthesis for electrically large antennas. The formulation of energy W rec in terms of field quantities can be found in [13] and an overview of its physical properties and more detailed exposition can be found in [58]. A first generalization of the concept to more arbitrary excitations of radiators can be found in [59].
D. System-level metrics not directly derived from stored energy
Determining the stored energy in a system is largely motivated by its approximate inverse proportionality 2 to frequency selectivity of a single resonant system, which is most commonly described by its fractional bandwidth (FBW) or Q-factor. There are however methods which attempt to evaluate Q-factor without knowledge of stored electromagnetic energy. The most well known are the Q-factors Q Z derived from the frequency derivative of an input impedance and Q FBW derived directly from the fractional bandwidth of the system. Both of these methods belong to the system-based class of approaches and share those properties. For comparison purposes, both methods will be calculated alongside Q-factors derived from stored energy.
1) Fractional bandwidth: The Q-factor Q FBW is calculated directly from the fractional bandwidth B as [19]
Q FBW = 2Γ 0 1 − Γ 2 0 1 B Γ0 ,(28)
where Γ 0 denotes the level of the reflection coefficient |Γ | at which the fractional bandwidth (FBW) B Γ0 is evaluated. The relation assumes that the system is matched and tuned to resonance at the evaluation frequency, i.e., Γ (ω) = 0.
The most important merit of the Q-factor Q FBW is its exact proportionality to fractional bandwidth. The major drawback of this method is its inability to evaluate Q-factor from data at a single frequency and its dependence on the choice of parameter Γ 0 .
2) Differentiated input impedance: The Q-factor Q Z has been derived [19] from Q FBW in the limit where Γ 0 → 0 and it represents the differential fractional bandwidth of the system. Similarly to Q FBW , it assumes the system is matched and tuned to resonance. It is most commonly defined as [19]
Q Z = ω 2R in ∂Z in ∂ω = ω ∂Γ ∂ω .(29)
Alternatively, Q Z can be viewed as the classical Q-factor (1) derived from a local approximation of an input impedance by a single resonance (RLC) circuit [19], [62] for which relation Q Z = Q ≈ Q FBW holds. The advantage of Q Z over Q FBW is its much simpler evaluation and its independence of the parameter Γ 0 . However, the cost of this simplification is the loss of a direct relation to fractional bandwidth [19], the possibility of predicting Q Z = 0 [60], [61], and the problematic interpretation in cases of closely spaced resonances [63]. The Q-factor Q Z can also be written solely in terms of source current density [64], [41] which relates it to the Q-factor based on energies W F and W reac , see Section IV-A.
E. Other methods
The list of methods discussed above is not complete and we have intentionally selected those which follow the definition (12) and at the same time exhibit generality. In this subsection we briefly comment on those not explicitly treated.
First concept is that of employing angular field decomposition, identifying stored energy with the energy of the evanescent (invisible) part of the spectra [65], [66]. A similar concept was proposed in [67] to evaluate Q-factors of electrically small dipole radiators and in [68] to evaluate Q-factors of arrays. This spectral decomposition method is an interesting scheme which gives important insight into the subtraction of the radiation part of unobservable energy. Its most important drawback is its applicability solely to planar radiators. A generalization to general radiators has been proposed in [69], [70], but has not been tested.
The second concept, proposed by Kaiser [71], bears similarity to the time domain version of the method of Collin and Rothschild [18] and claims to be its relativistic generalization. The major difference from (20) is the use of squared instead of linear subtraction which was introduced as an analogy to relativistic energy-momentum relation [72], [71]. The merit of this concept is positive semi-definiteness, coordinate independence, and the capability to deliver a local stored energy density. In canonical cases it leads to stored energy values [73] very close to (20), but its testing in more general scenarios is not available.
The last presented concept is based on a fact that the stored energy in a lossless network can be determined by differentiation of the input reactance X in or susceptance B in [39] as
W X in = 1 4 I H in ∂X in ∂ω I in and W B in = 1 4 V H in ∂B in ∂ω V in ,(30)
respectively. This formula is related to the Foster's reactance theorem [74] where a positive energy implies a positive slope of the reactance. The input resistance of antennas is, however, non-zero and the approximation (30) is hence generally inadequate. This is also concluded from (21), as (30) neglects the far-field term in (21). Moreover, it is necessary to include the input resistance to accurately estimate the fractional bandwidth as shown by Q Z expression in (29). Although the expression (30) has the same form as the differentiated reactance matrices in Sections III-B1 and III-B3 there are substantial differences. It is sufficient to know only the input-output relation for the lossless system in (30) whereas (22) requires knowledge of the internal dynamics of the system.
IV. ANALYTIC AND NUMERICAL COMPARISONS
In this section, two classes of comparisons are made between the methods described in the preceding section. First, we study the analytic relation between some methods under certain specific conditions. Following that, numerical examples are presented where the Q-factor of driven antennas are calculated and compared.
A. Analytical comparison of various methods
When methods from Table I are applied to fields and currents generated by PEC structures operating in the quasi-static limit where radiation is negligible, the stored energy predicted by them reduces to the electro-and magnetostatic expressions, see Box 4. They however start to differ for electrically larger structures. Here, the methods are analytically compared for canonical cases such as spherical geometries, PEC structures, and singleresonance models.
Spherical modes have dominated evaluation of stored energy and Q-factors since the publication by Chu [21]. Collin and Rothschild [17], see Section III-A1, presented closed form expressions of the Q-factor and stored energy W Pr for a single radiating spherical mode outside a sphere with radius a. Comparing the definitions of the methods in Table I for this case reveals the identities
W Pr = W F + a c 0 P rad = W P = W Z B in ,(31)
where the difference with aP rad /c 0 (ka for the Q-factor) for the subtracted far-field expression W F originates from the subtraction of the radiated power inside of the sphere in (21) and the equality for the Brune circuit follows from the circuit model of the spherical modes [21]. Thal [52] analyzed the corresponding case with electric currents by inclusion of the stored energy in standing waves inside the sphere. This case is identical to (31) for the field-based methods but with an added connection to W reac , i.e.,
W Pr = W F + a c 0 P rad = W P = W reac + a c 0 P rad ,(32)
where the spherical mode expansion in [28] is used for W reac in (23). The identity (32) can be generalized to arbitrary electric current densities on the sphere with exception for W P . When stored energy W F given by (21) is written as a bilinear form of source current density [28], it relates to energy (23) as W F = W reac + W coord , where coordinate-dependent term W coord is given by [28,Eq. 26]. The coordinate dependent part vanishes in the important case of equiphase current densities, i.e., |I T I| = I H I, which appear as a result of characteristic mode decomposition [36], minimum Q-factor modes [35], and often approximately for small self-resonant antennas. The equiphase case is also related to differentiation of the input admittance (30) for a fixed voltage source [41] revealing the following connection between the field, current, and port based methods:
W F = W reac = |W B in | ≈ Q Z P rad ω ,(33)
where the final step is valid for self-resonant cases for which the change of reactance dominates over the resistance. The MoM discretized version of (23) for PEC structures is also identical to the differentiated reactance matrix (22) and the state-space MoM (24), i.e.,
W reac = W X = W X .(34)
This equality is used for the presented numerical results in Section IV-B, where the energy W reac is used to indicate all three methods in (34). Finally, the system methods agree for single-resonance RLC circuit networks
Q Z B in = Q rec = Q Z ≈ Q FBW ,(35)
where the subscripts used are the same as for corresponding energies. The above comparison suggests that the proposed methods agree for many cases. However, the identities are based on specific assumptions and discarding the imposed restrictions on the geometries, equiphase currents, and single resonance can produce very different estimates of the stored energy. As an example, we generalize the single mode case (32) to single electric dipole mode (TM 01 ) originating from electric currents at two spherical shells with radii a 1 and a 2 > a 1 . Let the inner current have amplitude J 1 and normalize the outer current amplitude with J 0 such that J 2 = J 0 cancels the radiation from the inner surface. This non-radiation current has no dissipated power and hence an infinite Q-factor. Lowering the amplitude to J 2 = 0.5J 0 increases total the radiation as only half of the radiated field is canceled. Figure 5a depicts the case ka 1 = 0.75 with J 2 = (0.5 + 0.1j)J 0 , where the small imaginary part is added to invalidate the equiphase identity (33). In the figure, we observe that Q F ≈ Q reac ≈ Q Z as expected from (33) as the current is approximately equiphase. The Q-factors from the subtracted power flow (19) and (20) are substantially lower than the other Q-factors around ka 2 ≈ 4 and ka 2 ≈ 8. This is contrary to the expectation from the single mode case (32) and can be explained by the power flow between the spherical shells that is not subtracted by the far field in (21). The effects on the Q-factors of an increased phase shift between the current is depicted in Figure 5b, where ka 1 = 0.3 and J 2 = jJ 0 is used. Here, all considered methods produce different results. These simple examples illustrate the challenges to define stored energy and that the challenge increases with the electrical size of the object and phase variation of the current.
B. Numerical comparison of various methods
Numerical results for different antenna types are presented in this section. The examples are: a center fed cylindrical dipole, an off-center fed cylindrical dipole, a strip folded dipole, and a Yagi-Uda antenna. The tuned Q-factor (17) is chosen as an appropriate measure to compare the different methods, as it is only a renormalization of the stored energy along with an addition of a known tuning energy, see Section II. This permits us to compare and contrast methods for evaluating the stored energy with the methods in Section III-D which only calculate the tuned Q-factor, such as Q Z and Q FBW . All example structures are modeled as PEC in free space and are each fed by a single delta-gap voltage source. In this case many of the methods described in Section III are formally equivalent, see Section IV-A. Hence, only one representative of each such group is presented here. Each method follows the notation introduced in Table I. The frequency axis of all plots is expressed in the dimensionless quantity ka, where a is the radius of the smallest sphere that circumscribes each antenna. The Q-factor Q FBW has been calculated at the level Γ 0 = 1/3 ≈ −10 dB in (28).
1) Center fed cylindrical dipole: Figure 6 depicts the Q-factors calculated by the methods discussed in Section III for a hollow cylindrical dipole. All the methods agree well for low ka values, which are typical dimensions for electrically small antennas. The methods start to diverge for electrically larger structures, when ka 1.5. It should be noted that the relative difference in Q-factor is very small, even for larger structures. The only major divergence is the Q-factor from the recoverable energy W rec which predicts significantly lower values than the other methods for ka > 3. This, however, is to be expected as the recoverable energy is the lower bound to the stored energy, see (13).
2) Off-center fed cylindrical dipole: The dipole examined here is identical to the center fed dipole in Section IV-B1 except that its feeding point is shifted by a distance l = 0.23L from the center. This gives rise to a phase shift which changes the stored energy and Q-factor. If we compare Figures 6 and 7 we see that the Q-factors fluctuate much more than observed in the center fed dipole. However, the Q-factors retain the same behavior with respect to each other as for the center fed dipole for most of the simulated interval. They predict essentially the same results for low values of ka and diverge slightly for ka > 1.5. However, around ka = 6.2 the Q-factor Q Z has a dip which is not mimicked by the other methods. The recoverable energy W rec predicts lower values of Q-factor than the other methods but seems to follow the behavior of the curves with smaller fluctuations. 3) Strip folded dipole: In Figure 8, Q-factors are depicted for a folded strip dipole. Due to computational complexity the subtraction of the power flow |P |, the energy W P has not been calculated for this example. With exception of recoverable energy, the depicted methods shown agree well for ka < 4, above this point the Q-factors Q Z and Q FBW start to diverge from the other methods. 4) Yagi-Uda: Figure 9 depicts Q-factors calculated for a Yagi-Uda antenna, again the subtraction of the power flow, |P | has not been calculated due to computational complexity. All methods presented agree well over the entire interval, excluding a small dip from Q-factor Q Z at ka = 1.8 and some small divergence at ka > 6. This can be explained by the off resonance behavior of the Yagi-Uda antenna. When the parasitic elements are no longer active, the antenna essentially behaves as a center-fed dipole. Because of this simple behavior the relative difference between the methods becomes very small.
V. APPLICATIONS
Stored energy for radiating systems was initially used by Chu [21] to derive his classical antenna bounds for spherical shapes. Bounds have continued to be a major driving force for research into stored energy as antenna designers are, naturally, interested in how good their antennas are and how far they are from the optima [75], [76], [77]. The Chu bound was originally derived with a circuit model for spherical modes III-C1, see also [52], [78], [53]. The model was reformulated in fields (19) by Collin and Rothschild [17] and subsequently refined in [20], [79], [19], see [76], [77], [75] for an overview. Formulations as optimization problems has generalized the classical bounds on the Q-factor to a multitude of problems formulated as combinations of stored energy, radiated fields, induced currents, and losses [33], [32], [34], [35]. Many problems are formulated as convex optimization problems [33], [32], [80], [81], [35] which are efficiently solved with standard algorithms. Here, it is essential that the quadratic forms for the stored energy are positive semidefinite, see Table I. Unfortunately, several presented methods are indefinite for electrically large structures. This restricts the problems to sub-wavelength structures where the expressions are positive semidefinite. Apart from convex optimization and considering mainly sub-wavelength radiators, other techniques like parameter sweeps [46], [47], polarizabilities [48], [82], [83], or modal decomposition [49], [84], [85], [34] can be applied to determine bounds.
Although stored energy has so far mainly been used to determine physical bounds, stored energy has great potential to be an important concept also for an antenna design. The results by Chu [21] showed that small antennas are dipole radiators and the explicit shape of the current distribution can give insight to design. Thal [52] showed how the stored energy in the interior of a sphere contributes [86], [87]. The importance of the polarizability and its associated charge separation was shown in [88], [83]. With the current-based formulations in Section III-B and optimization of the current distribution we get suggestions for optimal currents for many antenna parameters [15], [33], [32], [34], [35].
Another direction from which the problem of minimization of Q-factor was attacked is characteristic mode theory [89] as it provides favorable separation of reactive stored energy (23), constituting thus modal Q-factors for arbitrary bodies [36]. Mixing rules similar to those used with spherical modes can be applied, leading to approximative, but straightforward rules for fundamental bounds on Q-factor of arbitrarily-shaped radiators. Stored energy expressions are also used to construct new type of modes with properties differing from those of characteristic modes. Energy modes formed from eigenvalue problems involving matrix X in (22) were introduced in [37]. These types of modes are also useful to determine and interpret the physical bounds discussed above [85], [32]. Moreover, as these modes are real-valued many of the proposed expressions for stored energy agree (33) and the resulting Q-factor is also a good estimate of the fractional bandwidth for single mode antennas.
Stored energy can also be used to simplify some antenna optimization by replacing simulations over a bandwidth with a single frequency calculation of the Q-factor [90]. This single frequency optimization increases the computational efficiency but is restricted to narrow band cases. A typical representative of an application which can enjoy this approach is a design and optimization of Radio Frequency Identification (RFID) tags with minimal mutual coupling [91], [92].
VI. SUMMARY
A definition of stored energy in a general electromagnetic system was proposed and discussed using the concept of unobservable energy. Various aspects of subtracting the unobservable energy have been pointed out in the examples of Zöbel's network, matched transmission lines, and, most importantly, radiating structures. It has been shown that a majority of the well-established concepts for evaluating stored energy in radiating systems can be categorized into three different groups -whether they used field quantities, source currents, or rely solely on knowledge in system as a whole without possibility to probe its internal structure. An important outcome of this paper is understanding that all existent concepts, in fact attempt do define unobservable energy. Nevertheless, the common association of unobservable energy purely with radiated energy is insufficient. By the proposed definition, the unobservable energy represents the difference between the total electromagnetic energy W EM and the stored energy W sto so that it contains the energy of all unobservable states.
Careful analysis of the presented results revealed good agreement between all evaluated methods for equiphased currents and electrically small (ka < 1.5) antenna structures, though simple analytically-constructed examples and larger objects revealed significant disagreements. The systematic difference between recoverable energy W rec and stored energy W sto is due to reciprocity of the resulting realizations. While the recoverable energy allows for nonreciprocal circuits, the stored energy approaches, as illustrated by Brune synthesis, deal with reciprocal systems only. Taking Q FBW as reference measure of fractional bandwidth, it is obvious that the Q-factor resulting from recoverable energy considerably overestimates the fractional bandwidth. The other presented methods have much better agreement with fractional bandwidth. However, from this point of view, the best predictor of bandwidth potential is Q-factor Q Z , but only when the system under study can be approximated as a single resonance system.
For practical aspects of stored energy evaluation, the method evaluating energy W reac or, alternatively, energy W X , gives precise approximation of stored energy for electrically small structures, offers simple implementation, and, in addition, is fully compatible with present approaches to minimization of Q-factor like convex optimization and pixeling. Whenever negative values of stored energy could be an issue, an alternative method, possibly Brune synthesis, is recommended since the breaking point at which stored energy W reac fails is not exactly known. As confirmed by all treated examples, Brune synthesis is capable of distilling the maximum amount of unobservable energy from the total energy, thus surpassing other contemporary approaches. However, complications in performing Brune synthesis for electrically large antennas may be an obstacle limiting its application.
Though many researchers have contributed to the study of stored energy with corresponding indisputable achievements, several fundamental questions remain open. The missing proof of the minimal reciprocal realizations generated by Brune synthesis as well as closely related reformulation of this circuit synthesis in terms of the electromagnetic quantities, may open the final stage to explicit, coherent, and exact definition and evaluation of unobservable energy. Additionally, further work is needed on the calculation, verification, and interpretation of stored energy in general dispersive media.
APPENDIX A STORED ENERGY IN DISPERSIVE MEDIA
The definition in Section II covers antennas in a non-dispersive background. Consider instead a radiator embedded in an isotropic dielectric material described by a Lorentz dispersion model
∂ 2 P ∂t 2 + Γ ∂P ∂t + ω 2 r P = ε 0 ω 2 p E,(36)
where P is the polarization, Γ is the loss factor, ω r is the resonance frequency of the material, and ω p is the coupling constant [10]. If we divide the energy analogously to (7), the material properties influence the heat and total energy terms [11]. The new heat term reads
W heat (t 0 ) = t0 −∞ V σ |E| 2 + Γ ε 0 ω 2 p ∂P ∂t 2 dV dt,(37)
and the total energy read
W EM (t 0 ) = 1 2 V ε 0 |E| 2 + µ 0 |H| 2 + 1 ε 0 ω 2 p ∂P ∂t 2 + ω 2 r |P| 2 dV.(38)
The stored energy definition (12) still applies, but the dispersion generally rise the energy of unobservable states. The subtraction of unobservable energy states becomes especially problematic in dispersive background since in a such case far field is no longer well defined and many classical methods break down. System based methods, see Table I, and engineering metrics Q Z and Q FBW are unaffected, in principle, but, in certain cases, they are more likely to predict unphysical results, see [41].
Fig. 2 :
2Sketch of electric field intensity E generated by dominant TM 10 spherical mode.
Fig. 3 :
3Illustration of surface current of dominant TM 10 mode on a spherical shell Ω.
Fig. 4 :
4Synthetized circuit for dominant TM 10 mode of a spherical shell with radius a[21].
Fig. 5 :
5Q-factors for concentric spherical current shells radiating the spherical TM 01 mode: a) ka 1 = 0.75 and J 2 = (0.5 + 0.1j)J 0 , b) ka 1 = 0.3 and J 2 = jJ 0 .
Fig. 6 :
6Q-factors of a hollow cylindrical dipole of length L and radius r = L/200, fed at its center. The gray solid and dashed vertical lines denote resonance and anti-resonances of the antenna.
Fig. 7 :
7Q-factors for a hollow cylindrical dipole of length L and radius r = L/200, with an off-center feed l = 0.23L from the center. The gray solid and dashed vertical lines denote resonance and anti-resonances of the antenna.
Fig. 8 :
8Q-factors for a folded strip dipole of circumscribing dimensions L × L/2, with strip width L/200. The gray solid and dashed vertical lines denote resonance and anti-resonances of the antenna.
Fig. 9 :
9Q-factors for a Yagi-Uda antenna specified in the upper right corner of the figure. All the dimensions of the Yagi-Uda antenna are normalized to the center dipole length L. The elements have been modeled as strips of width L/200. The gray solid and dashed vertical lines denote resonance and anti-resonances of the antenna.
TABLE I :
IMethods for evaluating stored energy. Rows are grouped by the data required for its evaluation, i.e.,
methods derived from fields (blue), source distributions (green), and systems (red). The final two uncolored methods
are metrics not generally related to stored energy which are used for comparison purposes.
Here we make an assumption that electric and magnetic fields are temporarily bandlimited and thus the radiation zone can be defined in a usual manner by the dominance of the 1/r field components.
Often, this inverse proportionality is taken for granted. It is, however, important to stress that a strict functional relation of Q-factor based on stored energy and fractional bandwidth does not exist[60], and the discrepancy from the inverse proportionality can in specific cases be enormous[61]. On the other hand, in many cases, including practically all electrically small radiators, the inverse proportionality is almost exact.
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respectively. Currently, he is a postdoctoral research fellow at North Carolina State University. His research interests include electromagnetic theory. 2011, and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois at. Portland, OR, USA; Urbana-Champaign, Champaign, IL, USAKurt Schab received the B.S. degree in electrical engineering and physics from Portland State Universityoptimized antenna design, and numerical methods in electromagneticsKurt Schab received the B.S. degree in electrical engineering and physics from Portland State University, Portland, OR, USA, in 2011, and the M.S. and Ph.D. degrees in electrical engineering from the University of Illinois at Urbana-Champaign, Champaign, IL, USA, in 2013 and 2016, respectively. Currently, he is a postdoctoral research fellow at North Carolina State University. His research interests include electromagnetic theory, optimized antenna design, and numerical methods in electromagnetics.
he was appointed Associate Professor at the Department of Electromagnetic Field at the same university. His research interests include wave propagation in complex media. Lukas Jelinek received his Ph.D. degree from the Czech Technical University in Prague, Czech Republicgeneral field theory, computational electromagnetics and optimizationLukas Jelinek received his Ph.D. degree from the Czech Technical University in Prague, Czech Republic, in 2006. In 2015 he was appointed Associate Professor at the Department of Electromagnetic Field at the same university. His research interests include wave propagation in complex media, general field theory, computational electromagnetics and optimization.
he was appointed Associate Professor at the Department of Electromagnetic Field at the same University. He leads the development of the AToM (Antenna Toolbox for Matlab) package. His research interests are in the area of electromagnetic theory, electrically small antennas, numerical techniques, fractal geometry and optimization. He authored or co-authored over 65 journal and conference papers. He is member of Radioengineering Society. Miloslav Capek (S'09, M'14) received his Ph.D. degree from the Czech Technical University in. Prague, Czech Republicregional delegate of EurAAP, and Associate Editor of RadioengineeringMiloslav Capek (S'09, M'14) received his Ph.D. degree from the Czech Technical University in Prague, Czech Republic, in 2014. In 2017 he was appointed Associate Professor at the Department of Electromagnetic Field at the same University. He leads the development of the AToM (Antenna Toolbox for Matlab) package. His research interests are in the area of electromagnetic theory, electrically small antennas, numerical techniques, fractal geometry and optimization. He authored or co-authored over 65 journal and conference papers. He is member of Radioengineering Society, regional delegate of EurAAP, and Associate Editor of Radioengineering.
he participated in and won the IEEE Antennas and Propagation Society Student Design Contest for his body area network antenna design. His research interests include antenna theory. SwedenCasimir Ehrenborg received his M.Sc. degree in engineering physics from Lund University ; Department of Electrical and Information Technology, Lund UniversityHe is currently a Ph.D. student in the Electromagnetic Theory Group. phase and radiation centers, as well as physical boundsCasimir Ehrenborg received his M.Sc. degree in engineering physics from Lund University, Sweden, in 2014. He is currently a Ph.D. student in the Electromagnetic Theory Group, Department of Electrical and Information Technology, Lund University. In 2015, he participated in and won the IEEE Antennas and Propagation Society Student Design Contest for his body area network antenna design. His research interests include antenna theory, phase and radiation centers, as well as physical bounds.
respectively. He is currently a Ph.D. student at Electromagnetic Theory Group, Department of Electrical and Information Technology at Lund University. His research interests are Physical Bounds, Small Antennas and Computational Electromagnetics. Doruk Tayli received his B.Sc. degree in Electronics Engineering from Istanbul Technical University and his M.Sc. in degree in Communications Systems from Lund UniversityDoruk Tayli received his B.Sc. degree in Electronics Engineering from Istanbul Technical University and his M.Sc. in degree in Commu- nications Systems from Lund University, in 2010 and 2013, respectively. He is currently a Ph.D. student at Electromagnetic Theory Group, Department of Electrical and Information Technology at Lund University. His research interests are Physical Bounds, Small Antennas and Computational Electromagnetics.
His interests are in the area of electromagnetic theory, computational electromagnetics, planar antennas and circuits, nano-electromagnetics, EM radiation, EMC, and bio-electromagnetics. His work has been published in ca. 265 papers in peer reviewed international journals and has led to ca. 365 presentations at international conferences. He is a former chair of the IEEE AP/MTT Benelux Chapter and currently, he leads the Working Group on Software within. A E Guy, EuRAAP. Dr. Vandenbosch is a Full Professor at KU Leuven. Vandenbosch is a fellow of the IEEE sinceGuy A. E. Vandenbosch is a Full Professor at KU Leuven, Belgium. His interests are in the area of electromagnetic theory, computational electromagnetics, planar antennas and circuits, nano-electromagnetics, EM radiation, EMC, and bio-electromagnetics. His work has been published in ca. 265 papers in peer reviewed international journals and has led to ca. 365 presentations at international conferences. He is a former chair of the IEEE AP/MTT Benelux Chapter and currently, he leads the Working Group on Software within EuRAAP. Dr. Vandenbosch is a fellow of the IEEE since January 2013.
He co-founded the company Phase holographic imaging AB in 2004. His research interests are in scattering and antenna theory and inverse scattering and imaging. He has written over 80 peer reviewed journal papers and over 100 conference papers. Prof. Gustafsson received the IEEE Schelkunoff Transactions Prize Paper Award 2010 and Best Paper Awards at EuCAP. Mats Gustafsson received the M.Sc. degree in Engineering Physics 1994, the Ph.D. degree in Electromagnetic Theory 2000, was appointed Docent 2005, and Professor of Electromagnetic Theory. Swedenall from Lund UniversityHe served as an IEEE AP-S Distinguished Lecturer for 2013-15Mats Gustafsson received the M.Sc. degree in Engineering Physics 1994, the Ph.D. degree in Electromagnetic Theory 2000, was appointed Docent 2005, and Professor of Electromagnetic Theory 2011, all from Lund University, Sweden. He co-founded the company Phase holographic imaging AB in 2004. His research interests are in scattering and antenna theory and inverse scattering and imaging. He has written over 80 peer reviewed journal papers and over 100 conference papers. Prof. Gustafsson received the IEEE Schelkunoff Transactions Prize Paper Award 2010 and Best Paper Awards at EuCAP 2007 and 2013. He served as an IEEE AP-S Distinguished Lecturer for 2013-15.
. Doruk L To R: Mats Gustafsson, Kurt Tayli, Lukas Schab, Jelinek, Miloslav Capek, Casimir Ehrenborg, Guy VandenboschL to R: Mats Gustafsson, Doruk Tayli, Kurt Schab, Lukas Jelinek, Miloslav Capek, Casimir Ehrenborg, Guy Vandenbosch
| {'fraction_non_alphanumeric': 0.052112972916008006, 'fraction_numerical': 0.028201074928865002, 'mean_word_length': 4.390615236039311, 'pattern_counts': {'":': 0, '<': 5, '<?xml version=': 0, '>': 9, 'https://': 1, 'lorem ipsum': 0, 'www.': 1, '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': 'Though commonly used to calculate Q-factor and fractional bandwidth, the energy stored by radiating systems (antennas) is a subtle and challenging concept that has perplexed researchers for over half a century. Here, the obstacles in defining and calculating stored energy in general electromagnetic systems are presented from first principles as well as using demonstrative examples from electrostatics, circuits, and radiating systems. Along the way, the concept of unobservable energy is introduced to formalize such challenges. Existing methods of defining stored energy in radiating systems are then reviewed in a framework based on technical commonalities rather than chronological order. Equivalences between some methods under common assumptions are highlighted, along with the strengths, weaknesses, and unique applications of certain techniques. Numerical examples are provided to compare the relative margin between methods on several radiating structures.', 'arxivid': '1705.07942', 'author': ['Kurt Schab ', 'Lukas Jelinek ', 'Miloslav Capek ', 'Casimir Ehrenborg ', 'Doruk Tayli ', 'Guy A E Vandenbosch ', 'Mats Gustafsson '], 'authoraffiliation': [], 'corpusid': 2716391, 'doi': '10.1109/access.2018.2807922', 'github_urls': [], 'n_tokens_mistral': 26837, 'n_tokens_neox': 23080, 'n_words': 15147, 'pdfsha': '8cf7eefddcbe735feb9bd5e4425c87ef7764cc43', 'pdfurls': ['https://export.arxiv.org/pdf/1705.07942v2.pdf'], 'title': ['Energy Stored by Radiating Systems', 'Energy Stored by Radiating Systems'], 'venue': []} |
arxiv |
Quantum computing online workshops and hackathon for Spanish speakers: A case study
Alberto Maldonado-Romo
Centro de Investigación en Computación
Department of Computer Science
Instituto Politécnico Nacional Mexico City
Mexico
Lia Yeh
University of Oxford Oxford
UK
Quantum computing online workshops and hackathon for Spanish speakers: A case study
Index Terms-quantum educationquantum computingwork- shophackathonSpanishLatin America
We discuss the challenges and findings of organizing an online event in Spanish, consisting of a series of introductory workshops leading up to a quantum hackathon for Latin America. 220 Spanish speakers were registered, 66% of whom self-identified as being at an introductory level of quantum computing. We gain a better picture of the impact of quantum computing in Latin America, and the importance of generating educational resources in Spanish about quantum computing. Additionally, we report results on surveying the participants by country; educational status; self-reported levels of quantum computing, linear algebra, and Python competency; and their areas of interest within quantum.This event was organized by Quantum Universal Education with the Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN) as the host institution, in collaboration with a number of organizations and companies: IBM Quantum, Xanadu, Multiverse Computing, Quantum Universal Education, Quantum Hispano, QMexico, Haq.ai, Dive in Learning. This was part of a larger event, the Qiskit Fall Fest 2021, as one of several hackathons organized around the world in a similar span of time. In each Qiskit Fall Fest hackathon, participants were challenged to form teams of up to 5, to develop in 5 days a project using the IBM Qiskit framework.
I. INTRODUCTION
Accompanying the rise in applications of quantum technology, there has been growing interest in quantum education and workforce development initiatives [1]. These beginning and/or grassroots efforts have the potential to be very impactful, particularly in this formative period, where traditional educational offerings have yet to catch up to train the skills demanded for in these new industries [2]. As with any emerging technical subject, most students and professionals across all ages and stages of learning who would like to begin to learn it, will not find a course offering at their institution to gain exposure to the subject. This makes it important that there be informal educational opportunities open to those who would like to learn, and welcoming to those new to the topic, especially as one must first be exposed to the topic in order to then become interested in learning it. Much progress has been made on addressing these needs in recent years, from online extracurricular courses for middle and high school students [3] to games and software tools designed to teach quantum science concepts [4]. This report serves to introduce the approach of hackathons as a gateway for quantum computing learning and outreach.
Hackathons are events where participants form teams of one to up to around five participants, who compete to create a functioning project from start to finish, over the course of the event, which can range from one day to a few weeks in duration. Hackathons are a widely adopted approach to facilitating learning and collaboration in a short but focused time period, with hundreds of hackathons taking place each year. Definitions of hackathons, or "hacking marathons," may vary greatly depending on whether the intended purpose(s) is innovation, collaboration, competition, business solutions, software prototyping, education, outreach, research, or fun [4], [5].
Despite the growing number of quantum hackathons worldwide, there is limited documentation and evaluation of their organization and effectiveness. A particular subcategory of quantum hackathons has been studied for educational purposes: quantum games hackathons, and their more relaxed and less competitive counterpart, quantum game jams. Ref. [4] presented a comprehensive overview of quantum gamesgames with quantum elements in their game mechanics -for education and outreach, with a section describing an online Quantum Games Hackathon. Ref. [6] evaluated each quantum game created in the course of five quantum game jams on criteria such as playability and educational value.
II. A SPANISH-LANGUAGE VIRTUAL HACKATHON FOR
LATIN AMERICA As an emerging area worldwide, quantum computing has seen recent growth in the educational materials accessible in a variety of forms, including but not limited to: formal courses, textbooks, self-paced online courses, video series, programming tutorials, workshops, games, and comics. However, all these learning resources and opportunities are predominantly restricted to the English language. For one to become interested in quantum and feel a sense of belonging, quantum education needs to recognize, and strive to attune to, each and every diversity of identity and background. In order to ©2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other works. drive equitable and inclusive education in quantum science and technology, the challenges of language, geographical, and socioeconomic barriers must be addressed.
In Mexico, the Cuarantena Cuantica (literal translation: Quarantine Quantum) quantum computing seminar [7] was held virtually in January 2021 by the Sociedad Científica Juvenil (SCJ), i.e. the Juvenile Scientific Society of Mexico. It was followed by the Qiskit Summer Jam 2021 Mexico quantum hackathon [8] held in August 2021 as a collaboration between SCJ and QMexico, a QCousins branch of the QWorld non-profit organization. These and the Quantum Latino 2021 event [9] were the direct antecedents of the workshop series and hackathon described in this report, as quantum computing events respectively centered on Mexico and Latin America.
With the importance of supporting more accessible pathways to exposure to quantum concepts in a collaborative and welcoming environment in mind, this event was organized with beginner friendliness as the key priority. The goal was to encourage participation at all levels, especially for those for whom this was their first quantum event or even exposure to technical quantum concepts and quantum programming. For this reason, we put together a series of workshops in Spanish and English in the week preceding the hackathon. By design, the first workshop, offered once in Spanish and once in English by the Quantum Universal Education not-forprofit organization, was Introduction to Quantum Computing; it began by explaining what a qubit is logically and what it can look like physically, with the quantum circuits introduced alongside small illustrations of colorful cats, accompanied by a live code demonstration in the quantum programming language Qiskit. As we conclude in the section on our participant survey findings, this decision of focusing on introductory workshops was crucial to the objective of increasing awareness of quantum computing in Latin America.
With these goals in mind, we proceed to describe the organizational structure, dissemination of event sign-up information, and other considerations for the planning of the event, in hopes of informing future Spanish language and/or hackathon quantum educational events.
A. Overview of the event organization
The hackathon was part of IBM Quantum Education's Qiskit Fall Fest 2021, which coordinated the initiative to encourage and support 18 hackathons organized in roughly the same time frame around the world. IBM Quantum Education provided guidance in planning a hackathon in the form of courses and a hackathon guide [10]. These courses supported the planning of this hackathon to design it online, and to make it open to all Spanish speakers. Topics covered included the dissemination of information, the tools to broadcast, and platforms to manage and link people together whether that be in-person or online. Some tools were Hype Innovation Management Software, a web platform used to manage, for each hackathon: team formation, project ideation, and project submission; and Discord, a community and messaging social media platform used by each hackathon for schedule, announcements, networking, mentoring, memes, and other communication. Considering the locations of countries in Latin America, to create a schedule that the most people across these time zones can realistically participate in, we used the UTC-6 time zone.
The event, while being run online, was hosted thanks to the Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN). For reasons due to the pandemic situation, the in-person conditions were not suitable. With regards to the CIC-IPN organization, the event was part of the CORE International Congress 2022, an international congress focused on computer science, organized by students from the Centro de Investigación en Computación (CIC) in Mexico City [11]. In conjunction with the Instituto Politécnico Nacional (IPN), this facilitated extending the scope of participation to different institutions in the city, country, and different countries.
To reach Spanish speakers online, information on the event and how to register were disseminated through a number of online channels, as a collaboration between various organizations and communities each with hundreds or thousands of people involved. Communities of special mention for the unique roles each played in supporting this event include Quantum Universal Education, Quantum Hispano, the QMexico chapter of the QWorld non-profit, and the Qiskit Advocates program of IBM Quantum Community. The social networks and messaging platforms utilized included Facebook, Twitter, LinkedIn, Discord, Whatsapp, Slack and others, with a majority of registrants having heard of the event through Facebook (see Figure 1). The announcement campaign was conducted in both English and Spanish, using phrases such as introduction to quantum computing; programming quantum algorithms; quantum computing hackathon; and create and learn your first quantum computing project. For there to be an expectation as to the prerequisite knowledge recommended, the suggested skills to have an introductory grasp of were: how to program in Python, to know linear algebra, and to know probability; to know about quantum mechanics or quantum computation was optional, given the purpose of the workshops.
The event was scheduled such that the workshops took place across six consecutive days. The last day of workshops focused on industry, where presenters from Multiverse Computing and Xanadu offered workshops by Spanish speakers where they indicated the aspects and requirements of a job in a quantum computing company, and different tips for starting out in this area. The week of workshops was followed by the hackathon, which took place across five consecutive days. Over the course of the hackathon, people new to quantum would have ample time to get to know each other and get started in the creation of their first quantum computing project with virtual sessions, and not feel rushed to ask general questions and concerns about quantum computing or about their projects.
B. During the Event
The event consisted of two parts in order to involve people who had no knowledge of quantum computing, but had some general knowledge of Python programming, linear algebra, and optionally general topics of quantum physics. For the event, a server was created on the Discord social community platform, in which only registered participants had access to the join link. After entering the Discord server, participants could select the option to attend the workshops, where they were then automatically provided access to the channels where the links, resources, and question and answer sessions regarding each workshop were posted. Likewise, participants could select the option to attend the hackathon, to access channels where they can introduce themselves, describe their areas of interest, form teams, and discuss possible projects and ideas. For each team to come up with their project proposal, they are given a channel for their team to work, visible to just their team members and to the hackathon volunteers and mentors to respond to any questions and concerns.
For all participants, whether they opted to take part in the workshops, hackathon, or both (See Figure 2 for these choices at the time of registration), there were general channels for: announcements about quantum computing news, sharing how to get started with tutorials and applications, suggesting project ideas, and memeing. These were used as spaces to motivate and support people to get comfortable, challenge themselves, learn from and with each other, and let their imagination take off. They were encouraged to ask or inquire more about topics in quantum, and to take a look at the linked open educational resources accessible to them geared towards those new to quantum concepts, for example from IBM Quantum Education and Quantum Universal Education.
C. Series of Workshops
The series of workshops was designed to introduce at a beginner level different applications of quantum computing, such as optimization, chemistry, algorithms, machine learning, and video games. These are areas focused on in previous Qiskit hackathons, as areas for which there are a variety of applications and more accessible learning materials. In addition to technical workshops, a panel was added to shed light on the landscape of quantum computing in Latin America, and opportunities for work and perspectives shared by Spanishspeaking people who are in the industry. The purpose of these workshops is to introduce the state of the art in quantum computing to Spanish speaking audiences, for which the original materials have to be adapted to Spanish and adapted to an introductory level. For this we considered the criterion of inviting people who meet one of three requirements: 1) be a Qiskit advocate with a particular interest in an area of quantum computing; 2) be someone who works in the quantum computing industry being people from for instance IBM Quantum, Xanadu and Multiverse Computing; or 3) have completed an internship or mentorship in one of the quantum computing communities such as QWorld. Each of the selected workshop speakers facilitated a workshop in their area of expertise. For accessibility of the learning in general, the workshops were coordinated to be introductory and limit the need for prerequisite knowledge of quantum where possible. Moreover, the more introductory topics were scheduled earlier in the week, and the programming examples were encouraged to use the Qiskit framework to ensure continuity of understanding between workshops. The series of workshops and their descriptions can be seen in Table 1.
D. Hackathon
In organizing a hackathon, a number of logistical components are involved. In this section, we describe the aspects of team formation and project ideation; mentors; and project evaluation and prizes. Additionally, we provide brief descriptions of and links to the winning projects. Companies and governments are investing in its research and development since its use represents the answer to tasks that involve excessively complex calculations in artificial intelligence, chemistry, cryptography, among others. Actors such as IBM and Microsoft are looking to train developers who can program their quantum computers. To this end, they make their platforms freely available to us. The challenge now is to know the resources, materials, and events that one can develop all their skills. This presentation will address the opportunity to have a vision of this large and complex world.
Estado actual y perspectiva de los juegos cuanticos Language: Spanish Workshop presenter: Anamaría García Hernández Description: The field of quantum games is developing rapidly. In this project we collect information about quantum games at present and classify them into different categories. The results obtained will be discussed, as well as future projections.
Creacion de oracles para algoritmos Language: Spanish Workshop presenter: Emilio Pelaez Description: In this workshop, we will explore the construction of oracles for different algorithms. Giving examples on concrete algorithms such as Grover's search algorithm and the 3-SAT problem, and defining the concept of an oracle formally and how we can translate it into a circuit efficiently.
Calculando observables físicos con VQE Language: Spanish Workshop presenter: Siddhartha Morales In this workshop, we will explore how to use variational algorithms to solve some interesting physical problems, such as the ground state of a molecule and of an atomic nucleus. We will see how to use the quantum variational algorithm, as well as how to create our own ansatz, test different optimizers and how to send the work to a real quantum computer.
Algoritmo BB84
Language: Spanish Workshop presenter: Luis Martínez Description: In this workshop, we will see a brief introduction to the area of quantum cryptography. With special emphasis on the BB84 key exchange algorithm. We will also review some primitives that are important in this field. For this workshop it is not necessary to know classical cryptography as the basic concepts will be reviewed in the workshop.
Knapsack Problem
Language: Spanish Workshop presenter: Claudia Zendejas Morales Having quantitative information to make decisions leads to more and better profits. Studying the knapsack problem (KP) allows us to find solutions to combinatorial optimization problems by modelling a situation analogous to filling a knapsack. Its applications range from transportation and logistics problems to financial investments. In this workshop, we will see how to solve the knapsack problem with quantum computing.
Criptografía murió RSA? Language: Spanish Workshop presenter: Daniel Sierra-Sosa Description: One of the areas of interest in Quantum Computing is information security. It involves the fundamental principles and techniques of quantum computing, notions of information theory, algorithms (Grover's search and Shor's factorization), and applications of Quantum Computing such as quantum encryption and key distribution. This workshop will aim to explore and discuss real-world scenarios related to information security in times of quantum technologies, participants will understand the opportunities and challenges in this area and will have a hands-on experience on the IBM Q Experience platform.
Quantum Game Development at an introductory level Language: English Workshop presenter: Wen-Sen Lu Description: It is our privilege to explore the cutting-edge quantum computational space during the NISQ era with QISKit. Looking back into the history, especially in the 1970's, arcade game developers already started the machine-level programming and prepared themselves as the future coders even if the hardware was still limited. In the meanwhile, game-driven breakthrough for the classical hardware, such as the first 3D acceleration chip Super FX in Nintendo super-NES home console, also demonstrated the possibilities where new hardware could be inspired by the game developers. In this talk, I will start with my personal experience to quickly walk us through the process of quantum game development: looking for ideas from existing games, selecting development tools, and putting together two example codes in PICO-8 (Lua) to quickly demonstrate the classical and quantum counterpart of the game dev, respectively.
Haq.ai
Language: English Workshop presenter: Adam Fattal Description: haq.ai is a platform oriented for everyone interested in the field of quantum computing that wants to sharpen their quantum programming skills. Through a wide collection of problems in many topics and with different difficulties, users can develop their abilities. They can learn to efficiently decompose quantum circuits through methods presented in literature, harness the power of numerical computation in the field of quantum information, explore fun problems that require using popular algorithms and protocols, and much more. We aim to make the journey through quantum computation more interactive, while we don't offer a whole educational component, we offer a great supplement to a conventional quantum computing education.
Introducción al Aprendizaje de Máquina Cuántico
Language: Spanish Workshop presenter: Alberto Maldonado Romo Description: In this workshop we will see a small introduction of how to pass classical information to qubits to be able to treat them and get to general models such as neural networks in their quantum version.
Situación de la computación cuántica en América Látina
Language: Spanish Workshop presenters: Jazmin Esteva, Bruno Ramírez, Dr. Javier Orduz Description: Quantum computing has had a great impact on the world and many companies have focused on this area, but what is happening in Latin America?
Introducción al QAOA Language: Spanish Workshop presenter: Victor Onofre Description: QAOA is one of the hybrid quantum-classical algorithms that have been proposed to take full advantage of current quantum resources. In this workshop, we will explain its application to the MAX-CUT problem.
Quantum Enhanced Monte Carlo Simulations
Language: Spanish Workshop presenter: Cristina Sanz Fernández Description: Case study of how quantum computing can be useful to us today. Monte Carlo simulations are a widely used tool both in research (physics, chemistry, etc.) and in practical applications in our day-today life (finance, meteorology, telecommunications, etc.). In this talk, I specifically explain how quantum leads to a quadratic improvement of Monte Carlo calculations.
Carreras en la computación cuántica Xanadu
Language: Spanish Workshop presenter: Catalina Albornoz Description: In this talk, we will discuss the different career opportunities in the field of quantum computing, and what skills can lead you to that dream job. specific channels for participants to search for teammates, with teams limited to 2 to 5 members. 48 hours before the hackathon, during team formation, mentors were involved in proposing and bouncing off ideas with the participants. This process included understanding, for each participant, what workshop they liked the most, what geographical location they are from, what skills they have and would like to learn, and their past project experiences. Suggested project ideas and feedback on proposed ideas were made to each participating team, especially in conversation with those who had doubts as to where to start with proposing their project. • Mentors: The mentors were selected from the Qiskit advocate program, most of them being the same people who gave workshops, or people who could speak or write in Spanish to provide one-on-one support to the teams. • Project evaluation and prizes: The hackathon judges evaluated the projects, consisting of the project files and a short recorded talk explaining the motivation and demoing the project, according to the following four criteria, each of equal weight.
-Technical Challenge: Did the team challenge themselves and try to learn and implement something new to them or to the area? -Impact: Does the project have a high potential for impact? E.g. industrial application, educational value, theoretical interest, etc. -Creativity: Does the project go beyond the scope of a typical hackathon project? Unlike the technical challenge criterion, the creativity criterion incentivizes imagination in forms beyond technical (eg. artistic, originality, user friendliness, etc.). -Presentation: Is the project functioning and thorough? Is the presentation of the project thoughtfully explained and easy to understand? There are some observed similarities between the above four hackathon judging criteria, and those of the quantum games hackathon by the Quantum AI Foundation of 1) Correctness, 2) Playability, 3) Originality, and 4) Quality and complete-ness [4].
To improve the learning experience, each team was asked which workshops they enjoyed, and what topics they were interested in, so that to the extent possible, they could be assigned a mentor(s) suiting those specializations.
E. Winning projects
A total of 10 projects were ideated by the 29 hackathon participants, of which 8 projects were submitted as code along with recorded presentation for judging [12]. Prizes were awarded to the top three projects, with an additional prize for the Best Education Hack awarded by Quantum Universal Education.
The winning projects along with brief description are in Table 2, and links to each project are in the bibliography. Second Place Description: The way to find the optimal resource allocation for users for which throughput is calculated based in resource blocks (chunks of frequency) and modulation (a sort of channel indicator).
Variational Quantum Circuits in a Protein Network Graph [15]
Third Place Description: The main idea of the proposal is to map biochemical interactions inside a 3D protein structure into a graph network.
Quantum classifier for medical data [16]
Best Education Hack Description: In this project, we propose an introduction to quantum machine learning using a variational quantum algorithm for classification, applying it to two medical datasets.
Additionally, we report a follow-up to the 14 winning participants six months after the conclusion of the hackathon.
First, we remark that of the 14, 9 expressed preference for future quantum computing events to be hybrid, in comparison to 2 for in-person and 3 for virtual.
In the response to the question How much time had passed between your first exposure to quantum science, computing, or engineering, and participating in this event (in October 2021)? 6 of 14 responded 0-6 months. Despite the high percentage of beginners, in response to the question On a scale of 1 (not interested) to 5 (very interested), what is your interest in a career in quantum? 2 of the 14 winning participants indicated a 3, 4 indicated a 4, and 8 indicated a 5.
Most notably, 9 out of the 14 hackathon winners responded Yes to the question Since participating in the Qiskit Fall Fest CIC-IPN Hackathon have you participated in any quantum-related event? This shows that the hackathon winners felt empowered to continue to actively learn quantum science following the event, which they did through participating in quantum-related school projects, challenges, summer schools, and other quantum hackathons.
III. SURVEY RESPONSES OF PARTICIPANTS
To guide inclusive and beginner-friendly hackathon organization, four questions were kept in mind [17]:
• Who is eligible to apply? • Who is it marketed to? • Who actually attends? • Who is it prepared to support? For participants for whom it is their first exposure to what they can do with quantum computing, it is understandable to feel not yet ready or comfortable to engage in a quantum hackathon. Regardless of whether they chose to participate in the hackathon (which anecdotally was a very positive experience for not just those experienced in quantum, but those new to quantum as well), the organizers considered the engagement and sincerity of participation in the workshops to be just as, if not more, important than the hackathon component.
A. Demographic information
The participation student status data indicates that the event, open to students and non-students of all levels of education and experience in quantum computing, attracted participation from all levels. This shows that there are no noticeable gaps in the demographic reached, compared to that targeted. We observe that amongst high school participants, the selfidentified level of quantum computing was comparable to that of other participants with other levels of education and occupational status. This raises the possibility of an alternate approach to quantum hackathons: Instead of targeting a certain level of education (e.g. high school) amongst which exposure to quantum computing may vary greatly, another option is to target beginners in quantum computing regardless of level of education, instead categorized by self-identified level of experience in quantum computing (see Figure 3). This approach is informed by the student-run hackathon 〈Womxn/Hacks〉, which in 2019 hosted ≈ 200 female-identifying and nongender-binary undergraduate and graduate students, ≈ 2/3 of whom self-identified as beginners in programming, and nearing 3/4 of whom were pursuing neither computer science nor computer engineering degrees -≈ 1/3 of the total participants were arts and humanities students [17]. The primary advantage of this approach is that individuals who have had their first exposure to or gained confidence to learn the topic at a later stage in life, who are disproportionately likely to be underrepresented in that topic, can have the opportunity to learn. For an approach like 〈Womxn/Hacks〉's which awarded prizes for both beginner and advanced levels, more advanced students in an earlier stage of education who may have exhausted the learning opportunities in quantum available to high schoolers have the option to seek the challenge to learn more by forming a team with others more experienced in other aspects and competing for advanced category prizes. Fig. 3: The survey data is first plotted by the number of participants for each self-reported level of quantum computing knowledge, ranging from 1 (little to none) to 5 (expert). Within each of these five levels, the data was then plotted by number of participants for each educational level (if they were a student) or occupation (for non-students). This shows that 66% of the participants identified as beginners in quantum computing -1 or 2 out of 5.
The country participation data indicates that the country with the most participation was Mexico. This is as the organizers anticipated, with a number of possible explanations that this could be attributed to. First, the advertisement for the event was distributed and shared by a number of Mexican organizations and reaching audiences of Mexican identity. Second, while this was a fully online event, there is the status of the host university and members of the event organizers being Mexican, and hence increased technical and administrative support for those time zones. Although the event time zones had workshops and final project presentations scheduled at more central times of day with respect to time zones around the world, the event being virtually hosted in Mexico may have resulted in people in time zones further from Mexico being less inclined to participate, due to anticipated inconvenience or impracticality of time zone difference.
With that said, the organizers were pleasantly surprised by the number of countries represented in the participants. This not only indicates that educational opportunities in quantum are appealing to students of all levels and from many geographical locations, but that channels to reach out to them all exist and should be utilized more. Combining this with the information that a majority of participants heard about the event through Facebook, this indirectly alludes to the fact that online Spanish-speaking communities for learning and interest in quantum encompass a greater diversity than previously realized (See Figure 4). Regarding the question guiding hackathon organization, "Who is it prepared to support?": To reach more persons in other countries, a recommendation for future online and international Spanish-language hackathons is to assemble an organizing team across more countries. However, the amount of overhead in supporting more time zones can make organization logistically more complex, and so a balance is needed. B. Self-reported amount of experience in quantum, linear algebra, python Comparison of the data on self-reported level of quantum, linear algebra, and Python competence establishes that participants of this event are generally more versed in linear algebra and Python, whilst being new to quantum computing. More participants identified as level 1 out of 5, the lowest level, in quantum computing, than any other level. While programming skills are expected of persons interested in a hackathon, the data on linear algebra being very similar to that is interesting. This supports the perception of both linear algebra and programming as prerequisite skills to learning quantum, as participants felt competent in these two skills whilst being beginners in quantum (see Figure 5).
It is worth noting that the participants are mostly concentrated at an introductory level in quantum computing, and most of them consider that they have intermediate to advanced knowledge about Python and linear algebra. This is in line with the communication about the event, where it was emphasized that beginners to quantum were very welcome, and it was recommended to have some familiarity with Python and linear algebra for the workshops.
Across all participants, the subject area the most participants were interested in was by far quantum algorithms, following by quantum machine learning and quantum cryptography. This finding is interesting because it shows correspondence with a survey of 57 companies in the quantum industry, which identified quantum algorithm development as the skill most relevant across job roles in quantum computing [18].
C. Feedback
We collected feedback from 16 participants a week after the hackathon. First, we note several comments regarding the data collection and storage to bear in mind when we run this workshop series and hackathon again. At the time, the objective of soliciting feedback was to gauge from a small sample of participants the general perception of the event, in addition to identifying any areas for improvement. In retrospect, it would have better informed future similar initiatives to collect feedback promptly at two points: at the conclusion of the week of workshops, and at the conclusion of the hackathon. By surveying participants a week after the hackathon, we were less likely to hear from participants of only the workshops part of the event. Another complications we had was inexperience with regards to data privacy laws to storing participant identifying information and thus opting to be on the safe side of not collecting sensitive information. For this reason, we did not have sufficient data to draw conclusions about changes before and after the hackathon, for instance in participants' self-reported quantum computing level. In the future, we would also like to know participants' field(s) of study for students undergraduate level and above, and aspiring topic(s) for high school students.
We report three observations pertaining to the participant feedback data. The first is the favorite workshop(s) of each surveyed participant, presented in Table III where each row lists the favorite workshop(s) of a particular participant, sorted from lowest (top of the table) to highest (bottom) by selfreported quantum computing level, and within the same level sorted by education level. As expected, the Introduction to quantum computing workshop was frequently favorited in the top half of the table but not in the bottom half. The most popular topics within the workshops were QML, QAOA, and Quantum Enhanced Monte Carlo simultation.
The second observation is the highly positive perception amongst participants about the event organization. 10 of the 16 participants answered the optional short reply question to solicit feedback. In spite of the question being about feedback, there were only two constructive criticisms received: that the Quantum Machine Learning workshop was too technical, and the online meeting links could be sent more in advance. All the responses were in Spanish. Their machine translation to English are provided in Table IV. Like Table III, Table IV is sorted from lowest (top of the table) to highest (bottom) by self-reported quantum computing level, and within the same level sorted by education level. We hope that by sharing their expressed wishes to see more workshops and events in Spanish, it can be seen that there is avid interest in learning quantum computing in Mexico and Latin America. Everything was excellent, I really noticed the effort of the organizers. I wish there were more events like this one. I was interested in the Quantum Machine Learning workshop, but it was very technical and I got lost. They were days of great learning in the quantum computing workshops, great organization of the event, I am very happy that I decided to propose my own initiative in the Hackathon, I want to live the experience and what better than in my language (Spanish) is really a plus that motivates me even more to continue learning. Thank you very much! Very good project, it is a breakthrough for quantum computing in the country. Nothing more. Thanks to the organizers, I appreciate the order that they had with everything and their effort is noticeable! I was delighted that these workshops were given in Spanish, it facilitates the dissemination of quantum computing in Latin America. You were very good, the only recommendation might be that the link was a little earlier because in some meetings it was 10 minutes earlier. I think this is an excellent initiative. Keep it up :D IV. CONCLUSION Science communication and outreach was done after the event, not only by the Qiskit Blog [19], but also outside the field of quantum reaching non-technical and Spanish speaking audiences. In the aftermath of the event, the main event organizer was interviewed in Spanish by a national television channel in Mexico to talk about the event, and the CORE International Congress published a magazine article in Spanish about the event [20]. Finally, the host institute IPN conducted an interview in Spanish with the local participants who won third place [21].
In addition to the benefits of raising literacy in quantum technologies, and encouraging students to interact with industry and researchers, these events motivate participants to gain confidence in further pursuits learning quantum. Furthermore, such events can connect participants with more opportunities. For example, at the hackathon, one of the first place winners was encouraged to apply to the Qiskit advocates program by IBM Quantum, which he has since become a part of after passing the Qiskit developer certification exam.
We conclude by stating that in the participant feedback survey, for the question, "Would you like to see more events like this in Spanish?", 100% of the responses were "Yes". There is much interest in learning quantum computing and gaining relevant skills from Latin America, and lessons were learned in how to develop opportunities to catalyze this.
Fig. 1 :
1Percentage of registered attendees by education level.
Fig. 2 :
2Percentage of registered attendees interested only in the hackathon (4.07%), only in the workshops (40.7%), and in both the workshops and the hackathon (55.2%).
Fig. 4 :
4Percentage of participants by country.
TABLE I :
ISeries of WorkshopsIntroduction to quantum computing Language: English / Spanish Workshop presenter: Lia Yeh / Alberto Maldonado Romo Description: There is a growing trend towards quantum computing.
TABLE II :
IIWinning Projects Threerra: A Qiskit module for three-level systems [13] First Place Description: Created a module to allow users to both create unitary operations acting onto three-level systems (qutrits) using Qiskit Pulse, and to execute them on real hardware available through the IBM Quantum platform. Quantum Radio Resource Scheduling for 4G and 5G simulators [14]
TABLE IV :
IVParticipant optional feedback to the organizers I wish there were more workshops and the resources were more open and available. Very interesting the topics presented.
• Team formation and choosing projects: The workshop session included activities such as a six-phase introduction, discussion of topics of interest, and the creation of
(a) Participants' knowledge of quantum computing compared to their interest in different areas of quantum computing.
ACKNOWLEDGMENTThis event would not have been possible without the active involvement and support of many people. We would like to thank our fellow hackathon organizers, Jose Navarro and David Pérez, along with CIC-IPN for their collaboration as the host organization. We would also like to thank all of the workshop presenters inTable IIand hackathon mentors. We thank IBM Quantum, Multiverse Computing, Xanadu, QMexico, QuantumHispano, Haq.ai, and Quantum Universal Education for sponsoring this event. We thank IBM Quantum Education for providing resources and guidance in organizing the hackathon as part of Qiskit Fall Fest 2021, especially Brian Ingmanson, Anamaría García Hernández, Katie Pizzolato, and Josie Kies. Finally, we thank the workshop series and hackathon participants for their feedback, and for their enthusiasm which made the event a joy to organize.(b) Participants' knowledge of Python compared to their interest in different areas of quantum computing.(c) Participants' knowledge of Linear Algebra compared to their interest in different areas of quantum computing.
Quantum Information Science and Technology Workforce Development National Strategic Plan. Subcommittee on Quantum Information Science. Committee On Science of the National Science & Technology Council"Quantum Information Science and Technology Workforce Development National Strategic Plan," Subcommittee on Quantum Information Sci- ence, Committee On Science of the National Science & Technology Council, February 2022. https://www.quantum.gov/wp-content/uploads /2022/02/QIST-Natl-Workforce-Plan.pdf
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Qubit by Qubit 2021 Impact Report. The Coding School"Qubit by Qubit 2021 Impact Report," The Coding School, December 2021.
Quantum games and interactive tools for quantum technologies outreach and education. Z C Seskir, P Migdał, C Weidner, A Anupam, N Case, N Davis, C Decaroli, İ Ercan, C Foti, P Gora, K Jankiewicz, B R La Cour, J Y Malo, S Maniscalco, A Naeemi, L Nita, N Parvin, F Scafirimuto, J F Sherson, E Surer, J R Wootton, L Yeh, O Zabello, M Chiofalo, 10.1117/1.OE.61.8.081809SPIE Opt. Eng. 618Z. C. Seskir, P. Migdał, C. Weidner, A. Anupam, N. Case, N. Davis, C. Decaroli,İ. Ercan, C. Foti, P. Gora, K. Jankiewicz, B. R. La Cour, J. Y. Malo, S. Maniscalco, A. Naeemi, L. Nita, N. Parvin, F. Scafirimuto, J. F. Sherson, E. Surer, J. R. Wootton, L. Yeh, O. Zabello, and M. Chiofalo, "Quantum games and interactive tools for quantum technologies outreach and education," SPIE Opt. Eng., vol. 61, no. 8, pp. 1-38, July 2022. https://doi.org/10.1117/1.OE.61.8.081809
Code camps and hackathons in education -literature review and lessons learned. J Porras, A Knutas, J Ikonen, A Happonen, J Khakurel, A Herala, 10.24251/hicss.2019.93352nd Hawaii International Conference on System Sciences. J. Porras, A. Knutas, J. Ikonen, A. Happonen, J. Khakurel, and A. Herala, "Code camps and hackathons in education -literature review and lessons learned," 52nd Hawaii International Conference on System Sciences, January 2019. http://dx.doi.org/10.24251/hicss.2019.933
Quantum game jammaking games with quantum physicists. A Kultima, L Piispanen, M Junnila, 10.1145/3464327.3464349Academic Mindtrek 2021. Association for Computing MachineryA. Kultima, L. Piispanen, and M. Junnila, "Quantum game jam - making games with quantum physicists," in Academic Mindtrek 2021, Association for Computing Machinery, pp. 134-144. https://doi.org/10 .1145/3464327.3464349
Semana de la computación cuántica y aplicaciones de la física cuántica, Sociedad Científica Juvenil. Cuarantena Cuantica, Cuarantena Cuantica: Semana de la computación cuántica y aplicaciones de la física cuántica, Sociedad Científica Juvenil, January 2021. https: //www.facebook.com/SCJ.MX/photos/3555268284521727
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CORE International Congress 2022, Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN). CORE International Congress 2022, Centro de Investigación en Com- putación del Instituto Politécnico Nacional (CIC-IPN), September 2021. https://www.core.cic.ipn.mx/
Recording of Workshop at Major League Hackcon VIII. E Martinez, L Yeh, L Zeng, N Y Lara, Developing a Beginner-Friendly HackathonE. Martinez, L. Yeh, L. Zeng, and N. Y. Lara, "Developing a Beginner- Friendly Hackathon," Recording of Workshop at Major League Hackcon VIII, September 2020. https://www.youtube.com/watch?v=EuBR6 jDk a8&t=196s
Assessing the Needs of the Quantum Industry. C Hughes, D Finke, D.-A German, C Merzbacher, P M Vora, H J Lewandowski, arXiv preprintC. Hughes, D. Finke, D.-A. German, C. Merzbacher, P. M. Vora, and H. J. Lewandowski, "Assessing the Needs of the Quantum Industry," arXiv preprint, August 2021. https://arxiv.org/abs/2109.03601
| {'fraction_non_alphanumeric': 0.03011472275334608, 'fraction_numerical': 0.00960368503389536, 'mean_word_length': 5.077512214446059, 'pattern_counts': {'":': 1, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 8, 'lorem ipsum': 0, 'www.': 4, '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': 'We discuss the challenges and findings of organizing an online event in Spanish, consisting of a series of introductory workshops leading up to a quantum hackathon for Latin America. 220 Spanish speakers were registered, 66% of whom self-identified as being at an introductory level of quantum computing. We gain a better picture of the impact of quantum computing in Latin America, and the importance of generating educational resources in Spanish about quantum computing. Additionally, we report results on surveying the participants by country; educational status; self-reported levels of quantum computing, linear algebra, and Python competency; and their areas of interest within quantum.This event was organized by Quantum Universal Education with the Centro de Investigación en Computación del Instituto Politécnico Nacional (CIC-IPN) as the host institution, in collaboration with a number of organizations and companies: IBM Quantum, Xanadu, Multiverse Computing, Quantum Universal Education, Quantum Hispano, QMexico, Haq.ai, Dive in Learning. This was part of a larger event, the Qiskit Fall Fest 2021, as one of several hackathons organized around the world in a similar span of time. In each Qiskit Fall Fest hackathon, participants were challenged to form teams of up to 5, to develop in 5 days a project using the IBM Qiskit framework.', 'arxivid': '2302.12119', 'author': ['Alberto Maldonado-Romo \nCentro de Investigación en Computación\nDepartment of Computer Science\nInstituto Politécnico Nacional Mexico City\nMexico\n', 'Lia Yeh \nUniversity of Oxford Oxford\nUK\n'], 'authoraffiliation': ['Centro de Investigación en Computación\nDepartment of Computer Science\nInstituto Politécnico Nacional Mexico City\nMexico', 'University of Oxford Oxford\nUK'], 'corpusid': 253803726, 'doi': '10.1109/qce53715.2022.00096', 'github_urls': [], 'n_tokens_mistral': 10384, 'n_tokens_neox': 9534, 'n_words': 6912, 'pdfsha': '2d35b63f62a48e91dc4a3c69fc114cfcd2ed4ee9', 'pdfurls': ['https://export.arxiv.org/pdf/2302.12119v1.pdf'], 'title': ['Quantum computing online workshops and hackathon for Spanish speakers: A case study', 'Quantum computing online workshops and hackathon for Spanish speakers: A case study'], 'venue': []} |
arxiv |
The SAMI Survey: Evidence for dynamical coupling of ionised gas and young stellar populations
2023
Caroline Foster
School of Physics
University of New South Wales
2052SydneyNSWAustralia
Sydney Institute for Astronomy
School of Physics
The University of Sydney
NSW
A28, 2006Australia
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Sam Vaughan
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Astronomy, Astrophysics and Astrophotonics Research Centre
Macquarie University
2109SydneyNSWAustralia
School of Mathematical and Physical Sciences
Macquarie University
2109NSWAustralia
Centre for Astrophysics and Supercomputing
School of Science
Swinburne University of Technology
3122HawthornVICAustralia
Amelia Fraser-Mckelvie
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
International Centre for Radio Astronomy Research
The University of Western Australia
35 Stirling Hwy6009CrawleyWAAustralia
Sarah Brough
School of Physics
University of New South Wales
2052SydneyNSWAustralia
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Julia J Bryant
Sydney Institute for Astronomy
School of Physics
The University of Sydney
NSW
A28, 2006Australia
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Astralis-USydney
School of Physics
University of Sydney
2006NSWAustralia
Scott M Croom
Sydney Institute for Astronomy
School of Physics
The University of Sydney
NSW
A28, 2006Australia
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Francesco D ' Eugenio
Kavli Institute for Cosmology
University of Cambridge
Madingley RoadCB3 0HACambridgeUnited Kingdom
Cavendish Laboratory -Astrophysics Group
University of Cambridge
19 JJ Thomson AvenueCB3 0HECambridgeUnited Kingdom
Brent Groves
International Centre for Radio Astronomy Research
The University of Western Australia
35 Stirling Hwy6009CrawleyWAAustralia
Iraklis S Konstantopoulos
Independent scholar Wellington
New Zealand 12 Research School of Astronomy and Astrophysics
Australian National University
Cotter Road, Weston Creek, ACT 2611Australia
Ángel R López-Sánchez
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Astronomy, Astrophysics and Astrophotonics Research Centre
Macquarie University
2109SydneyNSWAustralia
School of Mathematical and Physical Sciences
Macquarie University
2109NSWAustralia
Sree Oh
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Department of Astronomy and Yonsei University Observatory
Yonsei University
03722SeoulRepublic of Korea
Matt S Owers
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Astronomy, Astrophysics and Astrophotonics Research Centre
Macquarie University
2109SydneyNSWAustralia
School of Mathematical and Physical Sciences
Macquarie University
2109NSWAustralia
Sarah M Sweet
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
School of Mathematics and Physics
University of Queensland
4072BrisbaneQLDAustralia
Jesse Van De Sande
Sydney Institute for Astronomy
School of Physics
The University of Sydney
NSW
A28, 2006Australia
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Emily Wisnioski
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
Sukyoung K Yi
Department of Astronomy and Yonsei University Observatory
Yonsei University
03722SeoulRepublic of Korea
Henry R M Zovaro
ARC Centre of Excellence for All Sky Astrophysics in
Dimensions (ASTRO 3D
The SAMI Survey: Evidence for dynamical coupling of ionised gas and young stellar populations
MNRAS
0002023Accepted 02/2023.Preprint 15 February 2023 Compiled using MNRAS L A T E X style file v3.0galaxies: kinematics and dynamics -galaxies: stellar content
We explore local and global dynamical differences between the kinematics of ionised gas and stars in a sample of galaxies from Data Release 3 of the SAMI Galaxy Survey. We find better agreement between local (i.e., comparing on a spaxel-to-spaxel basis) velocities and dispersion of gas and stars in younger systems as with previous work on the asymmetric drift in galaxies, suggesting that the dynamics of stars and ionised gas are initially coupled. The intrinsic scatter around the velocity and dispersion relations increases with increasing stellar age and mass, suggesting that subsequent mechanisms such as internal processes, divergent star formation and assembly histories also play a role in setting and altering the dynamics of galaxies. The global (flux-weighted) dynamical support of older galaxies is hotter than in younger systems. We find that the ionised gas in galaxies is almost always dynamically colder than the stars with a steeper velocity gradient. In absolute terms, the local difference in velocity dispersion is more pronounced than the local difference in velocity, possibly reflecting inherent differences in the impact of turbulence, inflow and/or feedback on gas compared to stars. We suggest how these findings may be taken into account when comparing high and low redshift galaxy samples to infer dynamical evolution.
INTRODUCTION
Galaxies are complex structures composed of both dark and baryonic matter. The interaction of baryons in galaxies is thought to be regulated by a number of physical processes (e.g., gravitational interactions/collisions, gas inflow, outflow, star formation, etc; see e.g., Schaye et al. 2015;Taylor & Kobayashi 2015;Beckmann et al. 2017;Conselice et al. 2022) and these processes may apply to or involve multiple distinct "phases" of the baryons (i.e., stars, ionised gas, dust, molecular gas, e.g., Tumlinson et al. 2017). Studying galaxies through observations of multiple baryonic phases allows for a more ★ E-mail: [email protected] holistic picture and understanding of the physical mechanisms that have shaped galaxies throughout cosmic time.
Stars within galaxies are a compendium of successive generations. Some stars form within a galaxy, while others are accreted from neighbouring or merged systems (e.g., Ibata et al. 1994;Bell et al. 2008;D'Souza & Bell 2018;Helmi et al. 2018;Boecker et al. 2020;Casanueva et al. 2022;Remus & Forbes 2022). Over time, stellar orbits mix and evolve through interaction with external systems and secular star-to-star interactions. Enriched gas from previous stellar generations or externally accreted "pristine" gas can dissipate and form new stars usually by (re)forming a disc, thereby rejuvenating the system and changing its dynamics (e.g., Dekel et al. 2009;Wright et al. 2021). Combined studies of ionised gas and stellar kinematics have shown that the gas is usually kinematically colder and faster rotating than the stars (e.g., Pizzella et al. 2004).
Direct studies of the dynamical evolution of galaxies are limited by the relative brightness of the different phases. While the ionised gas can be detected out to higher redshifts ( = 0.7 − 2.5, e.g., Stott et al. 2016;Förster Schreiber et al. 2018;Wisnioski et al. 2019;Tiley et al. 2021), measuring spectra of the faint stellar continuum in galaxies is challenging/prohibitively observationally costly beyond ∼ 1 (e.g., van der Wel & van der Marel 2008;Belli et al. 2015;Mendel et al. 2015;Belli et al. 2017;Newman et al. 2018;Belli et al. 2019;Mendel et al. 2020). Furthermore, the most massive galaxies in the present day universe are quiescent. Since massive star-forming galaxies at high redshifts are often the progenitors of quiescent local galaxies (e.g., Guglielmo et al. 2015), understanding how coupled or otherwise the ionised gas is with the stars initially and over time is important for inferring the dynamical evolution of consistent galaxy populations across cosmic time (also see Straatman et al. 2022, for a study comparing dynamical mass estimates based on ionised gas and stars). It is thus desirable to further explore the validity of comparing ionised gas dynamics at high redshift with local stellar dynamics.
Recent findings tentatively suggest that the dynamical properties of stars of different ages in nearby galaxies may mirror that of the ionised gas across cosmic time. Using Schwarzschild orbital modelling (e.g., Schwarzschild 1979;van den Bosch et al. 2008;Zhu et al. 2020;Thater et al. 2022) of the lenticular galaxy NGC 3115, Poci et al. (2019) found that the stellar velocity dispersion as a function of population age steadily increases, mirroring that of the ionised gas in populations of galaxies across a broad range of redshifts (Kassin et al. 2012;Wisnioski et al. 2015;Übler et al. 2019). Older stars, like the ionised gas in high redshift galaxies, exhibit hotter dynamics (i.e., more turbulence or random motions) than their younger/local counterparts. This is surprising given the vastly different methods and samples used and does not imply the two are linked and much less causal. Indeed, this initially compelling picture has since been confounded by increased scatter when adding more systems into the local dynamically modelled sample (Poci et al. 2021), suggesting that other explanations and evolutionary mechanisms need to be considered.
The "asymmetric drift" observed in the Milky Way (e.g., Golubov et al. 2013), M31 (e.g., Quirk et al. 2019), MaNGA (Shetty et al. 2020) and DiskMass (Martinsson et al. 2013) galaxies demonstrates that older stars tend to be on hotter orbits than young stars in present day systems. This is consistent with expectations from large hydrodynamical simulations (e.g., Quirk & Patel 2020) and simulations of the Milky Way (Bird et al. 2013). In the latter, the dynamics of Milky Way stars are shown to reflect the local conditions at the time of their formation. This tendency of older stars being on hotter orbits is consistent with the findings that older galaxies have larger intrinsic flattening (i.e. indicative of globally hotter orbits) than their younger counterparts (van de Sande et al. 2018).
While the mirroring of stellar and gas dynamics across cosmic time may reflect the conditions of the interstellar medium (ISM) at that redshift it could also be a by-product of longer opportunities for internal processes of dynamical heating (e.g., stellar migration, radial mixing, etc). For example, Okalidis et al. (2022) and Hayden et al. (2020) suggest, using simulations and observations respectively, that stellar migration triggered by interactions with non-axisymmetric structures like bars explain aspects of the distribution of ages, metallicities and orbits of stars in the Solar neighborhood.
Older systems also have more time for externally triggered heating processes such as mergers and interactions to build up over time. Indeed, Minchev et al. (2013) suggest that both stellar migration and interactions may play a role. Mackereth et al. (2018) find that a significant episode of past interaction is required to produce the bimodal -element distribution observed in e.g. the Milky Way (Hayden et al. 2015) and other galaxies (e.g., Scott et al. 2021). In other words, a hybrid explanation where a mix of both initial conditions and latent dynamical heating may need to be invoked. Indeed, in eight Local Group galaxies, Leaman et al. (2017) find that the stellar velocity dispersion of local galaxies is largely set by the conditions at the redshift of formation with departure from this being dependent on stellar mass.
There is a myriad of observational evidence supporting the idea that the dynamics of gas and stars are initially coupled. Integral field spectroscopic surveys in particular have made it clear that there are dynamical and chemical evidence for a co-evolution of stars and gas. Using the MaNGA galaxy survey, Shetty et al. (2020) showed that the asymmetric drift exhibits more variations in stellar populations older than 1.5 Gyr compared to younger populations. In parallel, and also in MaNGA galaxies, Greener et al. (2022) find evidence for chemical co-evolution of the ionised gas and stars. Barat et al. (2019) compared rotational and central dispersion of gas and stars in the Sydney-AAO Multi Integral field (SAMI) Galaxy Survey and find a steeper slope for the 0.5 − log 10 * relationship in stars compared to gas kinematics. More recently, Oh et al. (2022) showed that the velocity dispersion of the ionised gas tends to be lower than that of the stars in SAMI galaxies, though caution that beam smearing may be playing a role in artificially enhancing differences. As the precursor to stars, cold gas kinematics can also be compared to stellar kinematics. Indeed, Quirk et al. (2019) find that the H and CO gas rotates slower when compared with the main sequence stars (∼ 7.5 Gyr) in the Andromeda galaxy.
In this work, we use SAMI data to explore how the dynamics of the ionised gas and stars compare locally within galaxies. We explore whether measured local and global dynamical differences are consistent with current thinking on the origin of asymmetric drift and the role of "latent" dynamical heating and/or the ISM conditions at the time of formation.
The paper is divided as follows. §2 presents the data, while our sample selection can be found in §3. §4 details our data analysis and results. A discussion and our conclusions can be found in §5 and §6, respectively.
Throughout this paper, we assume a ΛCDM cosmology with Ω m = 0.3, Ω = 0.7 and 0 = 70 km s −1 Mpc −1 , and a Chabrier (2003) initial mass function.
DATA
This work uses spatially resolved spectroscopy and ancillary data from the Sydney-AAO Multi Integral field (SAMI) Galaxy Survey (Croom et al. 2021, henceforth C21). Spectral cubes were observed with SAMI (Croom et al. 2012): a multi-object integral field instrument connected to the AAOmega spectrograph (Sharp et al. 2006) at the Anglo-Australian Telescope. SAMI had 13 hexabundles and 26 individual sky fibres deployable over a 1 degree field-of-view. Each hexabundle was a tightly packed bundle of 61 optical fibres over a 15 arcsec diameter with a 73 percent fill-factor. The SAMI Galaxy Survey observed ∼ 3000 unique galaxies in the nearby Universe (i.e., redshifts 0.004 < < 0.095). The selection of the Galaxy And Mass Assembly (GAMA, Driver et al. 2011) and clusters samples are detailed in Bryant et al. (2015) and Owers et al. (2017), respectively. The data used in this work are part of the latest and final public SAMI . Stellar mass (log 10 ( * / Sun )), left), effective radius ( e , middle-left), specific star formation rate (sSFR, middle-right) and visual morphological classification (right, where E: elliptical, S0: lenticular, eSp: early spiral and lSp: late spiral) distributions for the final sample (yellow), galaxies with both gas and stellar kinematic maps out to 1 effective circularised radius but before the seeing and stellar mass cut are applied (orange) and the full SAMI sample (white). About ∼ 5 percent of galaxies in full SAMI sample have uncertain morphological classifications and are not shown in the right panel. Due to the requirement of high quality, large spatial extent and good resolution of gas and stellar kinematic maps, the sample is necessarily biased towards star forming galaxies, intermediate morphological types (eSp) and higher effective radii. Star formation rate as a function of the stellar mass for the final sample (yellow), galaxies with both gas and stellar kinematic maps out to 1 effective circularised radius but before the seeing and stellar mass cut are applied (orange) and the full SAMI sample (grey). The requirement for both gas and stellar kinematics favours galaxies with significant star formation, although there are a handful of selected galaxies that lie significantly (> 2 standard deviations, dashed purple line) below the star forming main sequence defined by late-type spirals (purple dots) fitted as per Medling et al. (2018, solid purple line). Also shown as a red dashed line is the star forming main sequence as per Equation 3.
Galaxy Survey data release (SAMI DR3, C21). Thus, all data used in this work are available through this public data release.
SAMI data are reduced using the 2 pipeline, which performs the usual reduction steps: bias subtraction, wavelength calibration using CuAr arc frames and sky subtraction. Spectral extraction and reduction are detailed in Sharp & Birchall (2010) and Hopkins et al. (2013), respectively. Flux calibration and telluric absorption corrections are performed using standard stars. Each field is nominally observed with a 7-dither (minimum 6) pattern to deal with gaps between fibres within the hexabundle. Data are then combined and mapped onto a grid with flux and covariance carefully propagated as described in Sharp et al. (2015). Further detail on the data reduction can be found in Allen et al. (2015) and Sharp et al. (2015), with the latest improvements described in C21.
Production of the stellar kinematic maps is described in van de Sande et al. (2017b) with updates outlined in C21. Briefly, the lineof-sight velocity distribution (LOSVD) is parametrised as a Gaussian using the penalised pixel-fitting ( PXF; Cappellari & Emsellem 2004;Cappellari 2017) algorithm. A selection of templates are broadened and shifted through convolution to determine the most likely recession velocity ( rec ) and velocity dispersion ( ) for each spaxel. This is done in two steps as described in van de Sande et al. (2017b) to mitigate uncertainties associated with possible template mismatches, especially in the lower signal-to-noise spectra. The stellar kinematic position angles kin,stars is determined using the _ _ code, which is based on the method described in Krajnović et al. (2006, their appendix C).
Gas kinematics and H emission line flux maps are measured using (Ho et al. 2016) as described in Green et al. (2018), Scott et al. (2018) and C21. Briefly and as described in Owers et al. (2019), the continuum is first fitted to Voronoi-binned spaxels using single stellar population (SSP) templates from Vazdekis et al. (2010) and González Delgado et al. (2005). This continuum is subtracted from individual spaxels to leave "pure" emission line spectra, which are then fitted using . Although fits up to 3 independent velocity components, we only use the 1-component fit in this work. As for the stars, the kinematic position angle of the ionised gas ( kin,gas ) is determined using the _ _ code following Krajnović et al. (2006, their appendix C).
Global star formation rate (SFR) estimates are measured on elliptical 1 aperture spectra using the extinction-corrected H flux and use the Kennicutt et al. (1994) corrected to a Chabrier (2003) stellar initial mass function (Medling et al. 2018; C21, for more detail).
The light-weighted stellar population ages and metallicities ([ / ]) within 1 are derived as described in Vaughan et al. (2022). We give a brief summary here. First, the SAMI blue and red arm spectra are joined together and convolved with a Gaussian kernel (of variable width) such that the spectral resolution is a constant value at all wavelengths (following van de Sande et al. 2017a). We use PXF (Cappellari & Emsellem 2004;Cappellari 2017) to fit the MILES simple stellar population (SSP) models of (Vazdekis et al. 2015) to the spectrum of each galaxy extracted within 1 . The templates range in metallicity from −2.21 dex to 0.4 dex, in age from 30 Myr to 14 Gyr and in [ /Fe] abundance from 0.0 to +0.4 dex. The templates use the isochrones from the 'Bag of Stellar Tracks and Isochrones' models (BaSTI; Pietrinferni et al. 2004Pietrinferni et al. , 2006. During the fit, we also include gas emission line templates corresponding to the Balmer series (H , H , and H ) as well as the atomic species
SAMPLE SELECTION
Our target selection stringently keeps only those galaxies with the highest quality gas and stellar kinematic maps. Stellar kinematic map spaxels with stars,error > 30 km s −1 , stars,error > 0.1 + 25 km s −1 and signal-to-noise ratio < 3 are discarded. Gas kinematic maps spaxels with / ,error < 5 are discarded. We then only keep those galaxies where both the ionised gas and stellar kinematic maps have a minimum 85 percent fill factor within an elliptical aperture with circularised radius corresponding to 1 . As a result, all galaxies from the full SAMI sample of 3068 galaxies without significant and spatially extended emission lines are effectively removed. We select galaxies with log 10 ( * / Sun ) > 9.5 and require that the full width at half maximum (FWHM) seeing value for the cube not exceed half the effective radius (i.e., > 2 × FWHM) 1 . The above selection leaves 188 unique galaxies and an additional 22 duplicate observations. For targets with duplicates, we select the instance with the better seeing. Fig. 1 shows how the distributions in log 10 ( * / Sun ), , specific SFR (sSFR, i.e. the SFR per unit mass) and visual morphology of galaxies selected for this work compare with the full SAMI sample. Our requirement for high spatial resolution necessarily biases the sample towards high values and the requirement for reliable gas and stellar kinematics biases the sample towards star forming and intermediate to later galaxy types (eSp). Fig. 2 shows the star formation rate (SFR) vs. stellar mass of SAMI galaxies. We note that the full SAMI sample is not a volume limited survey and that it includes many small and low-mass filler targets that do not pass our selection criteria. The star-forming main sequence line is fit to the distribution of galaxies within the full SAMI sample with available SFRs as in Fraser-McKelvie et al. (2021), using the functional form introduced by Leslie et al. (2020). Over the stellar mass range 8 < log 10 ( * / Sun ) < 11.5, the best-fitting main sequence line was found to be: log 10 SFR = −0.352 − log 10 1 + 10 10.101
* / Sun .(1)
While the bulk of our selected galaxies lie along the star-forming main sequence, there are a considerable number that scatter to lower SFRs (Fig. 2), consistent with at least some galaxies in our sample containing gas ionised by other ionising mechanisms than young massive hot stars.
ANALYSIS AND RESULTS
Example gas and stellar velocity (V), dispersion ( ) and ( / ) maps for SAMI106717, a typical galaxy in our final sample, are shown in Table 1). The dashed lines represent Δ( / ) Re = 0. The Spearman correlation coefficient and the corresponding -value stated in the top right of each panel indicate a significant (anti-)correlation between age and Δ ( Δ ). There is increasing scatter in Δ and Δ as a function of age. A significant (anti-)correlation between stellar mass and Δ ( Δ ) is observed. Galaxies with lower stellar masses have more similar gas and star dynamics. In the majority of systems, the ionised gas rotates faster than the stars (i.e., most data points and the median in the bottom panel lie above the dashed line). Interpretation is given in §5. Fig. 3. For this galaxy, and as we will show below, for most galaxies in our sample, the ionised gas is visibly dynamically colder than the stars (i.e., ( / ) gas > ( / ) stars ).
We measure and subtract the systemic velocity ( sys ) for the gas and stars separately by performing a flux weighted mean of the velocity map within 1.5 arcsec (3×3 pixels).
To quantify the local dynamical differences between the ionised gas and stars, we define two parameters. We purposely avoid the central regions to ensure that differences aren't driven by beam smearing, whose effect is most pronounced in the centre. First, we take the mean absolute velocity difference between gas and stars Δ within an elliptical annulus of 0.9 − 1 around the SAMI cube centre, as:
Δ = 0.9 ≤ ≤1
gas, − stars,
0.9 ≤ ≤1 1 ,(2)
where gas, and stars, are the sys -subtracted recession velocity measured in the th spaxel for the ionised gas and the stellar kinematic maps, respectively. is the circularised radius of the th pixel. Similarly, the mean difference in velocity dispersion between the ionised gas and the stars, Δ , is computed as follows:
Δ = 0.9 ≤ ≤1 gas, − stars, 0.9 ≤ ≤1 1 ,(3)
where gas, and stars, is the velocity dispersion measured in the th spaxel for the ionised gas and the stellar kinematic maps, respectively.
To study the global nature (rotation vs pressure) of the dynamical support, we compute ( / ) for the ionised gas and stars within an elliptical 1 aperture following Cappellari et al. (2007):
( / ) , = 2 , 2 , = ≤ 2 , ≤ 2 , ,(4)
where X stands for either 'gas' or 'stars'. is the mean continuum or H flux for the stars or gas, respectively, in the th spaxel. We compute Δ( / ), the simple arithmetic difference between the ionised gas and stars ( / ), as follows:
Δ( / ) = ( / ) gas, − ( / ) stars, .(5)
When relevant, we use the Spearman correlation coefficient ( ) and associated -value to determine the significance of trends. We consider values of 0.01 ≤ ≤ 0.05 as weak trends, with < 0.01 being the threshold to call a trend statistically significant.
Throughout, we refer to Δ and Δ as measures of the local difference between the dynamics of the ionised gas and stars; and to Δ( / ) as a measure of global dynamical differences. In all cases, the uncertainties are propagated by neglecting covariance which is unavailable for the velocity and dispersion maps. We check the impact of ignoring covariance on our conclusions by repeating the analysis using every other spaxel. This effectively ensures that none of the spaxels used in the calculation are correlated (i.e., no covariance). This exercise shows only minimal absolute differences of order 1 km s −1 in the binned means and standard deviations of Δ and Δ compared to those listed in Table 1 and thus ignoring covariance does not affect our conclusions.
The local and global kinematic differences within 1 with respect to age and stellar mass are shown in Fig. 4. The amplitude of local differences between the velocity maps ( Δ R e ) are less scattered in younger stellar populations (see Table 1). Younger systems (∼ 1 Gyr) have a mean Δ ∼ 21 km s −1 , while older (∼ 9 Gyr) systems have mean Δ ∼ 43 km s −1 . The scatter away from this trend increases with stellar age with the standard deviation tripling from 11 km s −1 for ages ∼ 1 Gyr to 34 km s −1 for ages ∼ 9 Gyr. Similarly, low mass systems (log 10 ( * / Sun ) ∼ 9.9) have a mean Δ ∼ 23 km s −1 , while high mass (log 10 ( * / Sun ) ∼ 11.0) systems have mean Δ = 39 km s −1 . The scatter away from this trend more than doubles over the range of stellar masses probed (i.e. the standard deviation is 16 km s −1 for log 10 ( * / Sun ) ∼ 9.9 and 37 km s −1 for log 10 ( * / Sun ) ∼ 11.0).
The lower panels of Fig. 4 indicate that the dynamics of the ionised gas is usually colder than the stars in the majority of systems (i.e. all but a handful of targets lie above the dashed line). This is also shown in Fig. 6, which exhibits a steep correlation between ( / ) stars,R e and ( / ) gas,R e , where the bulk of the data points lie above the oneto-one line. The trend is also steeper than the one-to-one, with older and/or lower SFR and/or higher [N ]/H galaxies having lower and more similar ( / ) gas,R e and ( / ) stars,R e . The latter suggests a different mechanism (other than star-formation) is responsible for ionising the gas in those galaxies.
As per recent SAMI work by Ristea et al. (2022) and similarly to other studies (e.g., Davis et al. 2011;Lagos et al. 2015;Bryant et al. 2019;Casanueva et al. 2022), we define misaligned galaxies as those where the kinematic position angle of the gas and stars differ by more than 30 degrees (i.e. | kin,gas − kin,stars | > 30). We find a total of 10 misaligned galaxies amongst the 188 galaxies in our Figure 5. The relationship between the mean difference between the ionised gas and stellar velocity ( Δ Re ) and dispersion ( Δ Re ). Data points with yellow outlines show systems with kinematically misaligned gas and stars. The majority of data points lie below the line Δ Re = − Δ Re line (dashed). There is no clear trend with age or SFR. However, high mass galaxies tend to scatter lower in this space. Median values shown in Fig. 4 indicate that in most systems, it is the localised dispersion, rather than the velocity, that differs most between the ionised gas and stars. Interpretation is given in §5.
final sample (namely, SAMI IDs 202399, 208652, 238922, 279905, 321059, 39057, 517278, 536994, 560238 and 618993). Much of the increase in the scatter in Δ as a function of mass and age corresponds to a higher proportion of kinematically misaligned galaxies (yellow outlined symbols in Fig. 4). In other words, higher scatter tends to correspond to more massive and older galaxies, with all misaligned galaxies having flux weighted stellar ages > 3 Gyr. Fig. 5 shows how the local dispersion differs more than the local velocity in most galaxies (except for kinematically misaligned systems) and that this does not seem to be linked with stellar population age or SFR, but possibly weakly with stellar mass. . RHS: same as LHS, but using LOESS smoothing to highlight colour gradients. The ionised gas dynamics tend to be colder than the stars with a steeper slope than the one-to-one (dashed line). A colour gradient with lower values of ( / ) typically corresponding to older stellar ages and lower SFR on average is seen. A colour gradient suggests that higher values of [N ]/H (consistent with LINER and/or AGN ionising radiation) correspond primarily to lower values of ( / ) gas,Re . No colour gradient is observed with stellar mass (third from top panels). On average, the global dynamics of ionised gas and stars better agree at lower ( / ) and for galaxies that are older, less star forming and with higher [N ]/H . Interpretation is given in §5. . Top: comparison of the gas (∇ gas,Re ) and stellar (∇ stars,Re ) velocity gradients. Dashed line is the one-to-one. Middle: ratio of the stars and gas velocity gradients as a function of stellar age. Bottom: same as middle, but as a function of stellar mass. In the upper two panels, symbols are colour coded by stellar mass. Large purple symbols represent mean values binned along the x-axis with errorbars representing the standard deviation in each bin. The Spearman correlation coefficient and the corresponding -value stated in the top right of the bottom two panels indicate significant anti-correlations between stellar mass and the ratio of the velocity gradients. Velocity gradients best agree in low mass and younger systems. In high mass and older systems, the gas velocity gradient is on average steeper than the stellar velocity gradient. Interpretation is given in §5.
for both the ionised gas and stellar kinematics as follows:
∇ X,R e = 1 2 pix ≤ [ ( + 1, ) − ( − 1, )] 2 + [ ( , + 1) − ( , − 1)] 2 pix,R e ,(6)
where pix is the pixel scale (i.e., 0.5 arcsec/pix for SAMI) and pix,R e is the number of pixels within 1 . ∇ gas,R e and ∇ stars,R e reflect the spatial change within the effective radius of the velocity of galaxies. The velocity gradient has been used as a proxy for the impact of beam smearing on gas and stellar velocity dispersion. In Appendix A, we discuss how our results are robust against the impact of beam smearing and that trends with ∇ at least partly reflect real physical processes. Fig. 7 shows that the projected gas and stars velocity gradients best agree in low mass and younger systems. Conversely, in high mass and older systems, the gas velocity gradient is on average steeper than the stellar velocity gradient.
DISCUSSION
The main driver for this work is to better understand how gas and stars in galaxies correlate, and test whether one may safely assume that the two phases are coupled, at least initially, when forming stars. In other words, can we infer dynamical evolution of galaxies using high and low redshift surveys by comparing the dynamics of ionised gas in massive star forming galaxies to their local often quiescent (and without ionised gas) descendants? In what follows, we explore how this may be done along with stating significant caveats to this approach.
Despite our simplistic approach, our results are relevant to a number of topics in the literature, including asymmetric drift, dynamical heating, evolution of ISM dynamics, as well as our main goal, which is to test possible methodologies for quantifying the dynamical evolution of galaxies. As such, our results are relevant to both local group and high redshift studies. This discussion focuses on contrasting a selection of relevant previous work from the high and low redshift literature as well as local group studies to contextualise our findings.
As a first caveat, in galaxies with a complex mix of stellar populations, the stellar dynamics do not just represent those of the youngest stars. Hence, we do not necessarily expect an exact one-to-one correspondence between the ionised gas and stars in any of our systems. This is because on the one hand, newborn stars ionise and heat up the gas. On the other hand, the physics of gas and stars fundamentally differ, with gas being collisional and stars being collisionless, which affects the heating mechanisms of the two dynamical tracers. Furthermore, the ionising source of the gas can vary between systems and even spatially across a single galaxy. Ionising sources can include hot young and/or evolved stars, low-ionization nuclear emission-line region (LINER), shocks and Active Galactic Nuclei (AGN).
Using simulations of the Milky Way, Bird et al. (2013) show that the dynamics of the stars could reflect the ISM conditions of their surroundings at the time of their formation. Assuming massive galaxies are more dynamically "evolved", the trends we see in the mean Δ and Δ values (i.e., large purple symbols in Fig. 4) are consistent with a scenario in which gas and stars are initially partly coupled or in which heating mechanisms follow an overall trend across cosmic time. In other words, in young and low mass galaxies, the dynamics of the gas and stars could be more similar if the gas from which stars form is being ionised by (possibly those same) recently formed stars with possible subsequent heating enhancing differences over time. This suggests that the stars may continue to carry some of the kinematic information of the gas from which they formed and subsequently ionised. It is noteworthy that Poci et al. (2019, for a single galaxy) found an offset of about ∼ 10 km s −1 in the stellar dispersion compared to the ionised gas data from Wisnioski et al. (2015) at the same epoch, while Poci et al. (2021) found significant scatter in this offset when comparing with different systems.
In almost all cases, galaxies show colder gas dynamics than the stars, with the difference being more pronounced in high mass and older systems. Our finding that galaxies with younger stellar populations (or higher star formation rate, see Fig. 6) have more pronounced differences in dynamical support, with a steeper than one-to-one relationship between dynamical support of the ionised gas and stars, is consistent with a scenario wherein galaxy evolution becomes increasingly driven by dry mergers just as the global star formation drastically reduces (e.g., Madau & Dickinson 2014), thereby heating the stars through orbital mixing.
The fact that the most massive and older galaxies tend to scatter more than their lower mass and younger counterparts (top panel of Fig. 4 and bottom panel of Fig. 5) is consistent with the findings of Leaman et al. (2017) and with scenarios in which stars experience varying degrees of dynamical "latent" heating (i.e., dynamical heating after formation) depending on the galaxies' individual merging and formation histories. This interpretation that dynamical heating could be the cause of the increased scatter is corroborated by the fact that the most highly scattered points tend to be those with misaligned gas and stars kinematics, which is primarily associated with external gas accretion processes as recently argued by e.g. Ristea et al. (2022).
An important caveat and consideration is that we use luminosityweighted ages of mixed stellar populations, which are inherently uncertain and may over-represent the brighter younger populations as the mass-to-light ratio plateaus at older ages (e.g., Bruzual & Charlot 2003), hence leading to a higher scatter in age.
It may be possible to compare the local dynamical differences between gas and various stellar populations using orbital dynamical modelling for a subset of the galaxies in our sample (e.g., Poci et al. 2019;Santucci et al. 2022) and test the impact of using lightweighted ages on our conclusions, however this is beyond the scope of this work.
Comparing the dynamics of the ionised gas and stars locally (i.e., on a spaxel-by-spaxel basis) ensures that radial differences in asymmetric drift are implicitly accounted for (Shetty et al. 2020). The increased scatter in Δ R e at older ages is consistent with the MaNGA results of Shetty et al. (2020) that the asymmetric drift of stellar populations beyond 1.5 Gyr varies greatly, while the asymmetric drift of stellar populations younger than 1.5 Gyr is fairly stable with radius at or above 0 km s −1 . This is again consistent with and likely reflects a broad range of possible galaxy formation and evolutionary pathways that diverge slowly over time.
The measured local offsets (median Δ R e = +20 ± 1 km s −1 and median Δ R e = −40 ± 1 km s −1 ) are of comparable order of magnitude to those found in Quirk et al. (2019) for different gas phases in Andromeda. Namely, Quirk et al. (2019) find that the H and CO gas rotates faster by up to 18 km s −1 than the younger asymptotic giant branch stars (∼ 2 Gyr). The difference increases to 63 and 37 km s −1 for H and CO gas, respectively, when compared with red giant branch stars (∼ 4 Gyr).
The finding that the dispersion of the ionised gas and stars (median Δ R e is −40 km s −1 ) differs more (in absolute terms) than the velocities (median Δ R e is +20 km s −1 ; see Figs. 4 and 5) may be a result of inherent differences in the collisional properties of gas and stars and the impact of inflow and/or non star-forming feedback activity mainly on gas dynamics/heating (as described in e.g., Wellons et al. 2020). Part of this difference may also be explained by beam smearing effects (see Appendix A). Also using SAMI data, Oh et al. (2022) found that gas emission not directly associated with star-formation (e.g., AGN/LINER) is associated with higher gas dispersion. This finding could explain why, in galaxies with older stellar populations (i.e., those more likely to harbour AGN or LINER-like emission at the expense of star formation), the measured ( / ) gas is lower (see lower panel of Fig. 4). If the reason that the ionised gas dynamics are hotter in galaxies with older stellar populations than younger ones is because the gas is, on average, ionised by different sources, they should also exhibit higher [N ]/H , which is confirmed in the bottom panel of Fig. 6. The higher gas dispersion of the non star-forming gas could be associated with outflows in the case of AGN and shocks, but could also be associated with the diffuse ionised gas, that shows LINER-like emission and has both a higher dispersion and slower velocity structure when extra-planar (see eg, Levy et al. 2019;Belfiore et al. 2022;Micheva et al. 2022).
Another important dimension of this work is the observed difference in the velocity gradients (∇ ) of the ionised gas and stars. Our results indicate that in most galaxies, the stellar velocity gradient is steeper than the gas velocity gradient in those systems. This is true at nearly all masses, except in high mass galaxies where it is the ionised gas gradient that is steeper. An important caveat of using ∇ as defined here is that the "physical" pixel scale (i.e. kpc per pixel) changes with redshift and hence across the sample and this is not directly taken into account. While ∇ gas has been used as a beam smearing proxy (Oh et al. 2022;Varidel et al. 2016), Fig. 7 shows that there are statistically significant differences in the gradient of gas and stars that also reflect differences in the dynamics of gas and stars with stellar age and stellar mass discussed previously.
In Appendix A, we investigate the possible impact of beam smearing on our results using two separate proxies. Oh et al. (2022) suggested that gas velocity dispersion in SAMI is more impacted by beam smearing than stellar dispersion owing to steep gas velocity gradients using a different sample selection and Δ parameter definition. In the appendix, we argue that our stringent selection for the best resolved targets and different parameter definitions mitigate the impact of beam smearing and discuss that some of the trends seen with various beam smearing proxies are at least partly attributed to the real physical effects discussed.
As a sanity check, we also investigate possible trends with the uncertainties in the kinematics scale with age or stellar mass (not shown). As expected, we do find a shallow trend with stellar mass, but this trend does not explain the magnitude of the scatter shown in e.g., Fig. 4, and there is no trend of uncertainty with age.
SUMMARY AND CONCLUSIONS
In this work, we explore the difference in local and global dynamics of the ionised gas and stars. We select a sample of 188 SAMI galaxies with high quality kinematic maps and optimal spatial resolution. Local differences ( Δ and Δ ) are computed for a 1 elliptical annulus on a spaxel-by-spaxel basis, while global differences (Δ( / )) are computed using flux weighting within a 1 aperture.
We find that our findings are broadly consistent with observational and theoretical expectations from asymmetric drift (e.g., Quirk & Patel 2020) and detailed modelling of the stellar dynamics and populations in local galaxies (e.g., Poci et al. 2019Poci et al. , 2021.
Our main conclusions are summarised as follows:
• The local dynamics of the ionised gas better mimic those of the stars in galaxies with younger light-weighted stellar population ages or lower mass than in older and higher mass systems. This is consistent with the dynamics of ionised gas and stars being initially coupled (Figs. 4, 5).
• The two-to three-fold increase in scatter in local dynamical differences between gas and stars and the increased divergence of the gas and stellar velocity gradients with stellar mass and age are consistent with dynamical heating playing a key role and resulting from a broad range of formation and assembly histories among galaxies (Figs. 4, 7).
• The global dynamics of the ionised gas are typically colder than those of the stars in the star-formation dominated galaxies. In galaxies with AGN/LINER-like emission, the global dynamics are more similar (Figs. 4, 6).
• Older galaxies have hotter stellar and ionised gas dynamics on average than younger galaxies, and we argue that this difference is consistent with being driven by different merging and/or accretion histories and/or ionising mechanisms (Fig. 6).
• In absolute terms, the median local difference in the velocity dispersion of the ionised gas and stars is greater than the difference in velocity regardless of stellar age (i.e., | Δ R e ,median | − | Δ R e ,median | = 20 km s −1 ; Fig. 5). This may reflect inherent differences in the collisional properties of gas and stars or the impact of turbulence, inflow and/or non star-forming feedback on gas dynamics.
More generally, while younger stellar populations may initially better reflect the dynamical properties of the gas from which they formed, our findings suggest there would be inherent risks associated with comparing ionised gas kinematics at high redshifts with the stellar dynamics of local systems to infer the dynamical evolution of (now) quiescent galaxies. Considering sufficiently large samples and limiting the investigation to star-forming ionised gas can help mitigate the inherent scatter and counfounding factors identified in this work.
Galaxy Survey website is http://sami-survey.org/. The SAMI Galaxy Survey is supported by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013, the Australian Research Council Centre of Excellence for All-sky Astrophysics (CAASTRO), through project number CE110001020, and other participating institutions.
Based on data acquired at the Anglo-Australian Telescope under programs A/2013B/012 and A/2016B/16. We acknowledge the traditional owners of the land on which the AAT stands, the Gamilaraay people, and pay our respects to elders past and present.
GAMA is a joint European-Australasian project based around a spectroscopic campaign using the Anglo-Australian Telescope. The GAMA input catalogue is based on data taken from the Sloan Digital Sky Survey and the UKIRT Infrared Deep Sky Survey. Complementary imaging of the GAMA regions is being obtained by a number of independent survey programmes including GALEX MIS, VST KiDS, VISTA VIKING, WISE, Herschel-ATLAS, GMRT and ASKAP providing UV to radio coverage. GAMA is funded by the STFC (UK), the ARC (Australia), the AAO, and the participating institutions. The GAMA website is http://www.gama-survey.org/.
This work makes use of colour scales chosen from van der Velden (2020).
DATA AVAILABILITY
All SAMI DR3 data (Croom et al. 2021) . Kinematic maps used for the ionised gas are the "SAMI DR3 1-component ionised gas velocity map" and "SAMI DR3 1-component ionised gas dispersion map". Stellar kinematic maps used are "SAMI DR3 Stellar Velocity map (2 moments) default cube" and "SAMI DR3 Stellar Velocity dispersion map (2 moments) default cube". Stellar ages were contributed by co-author S. Vaughan based on the publicly available SAMI DR3 spectra as described in §2.
APPENDIX A: BEAM SMEARING
In this section, we explore the possible impact of beam smearing (or atmospheric seeing) on our results and conclusions. We investigate two separate proxies for the impact of beam smearing. First, we look for possible trends between the spatial resolution defined as the ratio of the size of the galaxy and the atmospheric seeing ( /HWHM). In Fig. A1, we find a negative trend between Δ and the spatial resolution ( /HWHM) and a weak trend with (Δ( / ) /( / ) stars,R e ). Importantly, these trends go contrary to what would be expected should beam smearing be the driver i.e., the dynamical differences between gas and stars are more pronounced in galaxies with higher and thus more favourable /HWHM. We suggest that given our strict selection, these trends are instead dominated by real correlations between the physical size of galaxies ( ) and dynamical differences between ionised gas and stars.
Finally, following Varidel et al. (2016) and Oh et al. (2022), we look for trends between the gas and stellar velocity gradients and our local and global kinematic measurements. Fig. A2 shows significant trends in Δ with velocity gradients, which can be attributed to real differences between the stellar and ionised gas velocity gradients (see Fig. 7). The correlation between Δ and velocity gradients is somewhat weaker, and less pronounced than those found in Oh 2022) for the dispersion ratio. This apparent discrepancy can be attributed to our markedly different choice of parameter definition and our more stringent sample selection. Given the results of previous tests in this Appendix, we instead attribute the bulk of the (admittedly sometimes weak) trends between dynamical differences and velocity gradients to real physical differences in the dynamics of the gas and stars already discussed in this work. The trend in Δ with gas velocity gradient is more pronounced and statistically significant than that of Δ . Trends with stellar velocity gradient are also less significant than those with gas velocity gradient. We argue these trends are at least partly due to real physical phenomena and cannot be entirely ascribed to beam smearing (see text).
Figure 1
1Figure 1. Stellar mass (log 10 ( * / Sun )), left), effective radius ( e , middle-left), specific star formation rate (sSFR, middle-right) and visual morphological classification (right, where E: elliptical, S0: lenticular, eSp: early spiral and lSp: late spiral) distributions for the final sample (yellow), galaxies with both gas and stellar kinematic maps out to 1 effective circularised radius but before the seeing and stellar mass cut are applied (orange) and the full SAMI sample (white). About ∼ 5 percent of galaxies in full SAMI sample have uncertain morphological classifications and are not shown in the right panel. Due to the requirement of high quality, large spatial extent and good resolution of gas and stellar kinematic maps, the sample is necessarily biased towards star forming galaxies, intermediate morphological types (eSp) and higher effective radii.
Figure 2 .
2Figure 2. Star formation rate as a function of the stellar mass for the final sample (yellow), galaxies with both gas and stellar kinematic maps out to 1 effective circularised radius but before the seeing and stellar mass cut are applied (orange) and the full SAMI sample (grey). The requirement for both gas and stellar kinematics favours galaxies with significant star formation, although there are a handful of selected galaxies that lie significantly (> 2 standard deviations, dashed purple line) below the star forming main sequence defined by late-type spirals (purple dots) fitted as per Medling et al. (2018, solid purple line). Also shown as a red dashed line is the star forming main sequence as per Equation 3.
[N ],[O ],[S ], and[O ]. We use a multiplicative Legendre polynomial of the order of 10 to correct for small differences in the shape of the observed and template spectra.The final stellar population parameters for each spectrum are calculated following McDermid et al. (2015). As described in D'Eugenio et al. (2021) and C21, photometric values such as effective radii, position angles and ellipticities are obtained by applying the Multi-Gaussian Expansion technique (MGE Emsellem et al. 1994) on -band Sloan Digital Sky Survey and Very Large Telescope Survey Telescope images. Stellar masses are estimated following equation 8 of Taylor et al. (2011) using -band absolute magnitudes and ( − ) colours.As described in(Cortese et al. 2016), morphologies for all SAMI galaxies are based on careful compilation of visual classifications from multiple team members. A small fraction have uncertain visual morphological types (∼ 5 percent).
Figure 3 .
3Ionised gas (top) and stellar (bottom) velocity (left), dispersion (middle) and ( / ) maps for a "typical" galaxy in our final sample: SAMI106717. This target has Δ( / ) Re = 0.88 ± 0.08, close to the median value for our final sample (i.e. Δ( / ) Re ,median = 0.82 ± 0.04
Figure 4 .
4Mean difference between the absolute gas and stellar velocity ( Δ , top) and dispersion ( Δ , middle) as a function of stellar age (left) and stellar mass (right). The bottom panels show global Δ( / ) Re as a function of stellar age (left) and stellar mass (right). Data points are colour-coded by stellar mass (left) and stellar age(right). Data points with yellow outlines in the top panel show systems with kinematically misaligned gas and stars. Red lines show median -axis values in each panel. Median uncertainties on individual measurements are shown in red in the top left of each panel. Age-binned (left) and stellar mass-binned (right) mean values are shown as purple symbols with errorbars corresponding to the standard deviation of each bin (values recorded in
Figure 6 .
6Following Varidel et al. (2016) and Oh et al. (2022), we compute the projected mean velocity gradient within an elliptical LHS: Comparison of gas and stellar ( / ) with points colourcoded by stellar population age (top), SFR (second from top), stellar mass (second from bottom) and [N ]/H (bottom)
Figure 7
7Figure 7. Top: comparison of the gas (∇ gas,Re ) and stellar (∇ stars,Re ) velocity gradients. Dashed line is the one-to-one. Middle: ratio of the stars and gas velocity gradients as a function of stellar age. Bottom: same as middle, but as a function of stellar mass. In the upper two panels, symbols are colour coded by stellar mass. Large purple symbols represent mean values binned along the x-axis with errorbars representing the standard deviation in each bin. The Spearman correlation coefficient and the corresponding -value stated in the top right of the bottom two panels indicate significant anti-correlations between stellar mass and the ratio of the velocity gradients. Velocity gradients best agree in low mass and younger systems. In high mass and older systems, the gas velocity gradient is on average steeper than the stellar velocity gradient. Interpretation is given in §5.
.
used in this work are publicly available through Data Central (datacentral.org.au). SAMI DR3 data tables used in this work are: The H flux and signalto-noise maps are based on "SAMI DR3 1-component line emission map: H ". The [N ] flux maps are taken from "SAMI DR3 1-component line emission map: [NII](6583Å)"
Figure A1 .
A1Average local difference in velocity (top) and velocity dispersion (middle), and average global Δ( / ) (normalised to the stellar ( / ), bottom) between the ionised gas and stars as a function spatial resolution /HWHM. Data points with yellow outlines in the top panel show systems with kinematically misaligned gas and stars. /HWHM-binned mean values are shown as purple symbols with errorbars corresponding to the standard deviation of each bin. Dashed line shows parity. Data points are colour coded by stellar age. The Spearman correlation coefficient and the corresponding -value are stated in the top right of each panel. While these show trends (albeit only a weak on in the top panel), these go contrary to expectations if they were caused by beam smearing (see text). et al. (
Figure A2 .
A2Local velocity (top) and velocity dispersion (bottom) as a function of stellar (left) and gas velocity gradient (right). Data points with yellow outlines in the top panels show systems with kinematically misaligned gas and stars. Stellar (left) and gas (right) velocity gradient-binned mean values are shown as purple symbols with errorbars corresponding to the standard deviation of each bin. The Spearman correlation coefficient and the corresponding -value are stated in the top right of each panel.
1 The median FWHM is 1.8 arcsec in our final sample. V gas (km/s) row(gasvelmap[[1]]$imDat)[, 1] col(gasvelmap[[1]]$imDat)[1, ] row(gasdispmap[[1]]$imDat)[, 1] col(gasdispmap[[1]]$imDat)[1, ]5 arcsec
− 150
− 100
− 50
0
50
100
150
σ gas (km/s)
). Red ellipses show the circularised effective radius aperture centered on the SAMI cube. In this example, the ionised gas exhibits higher rotational support than the stars. A scale is provided for reference on the top-left panel. North is up and East is left.Table 1. Age and stellar mass binned mean and standard deviation Δ and Δ values as shown inFig. 4Mean Age [Gyr]
1
2
3
5
9
Mean Δ
[km s −1 ]
21
18
22
27
43
std. dev. Δ
[km s −1 ]
11
6
11
29
34
Mean Δ
[km s −1 ]
-35
-37
-40
-47
-53
std. dev. Δ
[km s −1 ]
12
12
18
20
30
Mean log 10 ( * / Sun )
9.9
10.3 10.4 10.7 11.0
Mean Δ
[km s −1 ]
23
18
26
28
39
std. dev. Δ
[km s −1 ]
16
7
20
20
37
Mean Δ
[km s −1 ]
-36
-39
-38
-47
-54
std. dev. Δ
[km s −1 ]
15
14
12
23
28
C. Foster et al.
MNRAS 000, 1-11(2023)
This paper has been typeset from a T E X/L A T E X file prepared by the author.MNRAS 000, 1-11(2023)
Part of this research was conducted by the Australian Research Council Centre of Excellence for All Sky Astrophysics in 3 Dimensions (ASTRO 3D), through project number CE170100013.The SAMI Galaxy Survey is based on observations made at the Anglo-Australian Telescope. The Sydney-AAO Multi-object Integral field spectrograph (SAMI) was developed jointly by the University of Sydney and the Australian Astronomical Observatory. The SAMI input catalogue is based on data taken from the Sloan Digital Sky Survey, the GAMA Survey and the VST ATLAS Survey. The SAMI
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| {'fraction_non_alphanumeric': 0.0609191102414319, 'fraction_numerical': 0.06355212799631665, 'mean_word_length': 4.124834095266185, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 1, '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': 'We explore local and global dynamical differences between the kinematics of ionised gas and stars in a sample of galaxies from Data Release 3 of the SAMI Galaxy Survey. We find better agreement between local (i.e., comparing on a spaxel-to-spaxel basis) velocities and dispersion of gas and stars in younger systems as with previous work on the asymmetric drift in galaxies, suggesting that the dynamics of stars and ionised gas are initially coupled. The intrinsic scatter around the velocity and dispersion relations increases with increasing stellar age and mass, suggesting that subsequent mechanisms such as internal processes, divergent star formation and assembly histories also play a role in setting and altering the dynamics of galaxies. The global (flux-weighted) dynamical support of older galaxies is hotter than in younger systems. We find that the ionised gas in galaxies is almost always dynamically colder than the stars with a steeper velocity gradient. In absolute terms, the local difference in velocity dispersion is more pronounced than the local difference in velocity, possibly reflecting inherent differences in the impact of turbulence, inflow and/or feedback on gas compared to stars. We suggest how these findings may be taken into account when comparing high and low redshift galaxy samples to infer dynamical evolution.', 'arxivid': '2302.06712', 'author': ['Caroline Foster \nSchool of Physics\nUniversity of New South Wales\n2052SydneyNSWAustralia\n\nSydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia\n\nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n', 'Sam Vaughan \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n\nAstronomy, Astrophysics and Astrophotonics Research Centre\nMacquarie University\n2109SydneyNSWAustralia\n\nSchool of Mathematical and Physical Sciences\nMacquarie University\n2109NSWAustralia\n\nCentre for Astrophysics and Supercomputing\nSchool of Science\nSwinburne University of Technology\n3122HawthornVICAustralia\n', 'Amelia Fraser-Mckelvie \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n\nInternational Centre for Radio Astronomy Research\nThe University of Western Australia\n35 Stirling Hwy6009CrawleyWAAustralia\n', 'Sarah Brough \nSchool of Physics\nUniversity of New South Wales\n2052SydneyNSWAustralia\n\nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n', 'Julia J Bryant \nSydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia\n\nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n\nAstralis-USydney\nSchool of Physics\nUniversity of Sydney\n2006NSWAustralia\n', 'Scott M Croom \nSydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia\n\nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n', 'Francesco D ' Eugenio \nKavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUnited Kingdom\n\nCavendish Laboratory -Astrophysics Group\nUniversity of Cambridge\n19 JJ Thomson AvenueCB3 0HECambridgeUnited Kingdom\n', 'Brent Groves \nInternational Centre for Radio Astronomy Research\nThe University of Western Australia\n35 Stirling Hwy6009CrawleyWAAustralia\n', 'Iraklis S Konstantopoulos \nIndependent scholar Wellington\nNew Zealand 12 Research School of Astronomy and Astrophysics\nAustralian National University\nCotter Road, Weston Creek, ACT 2611Australia\n', 'Ángel R López-Sánchez \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n\nAstronomy, Astrophysics and Astrophotonics Research Centre\nMacquarie University\n2109SydneyNSWAustralia\n\nSchool of Mathematical and Physical Sciences\nMacquarie University\n2109NSWAustralia\n', 'Sree Oh \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n\nDepartment of Astronomy and Yonsei University Observatory\nYonsei University\n03722SeoulRepublic of Korea\n', 'Matt S Owers \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n\nAstronomy, Astrophysics and Astrophotonics Research Centre\nMacquarie University\n2109SydneyNSWAustralia\n\nSchool of Mathematical and Physical Sciences\nMacquarie University\n2109NSWAustralia\n', 'Sarah M Sweet \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n\nSchool of Mathematics and Physics\nUniversity of Queensland\n4072BrisbaneQLDAustralia\n', 'Jesse Van De Sande \nSydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia\n\nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n', 'Emily Wisnioski \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n', 'Sukyoung K Yi \nDepartment of Astronomy and Yonsei University Observatory\nYonsei University\n03722SeoulRepublic of Korea\n', 'Henry R M Zovaro \nARC Centre of Excellence for All Sky Astrophysics in\n\n\nDimensions (ASTRO 3D\n\n'], 'authoraffiliation': ['School of Physics\nUniversity of New South Wales\n2052SydneyNSWAustralia', 'Sydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Astronomy, Astrophysics and Astrophotonics Research Centre\nMacquarie University\n2109SydneyNSWAustralia', 'School of Mathematical and Physical Sciences\nMacquarie University\n2109NSWAustralia', 'Centre for Astrophysics and Supercomputing\nSchool of Science\nSwinburne University of Technology\n3122HawthornVICAustralia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'International Centre for Radio Astronomy Research\nThe University of Western Australia\n35 Stirling Hwy6009CrawleyWAAustralia', 'School of Physics\nUniversity of New South Wales\n2052SydneyNSWAustralia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Sydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Astralis-USydney\nSchool of Physics\nUniversity of Sydney\n2006NSWAustralia', 'Sydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Kavli Institute for Cosmology\nUniversity of Cambridge\nMadingley RoadCB3 0HACambridgeUnited Kingdom', 'Cavendish Laboratory -Astrophysics Group\nUniversity of Cambridge\n19 JJ Thomson AvenueCB3 0HECambridgeUnited Kingdom', 'International Centre for Radio Astronomy Research\nThe University of Western Australia\n35 Stirling Hwy6009CrawleyWAAustralia', 'Independent scholar Wellington\nNew Zealand 12 Research School of Astronomy and Astrophysics\nAustralian National University\nCotter Road, Weston Creek, ACT 2611Australia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Astronomy, Astrophysics and Astrophotonics Research Centre\nMacquarie University\n2109SydneyNSWAustralia', 'School of Mathematical and Physical Sciences\nMacquarie University\n2109NSWAustralia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Department of Astronomy and Yonsei University Observatory\nYonsei University\n03722SeoulRepublic of Korea', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Astronomy, Astrophysics and Astrophotonics Research Centre\nMacquarie University\n2109SydneyNSWAustralia', 'School of Mathematical and Physical Sciences\nMacquarie University\n2109NSWAustralia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'School of Mathematics and Physics\nUniversity of Queensland\n4072BrisbaneQLDAustralia', 'Sydney Institute for Astronomy\nSchool of Physics\nThe University of Sydney\nNSW\nA28, 2006Australia', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n', 'Department of Astronomy and Yonsei University Observatory\nYonsei University\n03722SeoulRepublic of Korea', 'ARC Centre of Excellence for All Sky Astrophysics in\n', 'Dimensions (ASTRO 3D\n'], 'corpusid': 256846821, 'doi': '10.1093/mnras/stad487', 'github_urls': [], 'n_tokens_mistral': 23650, 'n_tokens_neox': 18834, 'n_words': 10886, 'pdfsha': 'c2705673da724296c4d81e03315cf77683c0bc64', 'pdfurls': ['https://export.arxiv.org/pdf/2302.06712v1.pdf'], 'title': ['The SAMI Survey: Evidence for dynamical coupling of ionised gas and young stellar populations', 'The SAMI Survey: Evidence for dynamical coupling of ionised gas and young stellar populations'], 'venue': ['MNRAS']} |
arxiv |
**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Surprising Metallicity of a Newly Discovered M79 Post-AGB Star
29 Oct 2009
Timur Şahin
Department of Astronomy and The
University of Texas
78712AustinTXUSA
David L Lambert
Department of Astronomy and The
University of Texas
78712AustinTXUSA
W J Mcdonald Observatory
Department of Astronomy and The
University of Texas
78712AustinTXUSA
**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Surprising Metallicity of a Newly Discovered M79 Post-AGB Star
29 Oct 2009
A detailed chemical composition analysis based on a high-resolution (R ≃ 35, 000) CCD spectrum is presented for a newly discovered post-AGB star in the globular cluster M79 for the first time. The elemental abundance results of M79 Post-AGB star are found to be [C/F e] ≃-0.7, [O/F e] =+1.4, [α − process/F e] ≃0.5, and [s − process/F e] ≃-0.1. The surprising result is that the iron abundance of the star is apparently about 0.6 dex less than that of the cluster's red giants as reported by published studies including a recent highresolution spectroscopic analysis by Carretta and colleagues.
Introduction
In this study, published recently in full by Şahin & Lambert (2009), we report on an abundance analysis of the A-type m79 PAGB star discovered by Siegel & Bond (2009, in preparation) in the globular cluster M79 and compare its composition to that of the cluster's red giants. The initial mass of this star must have been slightly in excess of the mass of stars now at the main sequence turn-off, say, M ≃ 0.8M ⊙ . The star's composition may be referenced to that of the cluster's red giant stars for which abundance analyses have been reported. Comparison of abundances for the PAGB and RGB stars may reveal changes imposed by the evolution beyond the RGB; such changes are not necessarily attributable exclusively to internal nucleosynthesis and dredge-up. It was in the spirit of comparing the compositions of the PAGB and RGB stars that we undertook our analysis. For the RGB stars, we use results kindly provided in advance of publication by Carretta (2008, private communication).
Observations
Spectra for the abundance analysis were obtained on five nights between 2008 January 15 and March 3 with the 2.7 meter Harlan J. Smith reflector and its 2dcoudé cross-dispersedéchelle spectrograph (Tull et al. 1995). Full spectral coverage is provided from 3800Å to 5700Å with incomplete but substantial coverage beyond 5700Åto 10 200Å ; the effective short and long wavelength limits are set by the useful S/N ratio. A ThAr hollow cathode lamp provided the wavelength calibration. Flat-field and bias exposures completed the calibration files. Observations were reduced using standard procedures. A section of the final spectrum is shown in Figure 1. The heliocentric radial velocity measured form the final spectrum is 211±5 km s −1 with no evidence of a variation greater than about ±7 km s −1 over the observing runs. This velocity is consistent with the cluster's velocity of +207.5 km s −1 given by Harris (1996). This agreement between the PAGB star's velocity and that of the cluster confirms a result given by Siegel and bond (2009, in preparation).
Spectral Analysis
The abundance analysis was undertaken with models drawn from the ATLAS9 grid (Kurucz 1993) and the line analysis programme MOOG (Sneden 2002). The models are line-blanketed plane-parallel atmospheres in Local Thermodynamical Equilibrium (LTE) and hydrostatic equilibrium with flux conversation. A model is defined by the parameter set; effective temperature T eff , surface gravity g, chemical composition as represented by metallicity [F e/H] and all models are computed for a microturbulence ξ = 2 km s −1 . A model defined by the parameter set is fed to MOOG except that ξ is determined from the spectrum and not set to the canonical 2 km s −1 assumed for the model atmosphere. In Şahin & Lambert (2009), we discuss several methods in an attempt to find consistent values for the T eff and log g from photometry and spectroscopy. Application of photometric and spectroscopic indicators of the atmospheric parameters for the PAGB star led to the consensus choice of T eff = 6300 K and log g=0.8. A model with these parameters (and a micro-turbulence ξ = 3.4 km s −1 ) fits not only the indicators but also the locus in the T eff versus log g plane provided by the constraint on the star's luminosity and mass. Errors on these quantities are 300 K, 0.2 (cgs), and 0.5 km s −1 respectively. Synthetic spectrum fitting results for the 4481Å magnesium triplet and resonance strontium lines at 4077Å and 4215 A are presented in Figs. 2 and 3 as representative of α-and s-process elements. Table 1 summarizes the PAGB star's composition not only for the consensus model but also for other three different model atmospheres and contrasts it with the mean composition of the RGB stars. The abundance relative to iron [X/Fe] is also presented. The standard notation is used here, i.e. [X/F e] = log(X/X SUN − log(F e/F e SUN ). The PAGB star's Fe abundance is −0.5 dex lower than that of the RGB stars. For the majority of the investigated elements, the difference in abundance log ǫ(X) in the sense (Ours − Carretta) is within the range −0.5 ± 0.3 dex, i.e., the differences are equal to −0.5 dex to within measurement uncertainties.The exceptions are O, Na, Si, and Sr. A search (see Şahin & Lambert 2009) for an explanation of the composition difference between the PAGB and RGB stars in terms of nucleosynthesis and dredge-up, dust-gas winnowing, and the first ionization potential (FIP) effect proved negative.
Results
Many determinations of the metallicity [Fe/H] of cluster red giants have given estimates near [Fe/H]= −1.6 (Standard notation is used for quantities [X] where [X]=log(X) star −log(X) ⊙ ). For example, Zinn & West (1984) give [Fe/H]= −1.69 and Kraft & Ivans (2003) give [Fe/H]= −1.64. Recently from high-resolution UVES FLAMES spectra Carretta and colleagues (2008, private communication) performed an abundance analysis for 20 elements obtaining [Fe/H]= −1.58 for a sample of ten RGB stars. The exploration through quantitative spectroscopy of the newly discovered PAGB star in the globular cluster M79 has led to an unexpected and, therefore, fascinating result: the standard LTE analysis of the star has resulted in a metallicity different from that of the RGB stars analyzed also by standard LTE techniques by Carretta. The consensus model of (6300,0.8) provides a [Fe/H] of −2.0 but the RGB analysis gives a [Fe/H] of −1.5. The star does not show s-process enhancement. Oxygen and α-process elements are enhanced. Abundances relative to iron appear to be the same for the post-AGB star and the red giants for the 15 common elements. It is suggested that the explanation for the lower abundances of the post-AGB star may be that its atmospheric structure differs from that of a classical atmosphere; the temperature gradient may be flatter than predicted by a classical atmosphere.
Figure 1 . 3 Figure 2 .
132The spectrum for the PAGB star over the wavelength regions between 4467-4500Å (upper panel) and 4500-4530Å (lower panel). Selected lines are identified. Observed and synthetic spectra around the 4481Å Mg ii triplet lines.
Figure 3 .
3The observed spectrum around the Sr ii 4077.7Å and 4215.5Å resonance lines.
ii 0.89, −0.16 ≤ 0.7, ≤ +0.1 ≤ 0.9, ≤ +0.2 ≤ 1.1, ≤ +0.2 ≤ 1.4, ≤ +0.
Table 1 .
1Abundances of the observed species for M79 PAGB star are presented for four different model atmospheres. Also listed are abundances of the RGB stars in the same globular cluster analyzed by Carretta et al. Sr ii ..., ...(2008,
Acknowledgments. This research has been supported in part by the grant F-634 from the Robert A. Welch Foundation of Houston, Texas.
. W E Harris, AJ. 1121487Harris, W. E., 1996, AJ, 112, 1487
. R P Kraft, I I Ivans, PASP. 115143Kraft, R. P., Ivans, I. I., 2003, PASP, 115, 143
MOOG An LTE Stellar Line Analysis Program Şahin T. C Sneden, D L Lambert, MNRAS. 3981730Sneden, C., 2002, MOOG An LTE Stellar Line Analysis Program Şahin T., Lambert D. L. 2009, MNRAS, 398, 1730
. R G Tull, P J Macqueen, C Sneden, D L Lambert, PASP. 107251Tull, R. G., MacQueen, P. J., Sneden, C., Lambert, D. L., 1995, PASP, 107, 251
. R Zinn, M J West, ApJS. 5545Zinn, R., West, M. J., 1984, ApJS, 55, 45
| {'fraction_non_alphanumeric': 0.05533877261440227, 'fraction_numerical': 0.04422371999527019, 'mean_word_length': 4.224212476837554, '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': "A detailed chemical composition analysis based on a high-resolution (R ≃ 35, 000) CCD spectrum is presented for a newly discovered post-AGB star in the globular cluster M79 for the first time. The elemental abundance results of M79 Post-AGB star are found to be [C/F e] ≃-0.7, [O/F e] =+1.4, [α − process/F e] ≃0.5, and [s − process/F e] ≃-0.1. The surprising result is that the iron abundance of the star is apparently about 0.6 dex less than that of the cluster's red giants as reported by published studies including a recent highresolution spectroscopic analysis by Carretta and colleagues.", 'arxivid': '0910.5567', 'author': ['Timur Şahin \nDepartment of Astronomy and The\nUniversity of Texas\n78712AustinTXUSA\n', 'David L Lambert \nDepartment of Astronomy and The\nUniversity of Texas\n78712AustinTXUSA\n', 'W J Mcdonald Observatory \nDepartment of Astronomy and The\nUniversity of Texas\n78712AustinTXUSA\n'], 'authoraffiliation': ['Department of Astronomy and The\nUniversity of Texas\n78712AustinTXUSA', 'Department of Astronomy and The\nUniversity of Texas\n78712AustinTXUSA', 'Department of Astronomy and The\nUniversity of Texas\n78712AustinTXUSA'], 'corpusid': 118711298, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 2658, 'n_tokens_neox': 2214, 'n_words': 1380, 'pdfsha': '0d7d5d6f1a3dc9fbf5029e673c39f59224e62c06', 'pdfurls': ['https://arxiv.org/pdf/0910.5567v1.pdf'], 'title': ['**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Surprising Metallicity of a Newly Discovered M79 Post-AGB Star', '**FULL TITLE** ASP Conference Series, Vol. **VOLUME**, **YEAR OF PUBLICATION** **NAMES OF EDITORS** Surprising Metallicity of a Newly Discovered M79 Post-AGB Star'], 'venue': []} |
arxiv |
System Size and Energy Dependence of Elliptic Flow
6 Feb 2006
Alice C Mignerey
Department of Chemistry and Biochemistry
University of Maryland
20742College ParkMDUSA
System Size and Energy Dependence of Elliptic Flow
6 Feb 2006for the PHOBOS Collaboration 1
The elliptic flow v 2 is presented for the Cu+Cu collisions at √ s NN = 62.4 and 200 GeV, as a function of pseudorapidity. Comparison to results for the Au+Au collisions at the same energies shows a reduction of about 20% in the flow observed for a centrality selection of 0-40%. The centrality dependent flow, expressed as a function of the number of participants N part , is compared for the Cu+Cu and Au+Au systems using two definitions of eccentricity, the standard definition ε standard and a participant eccentricity ε part . The v 2 / ε part as a function of N part , for the Au+Au and Cu+Cu collisions are consistent within errors, while v 2 / ε standard gives unrealistically large values for Cu+Cu, especially for central collision.
INTRODUCTION
The azimuthal correlations of produced particles have proven to be a sensitive measure of the initial conditions and subsequent dynamics in relativistic heavy ion collisions. The elliptic flow v 2 , as inferred from the angular distribution of particles with respect to the reaction plane, provides important constraints on hydrodynamical descriptions about the evolution of the collision [1]. The large pseudorapidity coverage (|η| ≤ 5.4) and the near symmetric azimuthal acceptance for charged hadrons of the PHOBOS detector at RHIC make it excellent for the investigation of the systematics of the flow measurements, as a function of energy, system size, centrality and pseudorapidity. This contribution will concentrate on v 2 measurements for the Cu+Cu collisions at √ s NN =
RESULTS AND CONCLUSIONS
The pseudorapidity distributions of the elliptic flow, v 2 (η), for the Au+Au and Cu+Cu collisions over broad range of center-of-mass energies and for a centrality of 0-40% are shown in Fig. 1. The triangular shape first observed for Au+Au collisions [3] is also apparent in the Cu+Cu data [5]. The measurements of v 2 for Cu+Cu collisions is only about 20% lower than that for Au+Au results, even though the Cu+Cu system size is about a third of that of Au+Au. The measured pseudorapidity density of charged hadrons, dN ch dη , for Cu+Cu and Au+Au collisions at √ s NN = 200 GeV, is essentially the same at a given N part [6], but the elliptic flow is different due to the difference in system size. While N part of 100 corresponds to a 3-6% centrality selection for the Cu+Cu system, N part of 99 corresponds to only 35-40% central for Au+Au. This implies different initial geometrical overlaps for the two systems with the same energy density (as reflected in the near identical dN ch dη ). The measured v 2 for Au+Au and Cu+Cu collisions at the same energy, 200 GeV, expressed as a function of N part is shown in Fig. 2.
This discrepancy in elliptic flow for the two systems can be accounted for if v 2 is normalized by the eccentricity of the system. Two definitions of eccentricity have been used, based on a simple Glauber model [5,6]. The standard eccentricity, ε standard , is defined in the frame of the original impact parameter; a participant eccentricity, ε part , is obtained from the geometry of the participants that define N part , and is influenced by fluctuations in the participant positions. This is more important for the much smaller Cu+Cu system than for Au+Au, as seen in Fig. 3(a), where the mean eccentricities derived from the two approaches are compared. The v 2 normalized by the two eccentricities is shown in Fig. 3(b). While v 2 / ε part is nearly identical for the Cu+Cu and Au+Au systems, v 2 / ε standard gives unrealistically large values for the most central events. It is clear that a better understanding of the eccentricity relevant to the reaction dynamics is needed in order to meaningfully compare systems of such different sizes as Cu+Cu and Au+Au.
FIGURE 1 .
1Measured v 2 (η) from Au+Au and Cu+Cu collisions at RHIC energies and for the centrality range of 0-40%. Only the 1σ statistical error bars are shown for clarity. The full systematic errors can be found in Ref.[3] for the Au+Au data and Ref.[5] for the Cu+Cu data.62.4 and 200GeV and a comparison to previously reported Au+Au results. Details of the PHOBOS detector and the experimental technique used to extract the flow can be found in references[2,3,4].
FIGURE 2 .FIGURE 3 .
23The elliptic flow v 2 as a function of N part measured at midrapidity (|η| < 1), for Au+Au and Cu+Cu at √ s NN = 200 GeV. The Au+Au and Cu+Cu data are from Ref. [4] and [5], respectively. The error bars represent the 1σ statistical errors and the boxes are 90% C.L. systematic errors. Panel a): Mean standard and participant eccentricities, calculated using a Glauber model and panel b): elliptic flow v 2 normalized by the two eccentricities, for Au+Au and Cu+Cu collisions at √ s NN = 200 GeV, as a function of the number of participants N part . Only the 1σ statistical errors of v 2 are reflected in the error bars in panel b).
ACKNOWLEDGMENTS
T Hirano, arXiv:nucl-th/0601006Proceedings of PANIC (2005). PANIC (2005)T. Hirano, Proceedings of PANIC (2005), arXiv:nucl-th/0601006.
. B B Back, Nucl. Instrum. Methods A. 499603B. B. Back et al., Nucl. Instrum. Methods A 499, 603 (2003).
. B B Back, Phys. Rev. Lett. 94122303B. B. Back et al., Phys. Rev. Lett. 94, 122303 (2005).
. B B Back, Phys. Rev. C. 7251901B. B. Back et al., Phys. Rev. C 72 051901R (2005).
S Manly, PHOBOS CollaborationarXiv:nucl-ex/0510031Proceedings of QM (2005). QM (2005)S. Manly (for the PHOBOS Collaboration), Proceedings of QM (2005), arXiv:nucl-ex/0510031.
G Roland, PHOBOS CollaborationarXiv:nucl-ex/0510042Proceedings of QM (2005). QM (2005)G. Roland (for the PHOBOS Collaboration), Proceedings of QM (2005), arXiv:nucl-ex/0510042.
| {'fraction_non_alphanumeric': 0.05083319306514055, 'fraction_numerical': 0.041070526847332096, 'mean_word_length': 4.263064658990257, 'pattern_counts': {'":': 0, '<': 1, '<?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 elliptic flow v 2 is presented for the Cu+Cu collisions at √ s NN = 62.4 and 200 GeV, as a function of pseudorapidity. Comparison to results for the Au+Au collisions at the same energies shows a reduction of about 20% in the flow observed for a centrality selection of 0-40%. The centrality dependent flow, expressed as a function of the number of participants N part , is compared for the Cu+Cu and Au+Au systems using two definitions of eccentricity, the standard definition ε standard and a participant eccentricity ε part . The v 2 / ε part as a function of N part , for the Au+Au and Cu+Cu collisions are consistent within errors, while v 2 / ε standard gives unrealistically large values for Cu+Cu, especially for central collision.', 'arxivid': 'nucl-ex/0602008', 'author': ['Alice C Mignerey \nDepartment of Chemistry and Biochemistry\nUniversity of Maryland\n20742College ParkMDUSA\n'], 'authoraffiliation': ['Department of Chemistry and Biochemistry\nUniversity of Maryland\n20742College ParkMDUSA'], 'corpusid': 11573577, 'doi': '10.1063/1.2220212', 'github_urls': [], 'n_tokens_mistral': 1882, 'n_tokens_neox': 1522, 'n_words': 974, 'pdfsha': '2d4b9418b4c357a7bc4ed914bba3ef203c9cfddd', 'pdfurls': ['https://export.arxiv.org/pdf/nucl-ex/0602008v1.pdf'], 'title': ['System Size and Energy Dependence of Elliptic Flow', 'System Size and Energy Dependence of Elliptic Flow'], 'venue': []} |
arxiv |
A Moving Window Based Approach to Multi-scan Multi-Target Tracking
8 Oct 2022
Diluka Moratuwage *[email protected]
School of Electrical Engineering, Computing and Mathematical Sciences
Curtin University
Australia
Changbeom Shim †[email protected]
School of Electrical Engineering, Computing and Mathematical Sciences
Curtin University
Australia
Yuthika Punchihewa
School of Electrical Engineering, Computing and Mathematical Sciences
Curtin University
Australia
A Moving Window Based Approach to Multi-scan Multi-Target Tracking
8 Oct 2022
Multi-target state estimation refers to estimating the number of targets and their trajectories in a surveillance area using measurements contaminated with noise and clutter. In the Bayesian paradigm, the most common approach to multitarget estimation is by recursively propagating the multi-target filtering density, updating it with current measurements set at each timestep. In comparison, multi-target smoothing uses all measurements up to current timestep and recursively propagates the entire history of multi-target state using the multi-target posterior density. The recent Generalized Labeled Multi-Bernoulli (GLMB) smoother is an analytic recursion that propagate the labeled multi-object posterior by recursively updating labels to measurement association maps from the beginning to current timestep. In this paper, we propose a moving window based solution for multi-target tracking using the GLMB smoother, so that only those association maps in a window (consisting of latest maps) get updated, resulting in an efficient approximate solution suitable for practical implementations.
I. INTRODUCTION
In the Bayesian approach to single-target tracking, the current state (represented by a vector) of the target conditioned on the history of measurements is recursively propagated in time via the filtering density [1], [4]. On the other hand, in the multi-scan approach (to single-target tracking) the entire history of the single-target state (conditioned on the history of measurements) is recursively propagated via the smoothing density [28]. Analogously, in the multi-target state estimation, multi-target state is propagated using the multi-object filtering density [29] and multi-object smoothing density [33], where the multi-target state and observations are represented as Random Finite Sets (RFSs) [18] [19].
The Generalized Labeled Multi-Bernoulli (GLMB) filter [29], [30] is an analytic solution to the multi-object filtering density that can be used to estimate multi-target state with target labels (identities). An efficient implementation of the GLMB filter has been proposed in [31] by sampling its components using a Gibbs sampler [7]. This approach was demonstrated to track over a millions targets from about a billion detections in relatively challenging signal scenarios [2]. Due to its versatility and efficiency, the GLMB filter has been extended to track before detect [26], [16], distributed tracking [9], [15], tracking with lineage [24], [5], as well as applied to computer vision [14], [12], [25], Doppler tracking [8], [37], field robotics [20], [21], space situational awareness [34], [35], [10], [11], multi-object sensor scheduling and path planning [3], [22], [6], [23], [36].
A recent research avenue is GLMB smoothing [33], where Gibbs sampling was used to solve multi-dimensional assignment problem in multi-scan multi-object tracking. In addition, this Gibbs sampler was combined with importance to solve the multi-dimensional assignment problem in multisensor multi-object tracking [32]. The labeled multi-object states caters for trajectories, and smoothing improves the multi-object states in the all past timesteps as the smoothing density is updated with future information. However, even with single-target smoothing, the computation cost at each timestep increases, and may result in an unaffordable computational cost for practical implementation. In the multi-target state estimation, smoothing poses and even more difficult NP-hard multi-dimensional assignment problem that keeps increasing its complexity with every timestep. Therefore, it is desirable to come up with a multi-target smoothing approach with an approximately similar computational cost per each update step.
In this paper, we propose a novel technique to propagate the multi-object smoothing density with similar computational cost at each timestep. Instead of updating the entire history of the multi-object state trajectory, we only update the latest N scans of the multi-object history. Since the target trajectories in the GLMB smoother are characterised by association maps (see Section II), we propose to link the trajectories in the posterior using a moving window based method.
Section II summarizes the background information related to multi-object state estimation, Section III explains the implementation details of the proposed moving window based multitarget tracker and presents the pseudo code for the moving window based smoother. Section IV presents the numerical results, and Section V concludes the paper.
II. BACKGROUND
We follow the same convention as in [33], and summarize the symbols, notations and definitions in this section. For a given set S, F (S) denotes the class of finite subsets of S, 1 S (·) denotes the indicator function, δ X [Y ] denotes Kroneker-δ function, where δ X [Y ] = 1 if X = Y , 0 otherwise, and f, g denotes the inner product, i.e., f (x)g(x)dx, of two functions f and g. The list of variables X m , X m+1 , ..., X n is abbreviated as X m:n , the cardinality of a finite set X is denoted as |X|, and for some function f , the product x∈X f (x) is denoted by the (single scan) multi-object exponential f X , with f ∅ = 1.
A. Mulit-object States and Trajectories
Let X denote the single object state space, L k denote the space of all labels at time k, and B k denote the space of all birth labels at time k. Then, the label space for all objects up to time k is given by L k = k s=0 B s (note that L k = L k−1 ⊎B k ), and a labeled state of an object existing at time k is given by x = (x, ℓ) ∈ X×L k , where the vector x ∈ X is its kinematic state and ℓ = (s, ι) is a unique label, s is the time of birth, and ι is a unique index to distinguish objects born at the same time. A sequence of labeled states at consecutive times (s to
t) τ = [(x s , ℓ), (x s+1 , ℓ), ..., (x t , ℓ)],(1)
with the common label ℓ and states x s , x s+1 , ..., x t ∈ X is called a trajectory.
A labeled multi-object state at time i is a finite subset X i of X×L i with distinct labels. Let L : X×L i → L i is the projection defined by L((x, ℓ)) = ℓ, and L(X i ) is the set of lables of X i . Note that, a valid X i has distinct labels and results in the distinct label indicator ∆(X i ) δ |Xi| [|L(X i )|] to be equals to one. The labeled multi-object state can also be written as X i = {τ (i) : τ ∈S}, where S is a set of trajectories defined to have kinematic states and distinct labels at each timestep, and τ (i) denotes the labeled state of trajectory τ at time i. The trajectory of the object with label ℓ ∈ ∪ k i=j L(X i ) in a given sequence X j:k of labeled multi-object states in the interval {j : k} is given by:
x (ℓ) s(ℓ):t(ℓ) = [(x (ℓ) s(ℓ) , ℓ), ..., (x (ℓ) t(ℓ) , ℓ)],(2)
where s(ℓ) and t(ℓ) are respectively the earliest and latest times the label ℓ exists on the interval {j : k}, and x
(ℓ) i = (x (ℓ) i , ℓ) denotes the element of X i with label ℓ ∈ L(X i ) with unlabeled state x (ℓ)
i . The sequnce X j:k can thus be equivalently represented by the set of all trajectories
X j:k ≡ x (ℓ) s(ℓ):t(ℓ) : ℓ ∈ k i=j L(X i ) ,(3)
of all labels in ∪ k i=j L(X i ). Furthermore, for any function h : ⊎ I⊆{j:k} T I → [0, ∞), the multi-scan multi-object exponential [33] is defined as
[h] X j:k [h] {x (ℓ) s(ℓ):t(ℓ) :ℓ∈L(X j:k )} = ℓ∈L(X j:k ) h(x (ℓ) s(ℓ):t(ℓ) ),
where for any non-negative integer n and
i 1 < i 2 < ... < i n , T {i1,i2,...,in} X × L i1 × .... × X × L in , with T ∅ = ∅.
If j = k, the multi-scan exponential reduces to single-scan multi-object exponential h Xj defined above.
B. Multi-object System Model
Given the multi-object state X k−1 at time k − 1, each object with state x k−1 = (x k−1 , ℓ k−1 ) ∈ X k−1 either survies with probability P S,k−1 (x k−1 ) and moves to a new state x k = (x k , ℓ k ) with transition den-
sity f S,k|k−1 (x k |x k−1 , ℓ k )δ ℓ k−1 [ℓ k ], or dies with probability Q S,k−1 (x k−1 ) = 1 − P S,k−1 (x k−1 ) at time k. Further,
each object with label ℓ k in birth label space B k is either born with probability P B,k (ℓ k ) and state x k with probability density f B,k (x k , ℓ k ), or not born with probability Q B,k (ℓ k ) = 1 − P B,k (ℓ k ) at time k. Thus, the multi-object state X k at time k is the superposition of the suviving states and new birth states, and in the standard multi-object dynamic model, the birth and survival sets are independent of each other, and each object moves and dies independently of each other. The multi-object transition density f k|k−1 (X k |X k−1 ) captures the multi-object dynamic model, and see [33], [29] its detailed expressions.
C. Multi-object Observation Model
Let X k and Z k be the multi-object state at time k, and the set of measurements captured by the sensor. Each object x ∈ X k either generates a measurement z ∈ Z k (with detection probability P D (x)) on the measurement space Z with likelihood g k (z|x) or miss-detected (with probability Q D (x) = 1 − P D (x)). Additionally, the sensor also produces measurement clutter, which is modeled by a Poisson RFS with intensity function κ k on Z. Conditional on X k , the detections and measurement clutter are independent, and therefore multiobject observation Z k is the superposition of them.
A map of the form γ k : L k → {−1 : |Z k |} is called an association map if it is positive 1-1 (i.e., no two distinct labels are mapped to the same positive value). If ℓ generates the
γ k (ℓ)-th measurement γ k (ℓ) > 0, if ℓ is misdetected γ k (ℓ) = 0, and if ℓ does not exist γ k (ℓ) = −1. Then, the multi-object likelihood function is g k (Z k |X k ) ∝ γ k ∈Γ k δ L(γ k ) [L(X k )][ψ (γ k •L(·)) k,Z k (·)] X k , (4) where L(γ k ) {ℓ ∈ L k : γ k (ℓ) ≥ 0}
is the set of live labels of γ k , and Γ k is the space of all association maps, γ k • L(·) = γ k (L(·)) and
ψ (i) k,{z1:m} (x) = PD (x)g k (zi|x) κ k (zi) i > 0 Q D,k (x) i = 0 .(5)
D. Trajectory Posterior of a Single Object
The GLMB posterior is written in terms of single object periors, and the corresponding association weights [33]. Thus, it is informative to take a close look at the trajectory posterior and the association weight of an object with label ℓ ∈ L k , before the presenting the GLMB posterior. Recall that s(ℓ) and t(ℓ) respectively denote the earliest and latest times on {0 : k} such that ℓ exists. Assuming that ℓ generates the sequence of measurement indices α s(ℓ):k , its trajectory posterior at time k can be in one of four possible stages: (i) new born, s(ℓ) = k; (ii) surviving, t(ℓ) = k > s(ℓ); (iii) die at time k, t(ℓ) = k−1; (iv) died before time k, t(ℓ) < k − 1. Thus, its trajectory posterior at time k is given by,
= Λ (α k ) B,k (x k ,ℓ) Λ (α k ) B,k (ℓ) , s(ℓ) = k Λ (α k ) S,k|k−1 (x k |x k−1 ,ℓ)τ (α s(ℓ):k−1 ) 0:k−1 (x s(ℓ):k−1 ,ℓ) Λ (α s(ℓ):k ) S,k|k−1 (ℓ) , t(ℓ) = k > s(ℓ) Q S,k−1 (x k−1 ,ℓ)τ (α s(ℓ):k−1 ) 0:k−1 (x s(ℓ):k−1 ,ℓ) Q (α s(ℓ):k−1 ) S,k−1 (ℓ) , t(ℓ) = k − 1 τ (α s(ℓ):t(ℓ) ) 0:t(ℓ) (x s(ℓ):t(ℓ) , ℓ), t(ℓ) < k − 1 , where τ (α s(ℓ):k−1 ) 0:k (x s(ℓ):k−1 , ℓ) is the trajectory posterior at time k − 1, Λ (α k ) B,k (x, ℓ) = ψ (α k ) k,Z k (x, ℓ)P B,k (ℓ)f B,k (x, ℓ), (7) Λ (α k ) B,k (ℓ) = Λ (α k ) B,k (x, ℓ)dx,(6)Λ (α k ) S,k|k−1 (x k |x k−1 , ℓ) = ψ (α k ) k,Z k (x k , ℓ)P S,k−1 (x k−1 , ℓ) (9) × f S,k|k−1 (x k |x k−1 , ℓ), Λ (α s(ℓ):k ) S,k|k−1 (ℓ) = τ (α s(ℓ):k−1 ) 0:k−1 (x s(ℓ):k−1 , ℓ) (10) × Λ (α k ) S,k|k−1 (x k |x k−1 , ℓ)dx s(ℓ):k , Q (α s(ℓ):k−1 ) S,k−1 (ℓ) = τ (α s(ℓ):k−1 ) 0:k−1 (x s(ℓ):k−1 , ℓ) (11) × Q S,k−1 (x k−1 , ℓ)dx s(ℓ):k−1 .(8)
The association weight of ℓ is given by
η (α s(ℓ):k ) k|k−1 (ℓ) = Λ (α k ) B,k (ℓ), s(ℓ) = k Λ (α s(ℓ):k ) S,k|k−1 (ℓ), t(ℓ) = k > s(ℓ) Q (α s(ℓ):k−1 ) S,k−1 (ℓ), t(ℓ) = k − 1 Q B,k (ℓ), ℓ ∈ B k , α k = −1 V .(12)
E. Multi-object Bayes Recursion
Given the observation history Z 1:k , the multi-object posterior π 0:k (X 0:k ) π 0:k (X 0:k |Z 1:k ) captures all information about the set of objects in the surveillance region in the interval {0 : k}. It can be written in the recursive form π 0:k (X 0:k ) ∝ g k (Z k |X k )f k|k−1 (X k |X k−1 )π 0:k−1 (X 0:k−1 ).
(13) The GLMB smoother [33] was proposed to as an analytic solution to the multi-object posterior recursion. Assuming that there are no live objects at the beginning, i.e., π 0 (X 0 ) = δ 0 [L(X 0 )] with weight w (γ0) 0 = 1, the GLMB posterior at time k is given by [33] π 0:k (X 0:k ) ∝
∆(X 0:k )
w (γ0:j ) j =1 F (Bj⊎L(γj−1)) (L(γ j ))[η (γ0:j (·)) j|j−1 (·)] Bj ⊎L(γj−1) . (16) It is clear that the GLMB posterior is completely parameterized by the set of components {(w (γ 0:k ) 0:k , τ (γ 0:k ) 0:k )} indexed by γ 0:k . As per (13) the number of such components grow exponentially after each measurement update step, and to achieve tractability, truncation is performed and retain the highest weighted components [33].
III. COMPUTING THE GLMB POSTERIOR
In this section, we summarize the Gibbs sampler proposed in [33] to truncate the GLMB posterior.
A. Sampling Distributions
The GLMB posterior is truncated from some discrete probability distribution π of association maps γ 0:k , so that those maps with higher weights are more likely to be chosen. Starting with L(γ 0 ) = ∅, we consider
π(γ 0:k ) = k j=1 π (j) (γ j |γ 0:j−1 ) ∝ w (γ 0:k ) 0:k ,(17)
where
π (j) (γ j |γ 0:j−1 ) ∝ w (γ0:j ) j (18) ∝ 1 Γj (γ j )1 F (Bj⊎L(γj−1)) (L(γ j )) × [η (γ0:j (·))
j|j−1 (·)] Bj ⊎L(γj−1) . The term 1 Γj (γ j ) makes sure that γ j is in the space of all association maps at time j, and 1 F (Bj⊎L(γj−1)) (L(γ j )) makes sure that only values of γ j on B j ⊎L(γ j−1 ) need to considered.
Given a valid γ 0:k , the Gibbs sampler constructs a sequence of iterates, such that the next iterate γ
(ℓ n ),(20)M (S) β (α) δ β [α], α < 0 1, α = 0 (1 − 1 S (α) α > 0 ,
where a ∨ b denotes min{a, b}. Consider γ j of the valid association history γ 0:k , j ∈ {1 : k}. Then, for any ℓ n ∈ L j − (B j ⊎ L(γ j−1 )), the conditional (19) is given by,
π j,n (γ j (ℓ n )|γ j (ℓn), γj) (21) = δ −1 [γ j (ℓ n )]δ γ min{j+1,k} (ℓn) [γ j (ℓ n )],
and for ℓ n ∈ B j ⊎ L(γ j−1 ) π j,n (γ j (ℓ n )|γ j (ℓn), γj )
= η (γj ) j,n (γ j (ℓ n ))M (γj (ℓn)) γ min{k,j+1} (ℓn) (γ j (ℓ n )), and set γ j (ℓ n ) = −1 for all other ℓ n .
B. Computing the Posterior
It is clear that as the time interval, i.e., {0 : k}, grows, the dimensionality of the GLMB posterior (13) increases and it becomes impractical to compute the entire posterior at each timestep using (17). In this work, we propose to mitigate this computational complexity by smoothing over fixed windows, while linking the trajectory estimates between windows using their labels and corresponding association maps. This approach is illustrated in Fig. 1, which results in a computationally efficient approximate solution to the GLMB posterior propagation. Given an intial γ 0:k , a set of new samples can be generated based on this technique by using Algorithm 1. There are two methods to further improve the computational efficiency of Algorithm 1. In the first method, since there exist duplicate trajectories, i.e., those with the same association history and label combination, the corresponding trajectory and weight information can be stored and reused instead of calcuating them each time. The second method is to paralallelize Algorithm 1, so that each new sample based on γ 0:k is simultaneously generated instead of generating one after another.
Furthermore, Algorithm 1 can also be used with a smoothing-while-filtering approach as shown in Algorithm 2. Note that the function SampleFactors (in Algorithm 2) is given in [33].
IV. RESULTS
H k−1 h=1 , [T (h) ] H k−1 h=1 , T output: [G (h) 0:k ] H k h=1 for h = 1 : H k−1 [G (h,t) 0:k ]T (h) t=1 := Unique(SampleFactors(G (h) 0:k , T (h) )); end if k < N, keep H k best [G); keep H k best [G (h) 0:k ] H k h=1 ; normalize weights [w (h) 0:k ] H k h=1
as a Poisson RFS with the rate of 3 per scan and points are uniformely distributed over the surveillance region. The objects follow a constant velocity motion model, and the kinematic state of an object is represented by a 4D state vector consisting of 2D position and velocity given by x k = [p x,k ,ṗ x,k , p y,k ,ṗ y,k ]. The single object transition density is modeled by a linear gaussian given by f S,k|k−1 (x I 2 is the 2 × 2 identity matrix, ∆ = 1s is the sampling time, σ a = 1 m/s 2 , and ⊗ denotes the matrix outer product. Birth objects are modeled by a Labeled Multi-Bernoulli (LMB) Process having birth and spatial distribution parameters
(ℓ) k+1 |x (ℓ) k ) = N (x (ℓ) k+1 ; F k x (ℓ) k , Q k ) where F k = I 2 ⊗ 1 ∆ 0 1 , Q k = σ 2 a I 2 ⊗ ∆ 4 4 ∆ 3 2 ∆ 3 2 ∆ 2 ,{r B,k (ℓ i ), p (i) B,k (ℓ i )} 4 i=1 , where ℓ i = (k, i) ∈ B k , r B,k (ℓ i ) = 0.03, p (i) B,k (x (ℓi) , ℓ i ) = N (x (ℓi ) ; m (i) B,k , Q B,k ), m(1)
B,k = (500, 0, 500, 0) T , m [30,30] 2 ) and measurements are of the form z
k ) = N (z k ; H k x (ℓ) k , R k ), where R k = diag((v) k = [z (v)
x,k , z (v) y,k ] T . Algorithm 2 is executed with window sizes 5 and 20, and compare estimated tracks, cardinality, OSPA [27], OSPA (2) [2] with the GLMB filter. Fig. 2 compares the estimated tracks. The OSPA and OSPA (2) matrics are compared in Fig. 3 and Fig. 4, and it can be seen that smoothing with window sizes 5 and 20 produce smaller OSPA and OSPA (2) errors than filtering, and increasing the windows size results in smaller OSPA and OSPA (2) errors. The cardinality is compared in Fig. 6. It is clear that the larger the window size better the performance is, and a smaller window size with an acceptable running time can improve the results over filtering.
V. CONCLUSIONS
In this paper we introduce an approximate, practical approach to the multi-scan multi-target tracking problem using a moving window based technique. We adopt the recent GLMB smoother based on the Gibbs sampling based posterior truncation, and propose to recursively propagate the multitarget posterior using the most recent N scans, by linking the measurement association maps using labels. The efficicacy of the approach is demonstrated using a muli-target tracking simulation with 100 timesteps.
VI. ACKNOWLEDGEMENT
This work was supported by the Defence Science Centre Collaborative Research Grant (in 2020).
τ (α s(ℓ):k ) 0:k (x s(ℓ):t(ℓ) , ℓ)
ℓ 1:n−1 ), α, γ j (ℓ n+1:|Lj| ), γ j+1:k )(19) for each j ∈ {1 : k}, ℓ n ∈ {ℓ 1:|Lj| }. Let γj (γ 0:j−1 , γ j+1:k ),
Fig. 1 .
1Moving window approach with window size N . Maps shaded in brown are not updated.
A simulation was performed to evaluate performance of the proposed moving window based smoother. Births, deaths and movements of a set of 12 objects are simulated in a 2D surveillance area of [−1000, 1000] × [−1000, 1000]m 2 over 100 timesteps. The births occur at timesteps 1, 20 and 50 (respectively 4, 4 and 4), and the objects born at time 1 die at time 10, objects born at time 20 die at time 40, and objects born at time 50 die at time 90. The probability of survival of each object is set to P S (x (ℓ) , ℓ) = 0.95. The 2D positions of the objects are measured using a sensor that adds noise and measurement cluttter. The probability of detection of the sensor is set to P D (x (ℓ) , ℓ) = 0.3, and clutter is modeled j = k − N + 1 : k P j := |B j ⊎ L(γ j−1 )|; M j := |Z j |; c := −1 : M j ; γ ′ j = [ ]; for n = 1 : P j for α = −1 : M j κ(α) := π j,n (α|γ ′ j (ℓ 1:n−1 ), γ j (ℓ n+1:Pj ), γj); via (22) end γ ′ j (ℓ n ) ∝ Categorical(c,
Fig. 2 .
2Estimated trajectories (superimposed on the ground truth -solid lines) from GLMB filtering and GLMB smoothing with window sizes 5 and 20.
Fig. 3 .
3OSPA performance from GLMB filtering and GLMB smoothing with window sizes 5 and 20.
Fig. 4 .
4OSPA(2) (with 10 scans) performance from GLMB filtering and GLMB smoothing with window sizes 5 and 20.
Fig. 5 .
5Comparison of estimated cardinality from GLMB filtering and GLMB smoothing with window sizes 5 and 20.
B
,k = (−500, 0, 500, 0) T , m(3) B,k = (−500, 0, −500, 0) T , m (4) B,k = (500, 0, −500, 0) T , and Q B,k = diag([15, 15, 15, 15] 2 ). The measurements are modeled by a linear gaussian likelihood function of the form g k (z k |x (ℓ)
Fig. 6 .
6Run time comparison of GLMB filtering and GLMB smoothing with window sizes 5 and 20.
Fig. 5 .
5The actual time (in seconds) taken to simulate each timestep is compared in
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Generalized Labeled Multi-Bernoulli Multi-Target Tracking with Doppler-Only Measurements. Y Zhu, M Mallick, S Liang, J K Yan, Remote Sensing. 14132022Y. Zhu, M. Mallick, S. Liang, and J.K. Yan, "Generalized Labeled Multi- Bernoulli Multi-Target Tracking with Doppler-Only Measurements", Remote Sensing, 14(13), 2022.
| {'fraction_non_alphanumeric': 0.09543201692079427, 'fraction_numerical': 0.03478864791489093, 'mean_word_length': 3.8908445267870926, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 6, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, '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': 'Multi-target state estimation refers to estimating the number of targets and their trajectories in a surveillance area using measurements contaminated with noise and clutter. In the Bayesian paradigm, the most common approach to multitarget estimation is by recursively propagating the multi-target filtering density, updating it with current measurements set at each timestep. In comparison, multi-target smoothing uses all measurements up to current timestep and recursively propagates the entire history of multi-target state using the multi-target posterior density. The recent Generalized Labeled Multi-Bernoulli (GLMB) smoother is an analytic recursion that propagate the labeled multi-object posterior by recursively updating labels to measurement association maps from the beginning to current timestep. In this paper, we propose a moving window based solution for multi-target tracking using the GLMB smoother, so that only those association maps in a window (consisting of latest maps) get updated, resulting in an efficient approximate solution suitable for practical implementations.', 'arxivid': '2210.04008', 'author': ['Diluka Moratuwage *[email protected] \nSchool of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia\n', 'Changbeom Shim †[email protected] \nSchool of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia\n', 'Yuthika Punchihewa \nSchool of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia\n', 'Diluka Moratuwage *[email protected] \nSchool of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia\n', 'Changbeom Shim †[email protected] \nSchool of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia\n', 'Yuthika Punchihewa \nSchool of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia\n'], 'authoraffiliation': ['School of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia', 'School of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia', 'School of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia', 'School of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia', 'School of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia', 'School of Electrical Engineering, Computing and Mathematical Sciences\nCurtin University\nAustralia'], 'corpusid': 252780821, 'doi': '10.1109/iccais56082.2022.9990443', 'github_urls': [], 'n_tokens_mistral': 11278, 'n_tokens_neox': 10004, 'n_words': 5262, 'pdfsha': 'eb3de0fd655b7580ddad45f206a2ff8d8d6ff1da', 'pdfurls': ['https://export.arxiv.org/pdf/2210.04008v1.pdf'], 'title': ['A Moving Window Based Approach to Multi-scan Multi-Target Tracking', 'A Moving Window Based Approach to Multi-scan Multi-Target Tracking', 'A Moving Window Based Approach to Multi-scan Multi-Target Tracking', 'A Moving Window Based Approach to Multi-scan Multi-Target Tracking'], 'venue': []} |
arxiv |
The electron-phonon processes of the nitrogen-vacancy center in diamond
13 Mar 2015
Taras Plakhotnik
School of Mathematics and Physics
The University of Queensland
4072St LuciaQLDAustralia
Marcus W Doherty
Laser Physics Centre
Research School of Physics and Engineering
Australian National University
ACT 2601Australia
Neil B Manson
Laser Physics Centre
Research School of Physics and Engineering
Australian National University
ACT 2601Australia
The electron-phonon processes of the nitrogen-vacancy center in diamond
13 Mar 2015(Dated: March 16, 2015)
Applications of negatively charged nitrogen-vacancy center in diamond exploit the center's unique optical and spin properties, which at ambient temperature, are predominately governed by electronphonon interactions. Here, we investigate these interactions at ambient and elevated temperatures by observing the motional narrowing of the center's excited state spin resonances. We determine that the center's Jahn-Teller dynamics are much slower than currently believed and identify the vital role of symmetric phonon modes. Our results have pronounced implications for center's diverse applications (including quantum technology) and for understanding its fundamental properties.
The negatively charged nitrogen-vacancy (NV) center is a point defect in diamond [1] that has found diverse applications in quantum technology. The center is employed as a highly sensitive nanoscale sensor of electromagnetic fields [2][3][4][5][6][7], temperature [8][9][10][11][12][13] and pressure [14] that can operate in ambient and extreme conditions. Recent NV metrology proposals include gyroscopy [15][16][17][18] and the development of hybrid [19] and multi-mode [13] sensors. In quantum information science, the NV center is used to realize spin registers [20][21][22] at room temperature and spin-photon entanglement [23,24] at cryogenic temperatures. A new direction in NV quantum information science seeks to exploit spin-phonon coupling to enhance NV spin registers and develop novel quantum devices [25][26][27][28].
The applications of the NV center are based upon its remarkable optical and spin properties. The center's room temperature applications primarily rely upon its bright optical fluorescence, long-lived ground state spin coherence and methods of optical spin polarization and readout. The latter enable the optical detection of the center's magnetic resonances (ODMR) and are the consequence of spin-dependent phonon-mediated intersystem crossings (ISCs) [29,30]. The center's cryogenic applications also employ the coherence of the center's visible zero-phonon line (ZPL). The necessity of cryogenics arises from the temperature dependent electron-phonon induced dephasing and depolarization of the ZPL [36,37]. Electron-phonon coupling is also responsible for the motional narrowing of the center's excited state spin resonances, which determines their utility as an additional quantum resource for sensing and information processing [13,31]. Thus, a through understanding of the NV center's electron-phonon interactions is important to the continued advancement of its applications and may be generalized to similar defects with emerging quantum applications, such as the silicon-vacancy center in diamond [32,33] and centers in silicon carbide [34,35]. Here, we show that there exist several issues in the current under-standing and identify possible resolutions.
The electronic structure of the NV center is depicted in Fig. 1. The optical transitions of the visible ZPL occur between the ground 3 A 2 and excited 3 E spin triplet levels. The temperature dependent broadening of the ZPL was initially described [36] using the widely applicable model of quadratic electron-phonon interactions with A 1 phonon modes [38]. However, subsequent single center cryogenic measurements revealed that the broadening was more consistent with the characteristic ∝ T 5 temperature dependence of linear electron-phonon (Jahn-Teller) interactions with E phonon modes [37]. These interactions induce population transfer between the quasidegenerate orbital states (|X , |Y ) of the 3 E level (see Fig. 1), which dephases the optical transitions and leads to the depolarization of the ZPL fluorescence [37]. Applying their Jahn-Teller model, Fu et al [37] identified a factor of ∼ 2 inconsistency between the population transfer rates that describe the ZPL broadening and depolarization at low temperatures. By introducing a phonon cutoff energy, Abtew et al [39] attempted to extend Fu et al's model to describe the ZPL broadening up to room temperature. In doing so, they obtained a cutoff at 50 meV for E phonons, which is much lower than the diamond Debye energy ω D ≈168 meV [40] and the features of the NV electron-phonon spectral density extracted from the visible phonon sideband [41].
At temperatures < ∼ 30 K, the complicated six level 3 E fine structure (see Fig. 1) is observed via high resolution optical spectroscopy [29]. Above ≈ 150 K [42], the population transfer between the 3 E orbital states is sufficiently fast to dynamically average the 3 E fine structure so that ODMR resembles the simpler three level structure of the ground state [43]. The dynamically averaged fine structure is temperature dependent and is described by the spin Hamiltonian [13]
H = D (S 2 z − 2 3 ) − D ⊥ R(T )(S 2 x − S 2 y )(1)
where D = 1.42 GHz and D ⊥ = 0.775 GHz are the 3 E spin-spin parameters, R(T ) = (e hξ ⊥ /kB T − 1)/(e hξ ⊥ /kB T + 1) is the temperature reduction factor, h and k B are the Planck and Boltzman constants, respectively, and ξ ⊥ is the 3 E strain splitting. Note that the negligible contribution of the λ ⊥ spin-orbit term (see Ref. 13) to the fine structure is ignored here. The dynamical averaging is also expected to motionally narrow the 3 E ODMR, since the rapid population transfer decouples the orbit and spin degrees of freedom. Fuchs et al have measured the 3 E spin dephasing rate at room temperature [31]. They attributed the observed dephasing to the dynamical averaging process and, using a motional narrowing model, suggested that elevated temperatures or strain may decrease its rate. However, their proposal is yet to be tested by a systematic study of the motional narrowing effect. Furthermore, there is an inconsistency between Fuchs et al's observations and the current ZPL broadening model. If the population transfer rate (∼ 10 THz) at room temperature is inferred from the ZPL width [31], then the spin dephasing rate predicted by the motional narrowing model (∼ 1.2 MHz) is almost two orders of magnitude smaller than measured (∼ 92 MHz). The anomalously low cutoff of Abtew et al, the discrepancy identified by Fu et al, the conspicuous absence of interactions with A 1 modes, and the orders of magnitude larger than expected 3 E spin dephasing rate, all indicate problems in the current ZPL broadening model. The optical polarization and readout of the spin triplet levels is a result of spin-selective ISCs with the intermediate 1 A 1 and 1 E spin-singlet levels (see Fig. 1
3 A 2 3 E 1 A 1 3 E |Y |S 0 |X |S 0 |Y |S − |Y |S + |X |S − |X |S + |±1 |0 D + D + D − D − ξ ⊥ W ↓ , W ↑ 1.946 eV) [1].
Goldman et al [29,30] have developed a detailed model of the electron-phonon mechanisms that govern the ISC from 3 E to 1 A 1 . However, in order to validate Goldman et al's model and extend it to room temperature, more detailed and quantitative knowledge of linear E phonon interactions is required. This knowledge can be improved by extending the current measurements of the population transfer rates at cryogenic temperatures to room temperature and beyond.
In this paper, we report observations of the 3 E ODMR of NV centers in nanodiamond over the temperature range 295-550 K. We show that the ODMR is well described by a motional narrowing model and extract the population transfer rates. We establish that the rates are much slower than currently believed and do not account for the observed ZPL broadening at room temperature. We propose that quadratic A 1 phonon interactions contribute significantly to the ZPL already above 30 K. Finally, we back up and rectify the proposals of Fuchs et al, resolve the inconsistencies of the ZPL broadening model and provide valuable insight into electron-phonon coupling above cryogenic temperatures.
Our continuous wave ODMR experiments were performed using 532 nm laser excitation and fluorescence collection via an epifluorescence design. The nanodiamonds were spin coated on a silica substrate. The NV spin resonances were driven by radio-frequency (RF) magnetic field created by a gold wire deposited onto the substrate. The excitation laser spot overlapped with the wire and the optical heating of the wire was used to control the temperature of a chosen nanodiamond. See Ref. 13 for further experimental details. On average, the nanodiamonds had a diameter of ∼ 30 nm and contained ∼ 15 NV centers. We performed ODMR measurements on a total of 10 nanodiamonds. The results from one nanodiamond are presented here and are consistent with the rest of the sample and, as will be explained, measurements in bulk diamond. Previous optical spectroscopy has measured the 3 E strain splitting of the nanodiamond to be hξ ⊥ ∼ 4.7 meV [13]. This large strain splitting permits a simple 3 E fine structure (see Fig. 1) and application of the motional narrowing model. Note that the strain splitting in previous reports [31,37] have been much smaller.
Examples of ODMR spectra are shown in Fig. 2. Averaging over the unresolved 3 E hyperfine structure, the observed ODMR splitting is [13]
∆ ODMR = 2 3 D ⊥ R(T ) + 4 3 A 2 + D 2 ⊥ R 2 (T ) 1/2 ,(2)
where A ≈ 40 MHz is the isotropic hyperfine parameter. RF-power broadening is evident in Fig. 2. Similar to the analysis of the 3 A 2 ODMR in Ref. 44, a five-level model of the optical and spin dynamics yields the following expressions for the ODMR linewidth Γ ODMR and contrast C ODMR
Γ ODMR = Γ (inh) ODMR + Γ (h) ODMR 1 + 4πκP RF Γ (h) ODMR γ 1 1/2 C ODMR = C (max) ODMR 4πκP RF 4πκP RF + γ 1 Γ (h) ODMR ,(3)
where Γ ODMR are the homogenous and inhomogenous linewidths in the absence of power broadening, P RF is the RF-power, κ is a proportionality factor such that κP RF is the spin Rabi frequency, and γ 1 is the effective spin relaxation rate. The essential difference to Ref. 44 is a much weaker, but more complicated dependence of γ 1 on the laser power. At low laser powers, γ 1 ≈ kk ISC /(k + 0.5k ISC ) ≈ 22 MHz, where k ≈ 20 MHz [45] is the 3 E radiative decay rate in nanodiamond of the same type and origin as used in this work and k ISC ≈ 50 MHz is the average 3 E ISC rate [29,45]. Stress inhomogeneity and the unresolved hyperfine structure contribute to Γ
ODMR = Γ ∞ + Γ MN (T )
is the sum of the broadening due to the 3 E orbital decay rate Γ ∞ = (k + 0.5k ISC )/π and motional narrowing Γ MN (T ). Whilst the orbital decay rate increases at high temperature [8,46], this temperature dependence is ignored in the following because the contribution of Γ ∞ to the observed Γ ODMR changes little, from 14 MHz to 17 MHz between 295 K and 500 K. The spin bath dephasing contribution to Γ ∞ is also ignored because it has been assessed using the 3 A 2 ODMR to be negligibly small (1-2 MHz). In the fast exchange approximation of motional narrowing [47,48], where the population transfer rates (W ↑ , W ↓ ) are much larger than the jump in the spin resonances between the 3 E orbital states (2D ⊥ ), Γ MN (T ) ≈ β(T )2πD 2 ⊥ /W ↓ . The factor β(T ) = 8e −hξ ⊥ /kB T /(e −hξ ⊥ /kB T + 1) 3 is close to 1 above room temperatures. Thus, as W ↓ increases with temperature, Γ MN decreases.
In the temperature regime k B T ≫ hξ ⊥ , Raman scattering of E phonons dominate the population transfer rates which read [29]
W ↓ = B E T 5 Ω E k B T x ⊥ x 2 e x (x−x ⊥ ) 2 (e x −1)(e x−x ⊥ −1) dx(4)
and W ↑ = W ↓ e −hξ ⊥ /kB T , where x ⊥ = hξ ⊥ /k B T and Ω E is the cutoff energy for E phonons. The deformation potential and Debye density of states for acoustic phonons have been assumed such that the corresponding electron-phonon spectral density is J E (ω) ≈ η E ω 3 and the constant B E = 64 πh η 2 E k 5 B . Whilst in the simplest case Ω E = ω D , the cutoff is often considered as a phenomenological parameter which takes into account the departure from J E (ω) ∝ ω 3 . In the high temperature regime 1 ≫ x ⊥ ,hΩ E /k B T applicable to our work, the in-
ODMR = Γ ∞ + β(T )2πD 2 ⊥ /(QT 2 )
. Systematic measurements of the ODMR linewidth, contrast and splitting at different temperatures, RF and laser powers are presented in Fig. 3. The weak optical-power dependence [inset of Fig. 3(a)] supports the approximation γ 1 ≈ 22 MHz. The simultaneous fitting of the six data sets using the five parameters yields Γ (inh) ODMR = 33 ± 3 MHz, κ ≈ 210 ± 40 MHz 2 W −1 , C (max) ODMR = 16 ± 2%, Q = 0.83 ± 0.06 MHz K −2 , and hξ ⊥ = 4.6 ± 0.2 meV. The values of κ and hξ ⊥ are in reasonable agreement with the parameters of the RF wire and previous optical spectroscopy, respectively. The fitting yields Γ (h) ODMR = 55 MHz at room temperature. The dephasing rate measured by Fuchs et al at room temperature in bulk diamond corresponds to Γ (h) ODMR ∼ 29 MHz. Taking into account that the much smaller stress splitting ξ ⊥ of Fuchs et al's NV center will increase Q by ∼ 2, the two values are in agreement. Hence, we conclude that our nanodiamond measurements are consistent with bulk diamond and capture intrinsic phenomena of the NV center.
The previous measurements of the ZPL width [37] are plotted in Fig. 4 together with W ↓ /(2π) calculated here using the value of Q that we obtained by fitting our motional narrowing observations (rescaled to ξ ⊥ = 0 to match the stress splitting in Ref. 37). It is evident that the rates are orders of magnitude too small to account for the ZPL width alone. We propose that the additional width is due to quadratic interactions with A 1 modes that purely dephase the optical transitions [36,38]. In which case, the ZPL width is [49]
Γ ZPL = W ↓ 2π + W A π + γ 0 ,(5)
where W A is the additional dephasing rate and γ 0 is the approximately temperature independent contribution of the optical decay rate. As per a similar derivation of W ↓ ,
W A = B A T 7 Ω A k B T 0 e x x 6 (e x − 1) 2 dx,(6)
where B A is a constant and Ω A is the cutoff energy of A 1 phonons. We used Eqs. (4)(5)(6) and fitted the ZPL width measurements to obtain B E = 1.32 Hz K −5 , Ω E = 13 ± 1 meV, B A = 24 ± 4 µHz K −7 , Ω A = 37 ± 2 meV and γ 0 = 16.2 ± 0.5 MHz (in bulk diamond). We confirmed our parameters (B E and Ω E ) of the population transfer rates by also fitting the polarization visibility measurements of Ref. 37 (see Fig. 4). Our fit of the visibility curve is practically indistinguishable from Ref. 37 The phonon cutoffs that we obtained are much lower than expected. We attribute this to the inadequacies of the acoustic approximation of the phonon spectral density J(ω) ≈ η E ω 3 [36,38] and consider the cutoffs as phenomenological. Noting that Ω E /k B ∼ 155 K, these inadequacies are negligible at low temperatures, which 39. The red solid curve is the contribution of W ↓ to the ZPL width according to (4,5). The red dots show W ↓ /2π derived from ODMR data alone. Inset: the ZPL polarization visibility of two NV centers (red and blue points) from [37]. The solid curve is our fit obtained using the model V = (W ↑ − W ↓ ± r (1 − a) / (1 + a)) / (W ↓ + W ↑ + r), where a = 0.40 ± 0.02 and r = 80 MHz are defined in Ref. 37, and W ↓ and W ↑ are determined by our fit of the ZPL width. explains why they were not detected in previous cryogenic measurements [29,37]. Interestingly, Ω E is close to the calculated Jahn-Teller barrier energy ∼ 10 meV [39]. Note that the spectral density extracted, for example, from the visible phonon sideband represents contributions of E and A 1 phonons due to linear electronphonon interactions. It is difficult to distinguished the effects of A 1 and E modes on the phonon band experimentally and ab initio calculations therefore appear to be the best avenue for future advancement to resolve the puzzle.
FIG. 1 .
1The electronic and fine structures of the NV center at high stress. The 3 E sub-levels are labelled by their product of orbital (|X , |Y ) and spin (|0 , |S± ) states, where the spin states are solutions of (1). The 3 E fine structure splittings (D± = D ± D ⊥ ) are denoted in red. The 3 A2 sub-levels are denoted by their spin projection (ms = 0,±1). The optical transitions of the spin triplet and singlet levels are depicted as black solid arrows and the visible ZPL is at 1.956 eV. Blue dashed arrows represent the population transfers within the 3 E (rates W ↓ and W ↑ ). The black dashed arrows denote the allowed ISCs between the spin triplet and singlet levels.
. 3. A: ODMR linewidth as a function of temperature at RF-powers of 400, 200, and 50 mW (top to bottom). Inset shows the weak optical power dependence of the linewidth at 294 K and at two RF powers: 47 mW (bottom) and 380 mW (top). B and C show the RF-power dependence of the linewidth and contrast at 294 K and 100 mW optical power. D: ODMR splitting at different temperatures (50 mW RFpower). Error bars are determined by the statistics of repeated measurements. The plotted linewidth is the average width of the two lines.
FIG. 4 .
4Blue points are the ZPL width measured in Ref. 37. The black solid curve depicts the fit of our model and the black dashed curve is the extended Jahn-Teller model of Ref.
This work was supported by the Australian Research Council under the Discovery Project scheme DP0771676 and DP120102232. * [email protected] [1] M.W. Doherty, N.B. Manson, P. Delaney, F. Jelezko, J. Wrachtrup and L.C.L. Hollenberg, Physics Reports 528, 1 (2013). [2] D. Le Sage, K. Arai, D.R. Glenn, S.J. DeVience, L.M. Pham, L. Rahn-Lee, M.D. Lukin, A. Yacoby, A. Komeili and R.L. Walsworth, Nature 496, 486 (2013). [3] M.S. Grinolds, S. Hong, P. Maletinsky, L. Luan, M. D. Lukin, R. L. Walsworth and A. Yacoby. Nature Physics
FIG. 2. Example ODMR spectra at different temperatures (315 K upper, 455 K lower) and RF powers (440 mW left, 55 mW right). The narrowing and reduced splitting of the lines at higher temperature as well as power broadening at higher RF power can be seen. The lineshape fits (solid lines) are the sum of two Lorentzians and a linear background.1.2
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and our value of B E also agrees with the value ∼ 1.6 Hz K −5 obtained there. Unlike Ref. 37, we use the same value of B E for ZPL and visibility fits and our fit to ZPL data better describes the low and room temperature regions than the extended Jahn-Teller model presented in Ref.39. Most importantly, the ZPL broadening is fully consistent with our ODMR measurements at elevated temperatures.
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| {'fraction_non_alphanumeric': 0.08386358280308387, 'fraction_numerical': 0.04898495101504898, 'mean_word_length': 3.5670356703567037, 'pattern_counts': {'":': 0, '<': 1, '<?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': 11, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Applications of negatively charged nitrogen-vacancy center in diamond exploit the center's unique optical and spin properties, which at ambient temperature, are predominately governed by electronphonon interactions. Here, we investigate these interactions at ambient and elevated temperatures by observing the motional narrowing of the center's excited state spin resonances. We determine that the center's Jahn-Teller dynamics are much slower than currently believed and identify the vital role of symmetric phonon modes. Our results have pronounced implications for center's diverse applications (including quantum technology) and for understanding its fundamental properties.", 'arxivid': '1503.03956', 'author': ['Taras Plakhotnik \nSchool of Mathematics and Physics\nThe University of Queensland\n4072St LuciaQLDAustralia\n', 'Marcus W Doherty \nLaser Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 2601Australia\n', 'Neil B Manson \nLaser Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 2601Australia\n'], 'authoraffiliation': ['School of Mathematics and Physics\nThe University of Queensland\n4072St LuciaQLDAustralia', 'Laser Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 2601Australia', 'Laser Physics Centre\nResearch School of Physics and Engineering\nAustralian National University\nACT 2601Australia'], 'corpusid': 119234117, 'doi': '10.1103/physrevb.92.081203', 'github_urls': [], 'n_tokens_mistral': 11340, 'n_tokens_neox': 9504, 'n_words': 5160, 'pdfsha': '272d7390b5beacb17c728139a761669dd46ef980', 'pdfurls': ['https://arxiv.org/pdf/1503.03956v1.pdf'], 'title': ['The electron-phonon processes of the nitrogen-vacancy center in diamond', 'The electron-phonon processes of the nitrogen-vacancy center in diamond'], 'venue': []} |
arxiv |
9 Jun 2021 (Dated: June 10, 2021)
Damianos Iosifidis
Nurgissa Myrzakulov
Ratbay Myrzakulov International Centre for Theoretical Physics
010009Nur-SultanKazakhstan
Eurasian National University
010008Nur-Sultan
Ratbay Myrzakulov
Ratbay Myrzakulov International Centre for Theoretical Physics
010009Nur-SultanKazakhstan
Eurasian National University
010008Nur-Sultan
Kazakhstan
Institute of Theoretical Physics
Department of Physics
Aristotle University of Thessaloniki
54124ThessalonikiGreece
9 Jun 2021 (Dated: June 10, 2021)Metric-Affine Version of Myrzakulov F (R, T, Q, T ) Gravity and Cosmological Applications
We derive the full set of field equations for the Metric-Affine version of the Myrzakulov gravity model and also extend this family of theories to a broader one. More specifically, we consider theories whose gravitational Lagrangian is given by F (R, T, Q, T , D) where T , Q are the torsion and non-metricity scalars, T is the trace of the energy-momentum tensor and D the divergence of the dilation current. We then consider the linear case of the aforementioned theory and assuming a cosmological setup we obtain the modified Friedmann equations. In addition, focusing on the vanishing non-metricity sector and considering matter coupled to torsion we obtain the complete set of equations describing the cosmological behaviour of this model along with solutions.
Even though General Relativity (GR) is undeniably one of the most beautiful and successful theories of physics, recent observational data have challenged its status [1]. Probably the most important observations that cannot be explained within the realm of GR are the early time as well as the late time accelerated expansion of our Universe. This contradiction between theory and observations have lead to the development of a fairly large number of theories alternative to GR which collectively go by the name of Modified Gravity [2]. The search of a successful alternative has been proven to be both fruitful as well as constructive in regards with our understanding of gravity.
Among this plethora of modified gravities let us mention the metric f (R) theories , the Metric-Affine (Palatini) f (R) gravity [3][4][5], the teleparallel f (T ) gravities [6,7], the symmetric teleparallel f (Q) [8,9], Scalar-Tensor theories [10,11], etc and also certain extensions of them (see discussion on chapter IV ). Of course, the kind of modifications one chooses to adopt is highly a matter of personal taste. In our point of view, interesting and well motivated alternatives are those which extend the underlying geometry of spacetime by allowing a connection more general than * Electronic address: [email protected], [email protected], [email protected] the usual Levi-Civita one. In generic settings, when no a priori restriction is imposed on the connection and the latter is regarded as another fundamental field on top of the metric the space will be non-Riemannian [12] and possesses both torsion and non-metricity. These last geometric quantities can then be computed once the affine connection is found. The theories formulated on this non-Riemannian manifold are known as Metric-Affine theories of gravity [13,14].
In recent years there has been an ever increasing interest in the Metric-Affine approach [5,[15][16][17][18][19][20][21][22][23][24][25][26][27][28][29] and especially to its cosmological applications [30][31][32][33][34][35][36][37][38][39][40][41]. This interest is possibly due to the fact that the additional affects (compared to GR) that come into play in this framework have a direct geometrical interpretation. That is, the modifications are solely due to spacetime torsion and non-metricity. Furthermore, these geometric notions are excited by matter that has intrinsic structure [32,[42][43][44][45]. This inner structure-generalized geometry interrelation adds another positive characteristic to the MAG scheme. This is the framework we are going to consider in this study.
In particular, the paper is organized as follows. Firstly, we fix conventions and briefly review some of the basic elements of non-Riemannian geometry and the physics of Metric-Affine gravity. We then consider an extended version of the F (R, T, Q, T , D) theory [46]. To be more specific, working in a Metric-Affine setup, we consider the class of theories with gravitational Lagrangians of the form F (R, T, Q, T , D), where D is the divergence of the dilation current, the new add-on we are establishing here. Then, we obtain the field equations for this family of theories by varying with respect to the metric and the independent affine connection. Considering a linear function F we then present a cosmological application for this model and finally switching off non-metricity and considering a scalar field coupled to torsion we obtain the modified Friedmann equations and also provide solutions for this simple case.
II. CONVENTIONS/NOTATION
Let us now briefly go over the basic geometric as well as physical setup we are going to use and also fix notation. We consider a 4-dim non-Riemannian manifold endowed with a metric and an affine connection (M, g, ∇). Our definition for the covariant derivative, of a vector say, will be
∇ α u λ = ∂ α u λ + Γ λ βα u β(1)
We also define the (Cartan) torsion tensor by
S λ µν := Γ λ [µν](2)
and the non-metricity tensor as
Q αµν := −∇ α g αβ(3)
Contracting these with the metric tensor we obtain the associated torsion and non-metricity vectors
S µ := S ν µν(4)
Q µ := Q µαβ g αβ , q µ := Q αβµ g αβ (5) respectively. In addition, since we are in four dimensions, we can also form the torsion pseudo-vector according to
t µ := ε µαβγ S αβγ(6)
Given the above definitions for torsion and non-metricity one can easily show (see for instance [14]) the affine connection decomposition 1
Γ λ µν = N λ µν +Γ λ µν = 1 2 g αλ (Q µνα + Q ναµ − Q αµν ) − g αλ (S αµν + S ανµ − S µνα )(7)
where N λ µν is known as the distortion tensor. Continuing we define the curvature tensor as usual
R µ ναβ := 2∂ [α Γ µ |ν|β] + 2Γ µ ρ[α Γ ρ |ν|β](8)
and by a double contraction of the latter, we get the Ricci scalar
R := R µ νµβ g νβ(9)
Then by using decomposition (7) we obtain the post Riemannian expansion for the Ricci scalar [14] R
=R + T + Q + 2Q αµν S αµν + 2S µ (q µ − Q µ ) +∇ µ (q µ − Q µ − 4S µ )(10)
whereR is the Riemannian Ricci tensor (i.e. computed with respect to the Levi-Civita connection) and we have also defined the torsion and non-metricity scalars as
T := S µνα S µνα − 2S µνα S αµν − 4S µ S µ(11)
and
Q := 1 4 Q αµν Q αµν − 1 2 Q αµν Q µνα − 1 4 Q µ Q µ + 1 2 Q µ q µ(12)
respectively. Note that with the introduction of the superpotentials 2
Ω αµν := 1 4 Q αµν − 1 2 Q µνα − 1 4 g µν Q α + 1 2 g αµ Q ν(13)Σ αµν := S αµν − 2S µνα − 4g µν S α(14)
these can more compactly be expressed as
T = S αµν Σ αµν (15) Q = Q αµν Ω αµν(16)
Equation (8) is of key importance in teleparallel formulations. For instance by imposing vanishing curvature (which also implies R = 0) and metric compatibility (Q αµν = 0) one obtains from (7),
R = −T + 4∇ µ S µ(17)
which is the basis of the metric teleparallel formulation. In a similar manner the symmetric teleparallel (vanishing curvature and torsion) and also the generalized teleparalelism (only vanishing curvature) are obtained [47].
Let us now turn our attention to the matter content. In Metric-Affine gravity apart from the energy momentum tensor, which we define as usual
T µν := − 2 √ −g δ( √ −gL M ) δg µν(18)
one also has to vary the matter part with respect to the affine-connection. This new object, which is defined by
∆ µν λ := − 2 √ −g δ( √ −gL M ) δΓ λ µν(19)
is called hypermomentum [42] and encodes the microscopic characteristics of matter such as spin, dilation and shear.
In the same way that the energy momentum tensor sources spacetime curvature by means of the metric field equations, the hypermomentum is the source of spacetime torsion and non-metricity (through the connection field equations). Note that these energy related tensors are not quite independent and are subject to the conservation law
√ −g(2∇ µ T µ α − ∆ λµν R λµνα ) +∇ µ∇ν ( √ −g∆ µν α ) + 2S λ µα∇ν ( √ −g∆ µν λ ) = 0 ,∇ µ := 2S µ − ∇ µ(20)
which comes from the diffeomorphism invariance of the matter sector of the action (see [32]). In the above discussion we have briefly developed the geometric and physical setup needed for the rest of our study. Let us focus on the cosmological aspects of theories with torsion and non-metricity (i.e non-Riemannian extensions).
III. COSMOLOGY WITH TORSION AND NON-METRICITY
Let us consider a homogeneous flat FLRW Cosmology, with the usual Robertson-Walker line element
ds 2 = −dt 2 + a 2 (t)δ ij dx i dx j(21)
where i, j = 1, 2, 3 and a(t) is as usual the scale factor of the universe. As usual the Hubble parameter is defined as H :=ȧ/a. Now, let u µ be the normalized 4-velocity field and
h µν := g µν + u µ u ν(22)
be the projection tensor projecting objects on the space orthogonal to u µ . The affine connection of the non-Riemannian FLRW spacetime reads [32]
Γ λ µν = Γ λ µν + X(t)u λ h µν + Y (t)u µ h λ ν + Z(t)u ν h λ µ + V (t)u λ u µ u ν + ǫ λ µνρ u ρ W (t)δ n,4(23)
where the non-vanishing components of the Levi-Civita connection are in this casẽ
Γ 0 ij =Γ 0 ji =ȧaδ ij = Hg ij ,Γ i j0 =Γ i 0j =ȧ a δ i j = Hδ i j(24)
Continuing with the rest of the geometric objects, in this highly symmetric spacetime, the torsion and non-metricity tensors take the forms [32]
S (n) µνα = 2u [µ h ν]α Φ(t) + ǫ µναρ u ρ P (t) (25) Q αµν = A(t)u α h µν + B(t)h α(µ u ν) + C(t)u α u µ u ν(26)
respectively. The five functions Φ, P, A, B, C describe the non-Riemannian Cosmological effects. These, along with the scale factor, give the cosmic evolution of non-Riemannian geometries. Let us note that using the relations of the torsion and non-metricity tensors with the distortion tensor it is trivial to show that the functions
X(t), Y (t), Z(t), V (t), W (t) are linearly related to Φ(t), P (t), A(t), B(t), C(t) as [32] 2(X + Y ) = B , 2Z = A , 2V = C , 2Φ = Y − Z , P = W(27)
or inverting them
W = P , V = C/2 , Z = A/2 (28) Y = 2Φ + A 2 , X = B 2 − 2Φ − A 2(29)
Now, using the definitions (11) and (12) for the torsion and non-metricity scalars and the above cosmological forms for torsion and non-metricity we find for the former
T = 24Φ 2 − 6P 2 (30) Q = 3 4 2A 2 + B(C − A)(31)
respectively. These are the expressions for the torsion and non-metricity scalars in a homogeneous cosmological setup when no teleparallelism is imposed. Finally, using the post Riemannian decomposition of the Ricci scalar and the above forms of the torsion and non-metricity scalars we find
R =R + 6 1 4 A 2 + 4Φ 2 + Φ(2A − B) + 3 4 B(C − A) − 6P 2 +3 1 √ −g ∂ µ √ −gu µ B 2 − A − 4Φ (32) whereR = 6 ä a + ȧ a 2(33)
is the usual Riemannian part. The last decomposition will be very useful in our subsequent discussion. In this paper, we are going to study the Myrzakulov gravity [46] VIII (MG-VIII) 3 . Its action is given by [46] S[g,
Γ, φ] = S g + S m = 1 2κ √ −gd 4 x [F (R, T, Q, T ) + 2κL m ] ,(34)
where R stands for the Ricci scalar (curvature scalar), T is the torsion scalar, Q is the nonmetricity scalar and T is trace of the energy-momentum tensor of matter Lagrangian L m . The MG-VIII can be seen as some kind of unification of F (R), F (T ), F (Q) or F (R, T ), F (T, T ), F (Q, T ) theories (see [50][51][52] respectively). For instance, if one imposes flatness (i.e. R λ αµν ≡ 0) and metric compatibility (Q αµν ≡ 0) arrives at the f (T ) gravity [7,53]. Demanding flatness and a torsionless connection we get symmetric teleparallel f (Q) gravity [8,9]. More generally, imposing only teleparallelism we arrive at the recently developed generalized teleparallel scheme of f(G) [47,54] theories. If no restriction on the connection is assumed then (34) serves as a specific generalization of metric-affine f (R) gravity where where the energy momentum trace T and certain quadratic combinations of torsion and non-metricity are added as well. In fact in this generalized metric-affine setup one could consider also the presence of the hypermomentum analogue of the (metrical) energy momentum trace. Giving it a little thought we observe that similar to the trace T is the divergence of the dilation current as they appear in the trace of the canonical 4 energy momentum tensor (see for instance [32])
t = T + 1 2 √ −g ∂ ν ( √ −g∆ ν ) , ∆ ν := ∆ µν µ(35)
In this sense T and the divergence of ∆ ν are placed on equal footing as it is obvious from the above equation. Therefore, the scalar obtained by the divergence of the dilation current
D = 1 √ −g ∂ ν ( √ −g∆ ν )(36)
would be trace analogue for the hypermomentum. With this inclusion we may generalize the class of theories (34) to
S[g, Γ, φ] = S g + S m = 1 2κ √ −gd 4 x [F (R, T, Q, T , D) + 2κL m ] ,(37)
The field equations of the family of theories given by the the above action read as follows: g-Variation:
− 1 2 g µν F + F R R (µν) + F T 2S ναβ S αβ µ − S αβµ S αβ ν + 2S ναβ S βα µ − 4S µ S ν + F Q L (µν) +∇ λ (F Q J λ (µν) ) + g µν∇λ (F Q ζ λ ) + F T (Θ µν + T µν ) + F D M µν = κT µν(38)
where∇
λ := 1 √ −g (2S λ − ∇ λ )(39)Ω αµν = 1 4 Q αµν − 1 2 Q µνα − 1 4 g µν Q α + 1 2 g αµ Q ν(40)4L µν = (Q µαβ − 2Q αβµ )Q αβ ν + (Q µ + 2q µ )Q ν + (2Q µνα − Q αµν )Q α ) −4Ω αβ ν Q αβµ − 4Ω αµβ Q αβ ν(41)
Θ µν := g αβ δT αβ δg µν
M µν := δD δg µν (43) and we have also defined the densities
J λ µν := √ −g 1 4 Q λ µν − 1 2 Q λ µν + Ω λ µν (44) ζ λ = √ −g − 1 4 Q λ + 1 2 q λ(45)
Γ-Variation:
P µν λ (F R ) + 2F T S µν λ − 2S [µν] λ − 4S [µ δ ν] λ − M µνα λ ∂ α F D +F Q 2Q [νµ] λ − Q µν λ + (q ν − Q ν )δ µ λ + Q λ g µν + 1 2 Q µ δ ν λ = F T Θ µν λ + κ∆ µν λ (46) where P µν λ (F R ) = − ∇ λ ( √ −gF R g µν ) √ −g + ∇ α ( √ −gF R g µα δ ν λ ) √ −g + (47) 2F R (S λ g µν − S µ δ ν λ − S µν λ )
is the modified Palatini tensor and [46]. Here we derived the field equations with no restriction on the connection and also for the extended case F (R, T, Q, T , D). In the sequel we shall analyse further the linear case F = R + βT + γQ + µT + νD and also touch upon cosmological applications.
V. COSMOLOGICAL APPLICATIONS
A. The Cosmology of F = R + βT + γQ + µT + νD Theory Let us now analyse in more detail the linear case F = R + βT + γQ + µT + νD and also obtain the associated cosmological equations. To start with let us note that since √ −gD is a total divergence, the dilation current does not contribute to the field equations when included linearly. Therefore we can safely set ν = 0 for the rest of our discussion. In addition, in this linear case the metric field equations take the form
− 1 2 g µν F + R (µν) + β 2S ναβ S αβ µ − S αβµ S αβ ν + 2S ναβ S βα µ − 4S µ S ν + γL (µν) +∇ λ (γJ λ (µν) ) + g µν∇λ (γζ λ ) + µ(Θ µν + T µν ) = κT µν(49)
Taking the trace of the last equation, using the post Riemannian expansion (32) and also employing (25) along with (26) and after some long calculations we finally arrive aẗ
a a + ȧ a 2 + (1 + β)(4Φ 2 − P 2 ) + 1 8 2A 2 + B(C − A) + Φ(2A − B) +ḟ + 3Hf = −µ(Θ + T ) + κT(50)
where
f := 1 2 (1 − γ) B 2 − A − 4Φ , Θ := Θ µν g µν(51)
which is a variant of the modified Friedmann equation. As for the second Friedmann (acceleration) equation, its general form was derived in [31] for general non-Riemannian cosmological setups. It reads
a a = − 1 3 R µν u µ u ν + 2 ȧ a Φ + 2Φ + ȧ a A + C 2 +Ȧ 2 − A 2 2 − 1 2 AC − 2AΦ − 2CΦ(52)
One could then proceed by contracting (49) with u µ u ν in order to eliminate the first term (R µν u µ u ν ) and express everything in terms of the scale factor and the torsion and non-metricity variables. This results in a fairly complicated expression which we refrain from presenting it here since it goes beyond the scope of the present exposure. As a final note let us mention that in order to analyse in depth the above cosmological model one should consider an appropriate form of matter for which both the metrical energy momentum and hypermomentum tensors must respect the cosmological principle. The fluid with such characteristics was constructed in [32] (see also for a generalized version [45]) and goes by the name Perfect Cosmological Hyperfluid. The hypermomentum part of this fluid will then source the torsion and non-metricity variables Φ, P, A, ... etc. by virtue of the connection field equations. We note that scalar fields coupled to the connection belong (are certain subcases) to the aforementioned fluid description. For the sake of illustration, below we present such an example with a scalar field non-minimally coupled to the connection in the case of vanishing non-metricity and also study some of the cosmological implications of this theory.
B. Scalar Field Coupled to Torsion
We shall now focus on the vanishing non-metricity sector and also set γ = 0 , that is we will concentrate on the case F = R + βT . As for the matter part let us consider a scalar field. In the usual (i.e. purely Riemannian) case one would have the usual Lagrangian
L (0) m = − 1 2 g µν ∇ µ φ∇ ν φ − V (φ),(53)
for the scalar field φ. However, in the presence of torsion nothing prevent us to consider direct couplings of the scalar field with torsion. The most straightforward form of such a coupling is a torsion vector-scalar field derivative interaction of the form λ 0 S µ ∇ µ φ, where λ 0 is the coupling constant measuring the strength of the interaction. Including this term, our full matter Lagrangian now reads
L m = − 1 2 g µν ∇ µ φ∇ ν φ − V (φ) + λ 0 S µ ∇ µ φ(54)
Then, substituting this into (34) and varying the latter with respect to the scalar field, we obtain
1 √ −g ∂ µ √ −g(∂ µ φ − λ 0 S µ ) = ∂V ∂φ (55)
which is the evolution equation for the scalar field under the influence of torsion. In addition, the very presence of the interaction term λ 0 S µ ∇ µ φ produces a non-vanishing hypermomentum which is trivially computed to be
∆ µν λ = 2λ 0 δ [µ λ ∇ ν] φ(56)
With this result, starting from the connection field equations () which in our case read
P µν λ + 2β S µν λ − 2S [µν] λ − 4S [µ δ ν] λ = κ∆ µν λ(57)
and contracting in µ = λ we find
S µ = 3κλ 0 8β ∂ µ φ(58)
that is the presence of scalar field produces spacetime torsion 5 . In addition, contracting (57) with ε λ µνα it follows that
t α = 0(59)
Note that we can now plug back to (54) the above form of the torsion tensor to end up with
L m = − 1 2 1 − 3κλ 2 0 4β g µν ∇ µ φ∇ ν φ − V (φ)(60)
Interestingly, from the last equation we conclude that the scalar-torsion interaction changes the factor of the kinetic term for the scalar field. We also see that the is a crucial value for the coupling |λ 0 | = 2 β 3κ above which the kinetic term changes sign and for exactly this value vanishes identically. Since this last case would require severe fine tuning we shall disregard it and we shall also assume that λ 0 is under this bound so that the kinetic term keeps its original sign.
Up to this point, the above considerations were general. Let us now focus on the homogeneous FLRW cosmology of this theory. In this case, equation (59) implies that P = 0 and as a result, upon using (58) the full torsion tensor is given by
S µνα = 2u [µ h ν]α Φ(t) , Φ = − κλ 0 8βφ (61)
In the case of a free scalar field (i.e V (φ) = 0) inserting (58) into (55) we obtain
1 − 3κλ 2 0 8β ∂ µ √ −g∂ µ φ = 0 (62) which for |λ 0 | = 2 β 3κ implies thatφ = c 0 a 3(63)
On the other hand, the metric field equations in this case read
− 1 2 g µν F + R (µν) + β 2S ναβ S αβ µ − S αβµ S αβ ν + 2S ναβ S βα µ − 4S µ S ν = κT µν(64)
and by taking the trace, using the same procedure we outlined previously, we finally obtain
a a + ȧ a 2 = − κ 6 + (1 − β) κλ 0 4β 2 φ 2(65)
which is again a variant of the modified Friedmann equation. Let us now derive the acceleration equation for this case. First, we contract the above field equations with u µ u ν to obtain
R µν u µ u ν = 24βΦ 2 + κ 2 (ρ + 3p)(66)
which when substituted in (52) for vanishing non-metricity and the given scalar matter results in the acceleration equationä
a = −8βΦ 2 − κ 6 (ρ + 3p) + 2HΦ + 2Φ(67)
where ρ and p are the density and pressure associated to the scalar field Lagrangian (60). It is interesting to note that the first term on the right hand side of the acceleration equation has a fixed sign depending on the value of β. Intriguingly, for β < 0 the contribution from this term has always a fixed positive sign producing an accelerated expansion regardless of the sign of Φ (or equivalentlyφ). As for the last two terms, combining (61b) and (63) we observe thatΦ = −3HΦ which when substituted to the above acceleration equation yields
a a = −8βΦ 2 − κ 6 (ρ + 3p) + 4 3Φ(68)
We can conclude therefore that the last term aids to to acceleration whenΦ > 0 and slows it down wheneverΦ < 0.
From the above analysis we see that the non-Riemannian degrees of freedom play a crucial role on the cosmological evolution providing new interesting phenomena. Now using the latter form of the acceleration equation we can obtain the first Friedmann equation from (65) by eliminating the double derivative of the scale factor. For the simple case V (φ) = 0 we find
ȧ a 2 = κ 6 + (1 + β) κλ 0 4β 2 φ 2 − 4 3Φ (69)
as the modified first Friedmann equation. Note that on substituting (58b) in the above and completing the square in the resulting expression we easily find the power-law solution
a(t) ∝ t 1/3(70)
which is the stiff matter solution. We see that in the simplified case of a zero potential for the scalar we arrive at a known solution. However, we should remark that the situation changes drastically when one considers a non-vanishing potential. Note also that the torsion tensor in this case goes like 1/t and therefore its effect diminishes with time. Needless to say that when non-metricity is also included one gets more complicated expressions with a much richer phenomenology. It would be quite interesting to see exactly to what degree the simultaneous presence of torsion and non-metricity alters the cosmological evolution in such models. This would be the theme of a separate work.
VI. CONCLUSIONS
By working in a Metric-Affine approach (i.e. considering the metric and the connection as independent variables) we have considered a generalized version of the theory proposed in [46]. In particular, we derived the full set of field equations of the class of theories whose gravitational part of the Lagrangian is given by F (R, T, Q, T , D), where T , Q are the torsion and non-metricity scalars, T is the trace of the energy-momentum tensor and D is the divergence of the dilation current (one of the hypermomentum sources). The family of theories contained in our Lagrangian is fairly large since all, metric and Palatini f (R) theories, teleparallel f (T ), symmetric teleparallel f (Q) or even generalized teleparallel f (G) and generalizations of them such as f (R, T ), f (T, T ), f (Q, T ) can be seen as special cases of our theory.
Our contribution was two-fold. Firstly, we generalized the family of theories to those including also the divergence of the dilation current (which is the analogue of the energy-momentum trace for hypermomentum). Furthermore, as already mentioned above, we worked in a Metric-Affine framework, considering an independent affine connection as a fundamental variable along with the metric. This allows one not only to study the aforementioned theories (by restricting the connection one way or another), but also to analyse them in this general Metric-Affine scheme. Having derived the complete set of Metric-Affine F (R, T, Q, T , D) theories we then concentrated our attention on the linear case F = R + βT + γQ + µT + νD and obtained a variant version of the modified Friedmann equation. Finally, we focused on the vanishing non-metricity sector and also considered a scalar field coupled to torsion as our matter sector. In this case we derived both the first and second (acceleration) Friedmann equations and examined under what circumstances the presence of torsion can have an accelerating affect on the cosmological evolution. For this simple case we were also able to provide an exact power-law solution for the scale factor.
In closing let us note some further applications and additional developments of our study here. Firstly, it would be interesting to study in more detail the linear case especially in regard with its cosmological implications in the presence of the cosmological hyperfluid [32,45]. In addition, as we have already mentioned, it would be worthy to elaborate more on the coupled scalar field we presented when both torsion and non-metricity are allowed and direct couplings of the latter with the scalar field occur. Finally, it would be quite interesting to go beyond linear functions F of the new dilation current term we considered. In this way we will be able to investigate what exactly is the effect of this new addition/extension as well as its phenomenology especially with regards to its energy momentum trace counterpart.
III. Cosmology with Torsion and Non-metricity4 IV. MG-VIII model and Extension: The F (R, T, Q, T , D) Theories 5 V. Cosmological Applications 6 A. The Cosmology of F = R + βT + γQ + µT + νD Theory
.
If matter does not couple to the connection (e.g. classical perfect fluid with no inner structure) The above set of field equations constitutes an extended (with the divergence of dilation included) Metric-Affine version of the Myrzakulov gravities
From here onwards we shall use the tilde notation in order to denote Riemannian objects, that is objects computed with respect to the Levi-Civita connectionΓ λ µν .
Here we are using the conventions of[16].
See also[48,49] for some observational implications of this theory.4 Here t = t µν gµν is the trace of the canonical energy momentum tensor t µν .
Of course this is so because of the connection coupling which yields a non-vanishing hypermomentum. If no such coupling is included the scalar field cannot either feel or produce torsion.
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arxiv |
Coulomb drag and heat transfer in strange metals
24 Apr 2023
A L Chudnovskiy
Institut für Theoretische Physik
Universität Hamburg
Notkestraße 9D-22607HamburgGermany
Alex Levchenko
Department of Physics
University of Wisconsin-Madison
53706MadisonWisconsinUSA
Alex Kamenev
School of Physics and Astronomy
University of Minnesota
55455MinneapolisMinnesotaUSA
William I. Fine Theoretical Physics Institute
University of Minnesota
55455MinneapolisMinnesotaUSA
Coulomb drag and heat transfer in strange metals
24 Apr 2023(Dated: April 25, 2023)
We address Coulomb drag and near-field heat transfer in a double-layer system of incoherent metals. Each layer is modeled by an array of tunnel-coupled SYK dots with random inter-layer interactions. Depending on the strength of intra-dot interactions and inter-dot tunneling, this model captures the crossover from the Fermi liquid to a strange metal phase. The absence of quasiparticles in the strange metal leads to temperature-independent drag resistivity, which is in strong contrast with the quadratic temperature dependence in the Fermi liquid regime. We show that all the parameters can be independently measured in near-field heat transfer experiments, performed in Fermi liquid and strange metal regimes.
The electronic double layers -spatially separated and interactively coupled conducting circuits -provide a versatile array of low-dimensional quantum systems designed to directly probe electronic correlations via nonlocal transport measurements such as Coulomb drag [1]. Such double layers can be formed out of 0D quantum dots and point contacts [2][3][4], 1D nanowires [5][6][7][8] and topological edge states [9], and bilayers of 2DEG [10][11][12] or graphene [13][14][15]. These devices enable the exploration of various electron transport regimes and the identification of correlated electronic phases from the distinct temperature dependence of the drag resistance.
In the Fermi liquid (FL) regime the drag resistance is expected to scale quadratically with the temperature at the lowest temperatures. This result follows from the simple argument of the phase space available for the quasiparticle scattering that can be accurately established in the microscopic kinetic theory [16][17][18]. The interplay of screening and diffusion leads to the enhancement of drag resistance in the disordered systems [19][20][21]. At intermediate temperatures, dragging is dominated by the collective modes and resistance peaks at the energies of plasmons in 2D bilayers. The further fall-off of drag resistance at higher temperatures can be described by hydrodynamic effects and is governed by the electron liquid viscosity in clean systems [22,23]. All these features are well understood and rigorously described within the framework of the Fermi liquid theory.
There are known examples of essentially non-Fermi liquid behavior in systems where the quasiparticle concept breaks down. For instance, in Luttinger liquids kinematics of 1D collisions of electrons with linear spectrum dictates that drag is dominated by the inter-wire backscattering [24,25]. This translates into the signature power-law temperature dependence of drag resistance with the power exponent dependent on the strength of electron interaction. At the lowest temperatures, however, the trans-resistivity diverges, due to the formation of locked charge density waves. The enhancement of re-sistance occurs also in 2D layers provided that interactions are sufficiently strong and the electron system is on one of the possible microemulsion phases at the onset of Wigner crystallization [26]. Another notable example is the regime of drag between fractional quantum Hall liquids, where the trans-resistance is determined by the scattering and Coulomb screening effects of composite fermions [27][28][29][30]. Ultimately, the strong coupling limit may lead to pairing and inter-layer (indirect excitonic) superfluidity [31,32] that can be detected in the Coulomb drag counterflow setup.
In recent years much of the attention in the context of electronic transport is devoted to understanding the strange metal (SM) behavior in strongly correlated materials with and without quasiparticles, revealing the microscopic origin of the Planckian dissipation [33][34][35][36][37][38]. This broad interest facilitates the development of the corresponding transport theory for strange metal bilayers that may provide additional insights into the intricate physical properties of these systems. For that purpose, we use the paradigmatic Sachdev-Ye-Kitaev (SYK) model [39][40][41], which describes a strongly interacting quantum many-body system without quasiparticle excitations that is maximally chaotic, nearly conformally invariant, and exactly solvable in the limit of a large number of interacting particles. We derive analytical results for the drag resistance and near-field thermal conductance in bilayers of SYK arrays. Our analysis leads us to drastically different conclusions concerning the temperature dependence of the drag resistance in the SM phase as compared to the FL, and different from the earlier study based on the hydrodynamic-like holographic model of the strange metal [42].
To reveal the main qualitative features of the Coulomb drag in incoherent metals, we consider a theoretical model, which consists of two layers, dubbed by u and d, coupled by interactions. Each layer consists of an array of SYK dots, coupled by direct particle tunneling, see
H ν t + V int .(1)
Here the first term describes the set of isolated SYK dots
H νr SYK = N ij,kl J νr ij,kl c + νri c + νrj c νrk c νrl ,(2)
where J νr ij,kl are random couplings drawn from the Gaussian distribution with zero mean and the variances |J νr ij,kl | 2 = 2J 2 N 3 . The interactions in different dots are statistically independent of each other. The second term in Eq. (1) describes the inter-dot tunneling of electrons in each layer
H ν t = r,r N i t ν i c + νri c νr i + h.c. ,(3)
where t ν i denotes random tunneling amplitudes derived from the Gaussian distribution with zero mean and the variance |t ν i | 2 = t 2 0 , and the sum r, r runs over the nearest neighbors. The tunneling couplings in different layers are statistically independent. We associate the site index i, j, k, l within the SYK dot with a quantum number characterizing some quantum mechanical state (orbital), which is conserved by the tunneling. The last term in Eq.
(1) describes inter-layer interactions. Being guided by the random interactions within the SYK dot, we adopt the random inter-dot interaction between the on-site charge densities
V int = N i,j V ij r c + uri c uri c + drj c drj .(4)
The random interaction constants V ij have zero mean and are characterized by the variance Each isolated SYK dot provides a model of an incoherent metal that completely lacks electron or hole quasiparticles [39,40]. However, a weak electron hopping within an array of SYK dots changes the low-energy spectrum, restoring coherent quasiparticles. This, in turn, induces a crossover between the high-temperature incoherent SYK metal and low-temperature FL metal at a temperature of T 0 ∼ t 2 0 /J, which is determined by the electron escape rate from the SYK-grain [35,41,[43][44][45][46]. The model adopted here exhibits the same crossover. As we will show, the crossover between the SYK and FL regimes results in a qualitative change in the Coulomb drag resistance.
V 2 ij = V 2 N . 1d 2d N V 2 N 1u 2u N t 0 − 1d 2d 1u 2u a) N t 0 b) c) N N V 2 N u d
The trans-conductance is determined by the drag conductance between the two junctions, one from each layer, formed by the closest grains, which we assign with the numbers r = 1, r = 2, as shown in Fig. 1. The drag conductance is calculated according to the Kubo formula approach developed in Refs. [20,21]. The basic diagram describing the drag response between the layers u and d is shown in Fig. 2a). Solid lines in Fig. 2 denote the one-particle Green's functions of the SYK model. Since Coulomb drag is possible only if the particle-hole symmetry is violated, we assume that the SYK grains in both layers are away from half-filling. The charge asymmetry is parametrized by the parameters, introduced for SYK model in Ref. [47], E u,d ∝ −dµ u,d /dT , that are proportional to the temperature derivative of the corresponding chemical potentials. Detailed calculation outlined in the Supplemental Material [48] result in the following expression for the drag conductance in the strange metal (SM) regime
σ SM Drag ≈ 58.6N V 2 J 2 T 2 0 T 2 E u E d .(5)
The drag conductance diminishes with temperature as T −2 . At the same time, the DC conductance within the layer behaves as 1/T , [43], σ SM ≈ 0.886N e 2 h T0 T (for details see Supplemental material [48]). As a result the drag trans-resistance between the two strange metals is temperature-independent:
ρ SM Drag = σ SM Drag (σ SM ) 2 ≈ C SM N h e 2 V 2 J 2 E u E d ,(6)
where the numerical factor is estimated as C SM ≈ 74.7, and the SM regime is realized for T > T 0 . Furthermore, we find that in this regime drag resistance remains independent of the tunneling strength t 0 . This universality leads us to conclude that the validity of Eq. (6) extends beyond the specific microscopic model used in this paper. Besides the evident proportionality to the charge asymmetries in both layers, the only physical parameter governing the drag conductance is the ratio of the interand intra-layer interactions, V /J. Remarkably, this parameter can be independently determined through the measurement of the near-field heat transport [49][50][51][52], as we demonstrate below. The temperature-independent drag resistance of the strange metal stands in stark contrast to the drag resistance in the Fermi liquid, which is proportional to T 2 . In the Fermi liquid regime at T < T 0 < t 0 , tunneling is the most relevant term in the Hamiltonian. It smears the low-energy SYK singularity in the single-particle density of states, substituting it with a semi-circular energy band with a width of 4t 0 (at larger energy, 2t 0 < < J, the SYK-like tails remain). Assuming that the two chemical potentials fall within this central band, |µ u,d | < 2t 0 , the calculations of the drag conductance that are outlined in the Supplemental Material [48] result in
σ FL Drag ∝ N V 2 J 2 T 2 T 2 0 E u E d .(7)
Meanwhile, the intra-layer conductance in the FL regime is independent of temperature, σ FL = e 2 πh N . Therefore, the resulting drag resistance is given by
ρ FL Drag ≈ C FL N h e 2 V 2 J 2 T 2 T 2 0 E u E d ,(8)
where C FL ≈ 429.2 (for a detailed derivation of these results, see the Supplemental Material [48]). We conclude that the overall temperature dependence of the drag resistance rises as ∼ T 2 at low temperatures in the Fermi liquid regime and saturates to a temperature independent value at high temperatures in the SM regime. The drag resistances given by Eqs. (6), (8) become comparable in the range of temperature T ∼ T 0 = t 2 0 /J that marks the crossover between the Fermi-liquid and SM regimes. Since the numerical coefficient by the drag resistance in the Fermi liquid regime Eq. (8) is larger than the one in the SM regime Eq. (6), the estimation of the drag resistance in the two regimes at T = T 0 gives ρ FL Drag (T 0 ) > ρ SM Drag , which suggests that the overall temperature dependence may exhibit a maximum at temperatures about T 0 .
One may derive a phenomenological expression for the overall temperature dependence of the drag resistance based on the following physical picture. The energy spectrum in the tunnel-coupled SYK dots can be roughly separated into two regions. The states within the energy window of the order of the tunneling escape rate T 0 = t 2 0 /J form a quasi-Fermi liquid, contributing to the drag resistance according to Eq. (8). On the other hand, the energy states beyond the energy window of T 0 form the strange metal, leading to the drag resistance as given by Eq. (6). Both parts of the spectrum constitute the two liquids, contributing in parallel to the overall resistance. Since the high-energy states' population necessitates their thermal activation, the two liquids' contributions should be weighted by their corresponding thermal activation probabilities, resulting in the following expression for the inverse resistance:
1 ρ Drag = 1 − e −T0/T ρ FL Drag + e −T0/T ρ SM Drag .(9)
Qualitative temperature dependence of the drag resistance is shown in Fig. 3. It is important to note that the drag resistance calculation in the crossover regime necessitates exact form of the one-particle Green's functions of the tunnel-coupled SYK grains at the crossover temperature, which is currently unavailable to the best of our knowledge. Therefore, the question of whether the overall temperature dependence of the drag resistance exhibits a maximum remains open. Consider now the near-field heat transfer conductance in the model described by Eqs. (1)-(4). In the lowest order of interaction, the near-field heat transfer flux J h is given by the diagram shown in Fig. 2c), leading to the following result for the heat conductance in the SM regime:
κ SM = J h ∆T = 0.015N V 2 J 2 T,(10)
where T = (T u + T d )/2, ∆T = T u − T d , and we assume a small temperature difference ∆T T . Equation (10) allows one to define the near-field heat conductance as κ SM = J h /∆T , which is a linear function of temperature. The slope of the temperature dependence of the heat conductance is then directly related to the ratio V 2 /J 2 characterizing the interaction strength in the SM regime. Therefore, one can relate the drag resistance and the heat conductance as follows
ρ SM Drag = A SM N 2 h e 2 E u E d dκ SM dT ,(11)
where the constant A SM ≈ 4980. Eq. (11) provides a universal relation between the results of two different experiments in the incoherent metal.
Remarkably the same functional relation (11) between the drag resistance and the heat conductance holds in the Fermi liquid regime with a somewhat different numerical coefficient A FL ≈ 180. Indeed, the corresponding heat conductance is known to be [48][49][50][51] κ FL = 0.8N
V 2 T 3 t 4 0 = 0.8N V 2 J 2 T 3 T 2 0 .(12)
Along with Eq. (8) this leads to Eq. (11) with the aforementioned A FL . In summary, we studied the nonlocal electrical and thermal transport in the interactively coupled doublelayers of two strange metals. Each layer is modeled by the Hamiltonian of tunnel-coupled SYK quantum dots. This model is known to capture the physics of strange metal phases in the proper regime of parameters. If the temperature is smaller than the characteristic scale set by inter-grain tunneling and intra-grain interaction, we recover the FL regime with the quadratic temperature dependence of drag resistivity [Eq. (8)]. In the temperature range above that scale, we find trans-resistance approaching the limiting value, Eq. (6), from above. The latter fact reflects the interplay of Planckian intra-layer dissipation and interaction-mediated inter-layer dragging. Results obtained for our microscopic model differ from the recent study of the drag between two strange metal layers using the Einstein-Maxwell-dilaton model from holography, which claims ρ Drag ∝ T 4 [42]. Finally, we calculated near-field inter-layer thermal conductance. The established relationship, Eq. (11), between drag resistance and the near-field heat conductance that is free of parameters of the considered model suggests universality of this result.
In this Supplemental Material we provide details of the derivation of important formulas in the main text of the paper. In order to maintain a coherent and comprehensive presentation, we repeat here the Hamiltonian and the main diagrams (see Fig. 4) for the calculation of the drag conductance and the near field heat transfer. The Hamiltonian of the two SYK-array layers is given by
H = ν=u,d r H ν,r SYK + ν=u,d H ν t + V int .(13)
The Hamiltonian of an isolated SYK grain reads
H ν,r SYK = N ij,kl J νr ij,kl c + νri c + νrj c νrk c νrl ,(14)
where J νr ij,kl are random couplings drown from Gaussian distribution with zero mean and the variances |J νr ij,kl | 2 = 2J 2 N 3 . The interactions in different grains are statistically independent of each other. The inter-grain tunneling of an electron in a single layer is governed by the Hamiltonian
H ν t = r,r i t i c + νri c νr i + h.c.(15)
where t i denote random tunneling amplitudes derived from the Gaussian distribution with zero mean and the variance |t i | 2 = t 2 0 , and r, r denotes a pair of the nearest neighbor grains. The random inter-grain interaction between the on-site charge densities is given by
V int = i,j V ij r c + uri c uri c + drj c drj ,(16)
where the random interaction constants V ij have zero mean and are characterized by the variance V ij V kl = V 2 N .
One particle Green functions in the strange metal (high-temperature) regime
Since Coulomb drag is possible only if the particle-hole symmetry is violated, we assume that the SYK grains in both layers are away from half-filling. The charge asymmetry is parametrized by the parameters E u and E d correspondingly. Below we write down the single particle Green functions in the imaginary time τ and in the Matsubara frequency representations [41,47]
G(τ ) = − π 1/4 √ T e −2πET τ 2J sin(πT τ ) , (τ > 0),(17)G(τ ) = π 1/4 √ T e −2πE−2πET τ 2J sin(−πT τ ) , (τ < 0),(18)G(iω n ) = −i C(E) 2 √ πJT Γ 3 4 + n + iE(θ) Γ 5 4 + n + iE(θ) ,(19)
where ω n = 2πT (n + 1/2), and the constants C, E determine the charge asymmetry. They are related to each other as follows
C(E) = (1 + i) 1 + ie −2πE 2i .(20)
Analytical continuation to real frequencies results in the following retarded and advanced Green functions
G R (ω) = G A (ω) * = −ie −iθ √
2JT (π cos(2θ))
1/4 Γ 1/4 − i ω 2πT − E Γ 3/4 − i ω 2πT − E .(21)
Here the phase factor θ, −π/4 < θ < π/4 relates to the charge asymmetry parameters E and C as follows
C(θ) = π 1/4 √ J [cos(2θ)] 1/4 ,(22)E(θ) = 1 2π ln 1 + tan θ 1 − tan θ <=> tanh(2πE) = tan(θ).(23)
θ = 0 corresponds to the charge symmetry point (the half-filling). Note also, that in the SYK model, the charge asymmetry parameter relates to the chemical potential as
µ 0 − µ = 2πT E,(24)
where µ 0 denotes the chemical potential at zero temperature [34,41,47]. In what follows we use the expression for the Green function of the dimensionless frequency
x = ω 2πT G R/A (ω) = K syk g R/A syk (x),(25)
where
K syk = 1 2 √ πJT ,(26)g R syk (x) = g A syk (x) * = −ie −iθ (cos(2θ)) 1/4 Γ [1/4 − i (x − E)] Γ [3/4 − i (x − E)] .(27)
One particle Green functions in the Fermi liquid (low-temperature) regime
At temperatures less then single-electron tunneling rate between the two SYK-grains, T < T 0 = t 2 0 /J, the array of grains enters the Fermi liquid regime. In that regime, the transport properties of the array are determined by the energies less than T 0 . In turn, the single particle spectrum of each grain at those energies is determined by random inter-grain tunneling amplitudes, which leads to the semi-circular energy band with the width 4t 0 given by the variance of the random tunneling. Close to the center of the band, |ω + µ| 2t 0 , the Green functions can be written as
G R (ω) ≈ 2 ω + µ + i 2τ0 ,(28)
where the life time τ 0 is determined by the bandwidth
1 2τ 0 = 2t 0 .(29)
The advanced Green function is given by the complex conjugated expression. Similarly to the SYK regime discussed above, we further use the Green function of dimensionless frequency
x = ω 2πT G R/A (ω) = K fl g R/A fl (x).(30)
Here
K fl = 1 πT ,(31)g R fl (x) = g A fl (x) * = 1 x − E + i w ,(32)
where the dimensionless parameter w corresponding to the decay time is defined as w = 4πT τ 0 = πT /t 0 , and
E = −µ/(2πT ). 1d 2d N V 2 N 1u 2u N t 0 − 1d 2d 1u 2u a) b) N N V 2 N c) Layer d Layer u d1 d2 u1 u2 t i t i V ij V ij . . .
CALCULATION OF DRAG CONDUCTANCE
Here we calculate the drag conductance according to the Kubo formula approach developed in Ref. [20].
σ Drag = V 2 16πT N ∞ −∞ dω sinh 2 ω 2T Γ +− u (ω, ω)Γ +− d (ω, ω).(33)
The basic diagrams describing the drag-response between the layers u and d are shown in Fig. 4a). Solid lines in Fig. 4 denote the one-particle Green functions of the SYK model. The factors Γ +− u,d (ω, ω) denote the triangular parts of the diagram in Fig. 4a), each corresponding to the mathematical expression
Γ +− (ω, ω) = N t 2 0 4πi d tanh + ω 2T − tanh 2T (G A ( )) 2 − (G R ( )) 2 G R ( + ω)G A ( + ω) − {ω → −ω} .(34)
The one particle Green functions in Eq. (34) should be taken with the charge asymmetry parameters E u/d for the up and down layer respectively. Introducing dimensionless frequencies ξ = ω 2πT and x = 2πT we can cast the expression for Γ +− (ω, ω) to the form
Γ +− (ω, ω) = − i 2 N t 2 0 T K 4 [γ(ξ) − γ(−ξ)] ,(35)
where
γ(ξ) = dx [tanh (π(x + ξ)) − tanh (πx)] (g A (x)) 2 − (g R (x)) 2 g R (x + ξ)g A (x + ξ),(36)
where the constant K and the dimensionless Green functions g(x) are determined by Eqs. (26), (42) and Eqs. (31), (32) in the strange metal (SM) and Fermi liquid (FL) regimes respectively. Then, using dimensionless frequencies, Eq. (33) can be rewritten in the form
σ Drag = − N V 2 t 4 0 32 T 2 K 8 ∞ −∞ dξ sinh 2 (πξ) [γ u (ξ) − γ u (−ξ)] [γ d (ξ) − γ d (−ξ)] .(37)
Small-E expansion of the drag conductance
The lowest order term in the small -E expansion of the drug conductance is obtained from Eq. (37) as
σ Drag ≈ −E u E d N V 2 t 4 0 32 T 2 K 8 ∞ −∞ dξ sinh 2 (πξ) [γ (ξ) − γ (−ξ)] 2 ,(38)
where
γ (ξ) = ∂γ(ξ) ∂E E=0 .(39)
Drag conductance in the SM regime Substituting explicit expression Eq. (27) in Eq. (36), we obtain
γ(ξ) = dx [tanh(π(x + ξ)) − tanh(πx)] cos(2θ) e 2iθ Γ(1/4 + i(x − E)) Γ(3/4 + i(x − E)) 2 − e −2iθ Γ(1/4 − i(x − E)) Γ(3/4 − i(x − E)) 2 × Γ(1/4 − i(x + ξ − E)) Γ(3/4 − i(x + ξ − E)) Γ(1/4 + i(x + ξ − E)) Γ(3/4 + i(x + ξ − E)) .(40)
For the explicit calculation of derivative over E it is convenient to shift the integration variable x → x + E in Eq. (40). Then, in the SM regime we obtain
γ (ξ) = ∞ −∞ dx π cosh 2 [π(x + ξ)] − π cosh 2 (πx) (g R syk (x)) 2 − (g A syk (x)) 2 g R syk (x + ξ)g A syk (x + ξ) + 4πi ∞ −∞ dx [tanh[π(x + ξ)] − tanh(πx)] (g R syk (x)) 2 + (g A syk (x)) 2 g R syk (x + ξ)g A syk (x + ξ),(41)
where the functions g
R/A
syk (x) are taken for E = 0,
g R syk (x) = g A syk (x) * = Γ(1/4 − ix) Γ(3/4 − ix)(42)
Furthermore, evaluating the derivative over E, and using Eq. (38) together with Eq. (26), we obtain the expression for the drag conductance in the final form
σ SM Drag ≈ C N t 4 0 V 2 J 4 T 2 E u E d = CN V 2 J 2 T 0 T 2 E u E d ,(43)
where the constant C is evaluated numerically as C ≈ 58.6, and we introduced the crossover temperature T 0 = t 2 0 /J in the second equation.
One particle conductance in the SM regime
We calculate one particle conductance as the tunneling conductance between the two SYK grains according to the formula
σ SM 1 = e 2 N t 2 0 dω 2π ν 2 SYK (ω) 4T cosh 2 ω 2T ,(44)
where ν SYK denotes the one particle density of states in the SYK grain at the Fermi energy, which can be obtained from the imaginary part of the one particle Green function Eq. (21). Since we calculate the drag resistance in the lowest order of the charge asymmetry parameter E, the calculation of the one particle conductance can be performed for the charge symmetric point E = 0. Then we obtain
ν SYK = − 1 π ImG R (ω, E = 0) = √ 2 π 1/4 √ JT Re Γ 1/4 − i ω 2πT Γ 3/4 − i ω 2πT ,(45)σ SM 1 = e 2 h N t 2 0 2 √ πJT ∞ −∞ dx cosh 2 (πx) Re Γ(1/4 − ix) Γ(3/4 − ix) 2 ≈ 0.886 e 2 h N t 2 0 JT = 0.886N e 2 h T 0 T ,(46)
which is in accord with results of Ref. [43] Drag resistance in the SM regime
The drag resistance is obtained as
ρ SM Drag = σ SM Drag σ SM 1 2 ≈ C SM h e 2 V 2 N J 2 E u E d ,(47)
where the numerical factor C SM is estimated as C SM ≈ 74.7. Therefore, the SM drag resistance is independent of temperature.
Drag conductance in the Fermi liquid regime
It is the chemical potential µ rather than the dimensionless parameter E that determines the filling fraction in the Fermi liquid regime. For small charge asymmetry (close to the half-filling), the chemical potential is much smaller than the random energy bandwidth, µ 4t 0 . In contrast to the linear temperature dependence of the chemical potential in the SM regime, in the FL regime the chemical potential µ at a constant filling is only weakly dependent on temperature. Yet for the sake of technical convenience we use Eqs. (31), (32), formulated in terms of dimensionless quantities for the evaluation of the drag conductance in the Fermi liquid regime. We note that despite E = −µ/(2πT ) can become large for temperatures close to zero, it still remains much smaller than 1/w = t 0 /(πT ). It follows that the product Ew = −µ/(2t 0 ) plays the role of the small parameter for the expansion close to the charge-symmetry point (half-filling).
Substituting Eq. (32) in Eq. (36), and shifting the integration variable x − E → x we obtain
γ(ξ) − γ(−ξ) = ∞ −∞ 4ix/w x 2 + 1 w 2 2 [tanh[π(x + E + ξ)] − tanh[π(x + E)]] 1 (x + ξ) 2 + 1 w 2 − [tanh[π(x + E − ξ)] − tanh[π(x + E)]] 1 (x − ξ) 2 + 1 w 2 .(48)
Furthermore, changing the integration variable x → −x in the second line, and rescaling x → x w , we cast Eq. (48) into the form suitable for the expansion in small wE
γ(ξ) − γ(−ξ) = ∞ −∞ dx 4iw 3 x (x 2 + 1) 2 1 (x + wξ) 2 + 1 tanh π w (x + wξ + wE) − tanh π w (x + wξ − wE) − tanh π w (x + wE) − tanh π w (x − wE) ≈ 8iπw 3 E ∞ −∞ xdx (x 2 + 1) 2 [(x + wξ) 2 + 1] 1 cosh 2 π w (x + wξ) − 1 cosh 2 π w x ≈ 8iπw 3 E w/π −w/π (−wξ)dx (w 2 ξ 2 + 1) 2 (x 2 + 1) − w/π −w/π xdx (x 2 + 1) 2 1 w 2 ξ 2 + 1 = −i 16w 5 Eξ (1 + w 2 ξ 2 ) 2 .(49)
Substituting Eq. (49) in Eq. (37), we obtain the integral over ξ in the leading order in w in the form
∞ −∞ dξ sinh 2 (πξ) [γ (ξ) − γ (−ξ)] 2 = −2 8 w 10 ∞ −∞ dξ sinh 2 (πξ) ξ 2 (1 + w 2 ξ 2 ) 2 ≈ − 2 8 3π w 10 .(50)
Finally, restoring all factors in Eq. (38), using w = πT τ 0 = πT /t 0 in Eq. (50), and taking into account the relation between E and the chemical potential µ, we obtain
σ FL Drag ≈ 2 3 N V 2 T 2 t 6 0 µ u µ d .(51)
One particle conductance in the Fermi liquid regime
The one particle conductance is calculated as the tunneling conductance between the nearest neighbor grains, using the simplified expression for the Green functions Eq. (28), which results in [43]
σ 1 = N e 2 πh .(52)
Therefore, the one particle conductance is independent of tunneling strength and temperature (at low temperatures).
The independence of the one-particle conductance of the tunneling strength is explained by the fact that the bandwidth (and thus the one-particle density of states) is determined by the variance of the randomized inter-grain tunneling, which leads to the exact compensation between the one particle density of states and the inter-grain tunneling amplitude.
Drag resistance in the Fermi liquid regime
The drag resistance is obtained as
ρ FL Drag = σ FL Drag σ 2 1 ≈ 2π 2 3 h e 2 V 2 T 2 N t 6 0 µ u µ d .(53)
Therefore, the drag resistance in the Fermi liquid regime is proportional to the temperature squared.
Drag conductance in the crossover regime
Here we compare the values of the drag conductance in the SM and in the FL regime at the crossover temperature T 0 = t 2 0 /J. We show that the estimations for the two regimes differ at the crossover temperature only by the numerical factor. We assume that the charge asymmetry remains constant for all temperatures.
In the SM regime, the charge asymmetry is determined by the asymmetry factor E. At small charge asymmetry (close to the half-filled system), the deviation from the half filling is given by the expression [41,47]
Q − 1/2 ≈ −E(1 + π/2).(54)
In the Fermi liquid regime, the charge asymmetry is determined by the chemical potential. Close to the half filling at low temperature, the deviation from the half filling is given by
Q − 1/2 = dων 1 (ω) 1 e (ω−µ)/T + 1 − 1 2 ≈ ν 1 (0)µ,(55)
where ν 1 (ω) denotes the single-particle density of states. The one particle density of states (DOS) ν 1 (0) at the middle of the semicircular band is given by ν 1 (0) = 4τ 0 /π = 1/(πt 0 ),
which leads to the relation
Q − 1/2 ≈ µ πt 0 .(57)
Equating the expressions for the filling in the strange metal and in the Fermi liquid regime given by Eqs. (54) and (57), we obtain the relation between the chemical potential in the Fermi liquid and the asymmetry parameter in the SYK, which reads µ FL = −πt 0 (1 + π/2)E SM .
(58)
Now we use Eq. (58) to compare the estimations for the drug conductance in the SM and in the FL regimes at the temperature T 0 = t 2 0 /J. In the FL regime we can represent Eq. (51) in the form
σ Drag ≈ 2 3 N π 2 (1 + π/2) 2 V 2 T 2 t 4 0 E u E d = 2 3 N π 2 (1 + π/2) 2 V 2 J 2 T T 0 2 E u E d .(59)
At T = T 0 we obtain
σ FL Drag (T 0 ) ≈ 43.5N V 2 J 2 E u E d .(60)
In the SM regime we obtain from Eq. (5) σ SM Drag (T 0 ) ≈ 58.6N
V 2 J 2 E u E d .(61)
One can see that the estimation for the drag conductance at the crossover temperature T 0 differ by the numerical prefactor only. Furthermore, expressing the chemical potentials through the spectral asymmetry parameters in Eq. (8), and introducing the crossover temperature T 0 = t 2 0 /J we obtain the drag resistance in the Fermi liquid regime in the form
ρ FL Drag ≈ C FL N h e 2 V 2 J 2 T 2 T 2 0 E u E d ,(62)
where C FL ≈ 429.2
CALCULATION OF NEAR FIELD HEAT TRANSFER FLUX
In this section we calculate the near field heat transport between the up and down layers. We assume that the heat transport takes place between the nearest grains in the different layers, that is between the grains with the same number (1d ↔ 1u), (2d ↔ 2u). The general expression for the near field heat transfer flux, illustrated by the diagram Fig. 4b), is given by [53]
J h = V 2 N dω 2π (ImΠ R u (ω))(ImΠ R d (ω))ω [n B (ω/T u ) − n B (ω/T d )] .(63)
Here n B (ω/T ) denotes the Bose distribution at temperature T , and ImΠ R u/d (ω) denotes the imaginary part of the polarization operator. Since no charge asymmetry is required for a finite heat transfer, we perform calculations at the charge symmetric point E u = E d = 0.
We calculate the polarization operator using the Keldysh formalism, where it is defined as [53] Π
R (ω) = i 2 N d 2π G R ( + ω)G K ( ) + G K ( + ω)G A ( ) .(64)
Here G K ( ) = tanh 2T G R ( ) − G R ( ) denotes the Keldysh Green function at temperature T . Introducing dimensionless frequencies x = 2πT , ξ = ω 2πT , and using the definitions of the Green functions Eqs. (25), (30), we cast Eq. (64) in the form
Π R (ω) = i 2 N K 2 T κ(ξ),(65)
FIG. 2
2. a) Diagrams for the drag trans-conductance. Full lines represent interacting SYK Green functions, wavy linesinter-layer interactions, and crossed circles -intra-layer tunneling; b) Diagram for the inter-dot conductance within a single layer. c) Diagram describing the heat current between the SYK dots in the up (u) and down (d) layers. Factors of N are indicated explicitly.
FIG. 3 .
3Temperature dependence of the drag resistance (in units of the drag resistance at high temperature ρ∞): the T 2 increase of resistance in the low temperature FL regime changes to saturation in the high temperature SM regime.
FIG
. 4. a) Diagrams for the drag response; b) Diagram for the near field heat transfer response; c) Scheme of the two layer system.
arXiv:2304.12221v1 [cond-mat.str-el] 24 Apr 2023Layer d
Layer u
d1
d2
u1
u2
t i
t i
V ij
V ij
. . .
. . .
. . .
. . .
FIG. 1. Schematic representation of the SYK double layer
setup. The four depicted dots is the minimal set needed to
evaluate the drag trans-conductance.
Fig. 1. The Hamiltonian of the adopted model reads
H =
ν=u,d r
H νr
SYK +
ν=u,d
We thank A. Patel for the communication regarding Ref.[35]. This work at UW-Madison was financially supported by the National Science Foundation Grant No. DMR-2203411 (A.L.). A.K. was supported by the NSF Grant No. DMR-2037654. A.C. thanks the Fine Theoretical Physics Institute at the University of Minnesota for hospitality and support.whereFor the further evaluation we assume small temperature difference between the layers, T u = T − (∆T )/2, T d = T + (∆T )/2. Then, substituting Eqs. (65), (66) in Eq. (63) and expanding the difference of Bose distribution functions in ∆T , we obtainStrange metal regimeSubstituting K = K syk as given by Eq.(26)and g R/A = g R/A syk as given by Eq.(27)in Eq. (67), we obtainFurther numerical evaluation of the integral in Eq. (67) results in the expression for the heat transfer fluxfrom which it follows for the heat conductanceFermi liquid regimeIn the Fermi liquid regime we use Eqs.(31)and(32). Substituting them into Eq. (67), we obtainSince there is no scale invariance of the Green functions in the FL regime, the last integration does not reduce to a number. Rather it is a function of the dimensionless density of states in the center of the random energy band w = πT /t 0 . Explicitly we obtain for κ(ξ) from Eq. (66)For the further evaluation we note, that T t 0 in the FL regime hence w 1, and that the actual values of ξ are restricted by |ξ| 1 by the following integration over ξ in Eq. (67). Those conditions justify the evaluation of κ FL in the leading order in w and ξ, which contributes to the neglecting the terms wξ in the denominator of Eq. (72) and replacing tanh(2πy/w) ≈ sign(y). Under those approximations we obtainSubstituting Eq. (73) in Eq. (71) and performing integration over ξ, we obtain the heat transfer flux in the FL regime in the form
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| {'fraction_non_alphanumeric': 0.07444348368155403, 'fraction_numerical': 0.05373122322990609, 'mean_word_length': 3.8155321003195506, 'pattern_counts': {'":': 0, '<': 10, '<?xml version=': 0, '>': 4, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, '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': 'We address Coulomb drag and near-field heat transfer in a double-layer system of incoherent metals. Each layer is modeled by an array of tunnel-coupled SYK dots with random inter-layer interactions. Depending on the strength of intra-dot interactions and inter-dot tunneling, this model captures the crossover from the Fermi liquid to a strange metal phase. The absence of quasiparticles in the strange metal leads to temperature-independent drag resistivity, which is in strong contrast with the quadratic temperature dependence in the Fermi liquid regime. We show that all the parameters can be independently measured in near-field heat transfer experiments, performed in Fermi liquid and strange metal regimes.', 'arxivid': '2304.12221', 'author': ['A L Chudnovskiy \nInstitut für Theoretische Physik\nUniversität Hamburg\nNotkestraße 9D-22607HamburgGermany\n', 'Alex Levchenko \nDepartment of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n', 'Alex Kamenev \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n\nWilliam I. Fine Theoretical Physics Institute\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n', 'A L Chudnovskiy \nInstitut für Theoretische Physik\nUniversität Hamburg\nNotkestraße 9D-22607HamburgGermany\n', 'Alex Levchenko \nDepartment of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA\n', 'Alex Kamenev \nSchool of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n\nWilliam I. Fine Theoretical Physics Institute\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA\n'], 'authoraffiliation': ['Institut für Theoretische Physik\nUniversität Hamburg\nNotkestraße 9D-22607HamburgGermany', 'Department of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA', 'School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA', 'William I. Fine Theoretical Physics Institute\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA', 'Institut für Theoretische Physik\nUniversität Hamburg\nNotkestraße 9D-22607HamburgGermany', 'Department of Physics\nUniversity of Wisconsin-Madison\n53706MadisonWisconsinUSA', 'School of Physics and Astronomy\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA', 'William I. Fine Theoretical Physics Institute\nUniversity of Minnesota\n55455MinneapolisMinnesotaUSA'], 'corpusid': 258298288, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 17769, 'n_tokens_neox': 14754, 'n_words': 8370, 'pdfsha': 'a7439edf00080f49c582330dfba232a81500b356', 'pdfurls': ['https://export.arxiv.org/pdf/2304.12221v1.pdf'], 'title': ['Coulomb drag and heat transfer in strange metals', 'Coulomb drag and heat transfer in strange metals', 'Coulomb drag and heat transfer in strange metals', 'Coulomb drag and heat transfer in strange metals'], 'venue': []} |
arxiv |
Evidence for Transition Temperature Fluctuation Induced Pinning in MgB 2 Superconductor
10 Aug 2001
M J Qin
Institute for Superconducting and Electronic Materials
University of Wollongong
2522WollongongNSWAustralia
X L Wang
Institute for Superconducting and Electronic Materials
University of Wollongong
2522WollongongNSWAustralia
H K Liu
Institute for Superconducting and Electronic Materials
University of Wollongong
2522WollongongNSWAustralia
S X Dou
Institute for Superconducting and Electronic Materials
University of Wollongong
2522WollongongNSWAustralia
Evidence for Transition Temperature Fluctuation Induced Pinning in MgB 2 Superconductor
10 Aug 2001
The magnetic field dependent critical current density j c (B) of a MgB 2 bulk sample has been obtained by means of magnetization hysteresis measurements. The j c (B) curves at different temperatures demonstrate a crossover from single vortex pinning to small-bundle vortex pinning, when the field is larger than the crossover field B sb . The temperature dependence of the crossover field B sb (T ) is in agreement with a model of randomly distributed weak pinning centers via the spatial fluctuations of the transition temperature (δT c -pinning), while pinning due to the mean free path fluctuations (δl-pinning) is not observed. 74.70.Ad, 74.25.Ha, 74.25.Dw Typeset using REVT E X
The recent discovery of superconductivity in the intermetallic compound MgB 2 [1] with transition temperature at 39K has led to intensive experimental and theoretical activities [2][3][4][5][6][7][8][9][10][11][12][13][14][15], with the purpose of understanding the basic mechanism of superconductivity and the vortex pinning mechanism governing the critical current density j c in this new superconductor. Although the critical current density has been improved greatly since its discovery, the underlying pinning mechanism is still under investigation.
In type-II superconductors, the most important elementary interactions between vortices and pinning centers are the magnetic interaction and the core interaction [16][17][18][19][20][21][22]. The magnetic interaction arises from the interaction of surfaces between superconducting and non-superconducting material parallel to the applied magnetic field. In technical type-II superconductors with a high Ginzburg-Landau (GL) parameter κ, the magnetic interaction is usually very small and disappears with increasing magnetic field. The core interaction is usually more effective in technical type-II superconductors due to the short coherence length and the larger penetration depth (high κ). This interaction arises from the coupling of the locally distorted superconducting properties with the periodic variation of the superconducting order parameter. Two mechanisms of core pinning are predominant in type-II superconductors, i.e. δT c -and δl-pinning. Whereas δT c -pinning is caused by the spatial variation of the GL coefficient α associated with disorder in the transition temperature T c , variations in the charge carrier mean free path l near lattice defects are the main cause of δl-pinning.
It has been reported by Griessen et al. [16] that the δl-pinning mechanism is dominant in both YBa 2 Cu 3 O 7 and YBa 2 Cu 4 O 8 thin films. For the new superconductor MgB 2 , a high κ value of 26 has been reported [10], it is therefore expected that the magnetic interaction is negligible, while the core interaction is more important. However, it has not been experimentally determined whether the δl-pinning or the δT c -pinning is the dominant mechanism in MgB 2 . The purpose of this Letter is to report measurements of the critical current density of this new material to achieve an understanding of the vortex pinning mechanism and to demonstrate that in MgB 2 governed by bulk pinning, δT c is the only important pinning mechanism.
All measurements have been performed on a MgB 2 bulk sample, which was prepared by conventional solid state reaction [23]. High purity Mg and B (amorphous) with a nominal composition ratio of Mg:B=1.2:2 were mixed and finely ground, then pressed into pellets 10 mm in diameter with 1-2 mm thickness. Extra Mg was added in order to make up for loss of Mg at high temperatures. These pellets were placed on an iron plate and covered with iron foil, then put into a tube furnace. The samples were sintered at temperatures between 700 and 1000 o C for 1-14h. A high purity Ar gas flow was maintained throughout the sintering process. A sample with T c = 38.6 K and dimensions of 2.18 × 2.76 × 1.88 mm 3 was cut from the pellet. Phase purity was determined by XRD [24] and grain size by SEM. Only a small level of MgO (less than 10%) was found and the grain size was determined to be about 200
µm. Fig. 2 as different symbols. As can be seen from the plateau at low magnetic field, j c initially has a weak dependence on the field. When the magnetic field is increased beyond a crossover field, it then begins to decrease quickly. The crossover field decreases with increasing temperature. Further increasing the magnetic field results in a faster drop in j c near the irreversibility line, which is obtained by using a criterion for the critical current density j c = 100 A/cm 2 . The results are shown as open circles in Fig. 3.
The best fitting of the data yields the result,
B irr (t) = B irr (0) 1 − t 2 3 2 ,(1)
shown as solid line in Fig. 3, with t = T /T c . For high temperature superconductors, a
(1 − t) 3/2 behavior is usually observed [25] and has been explained by means of giant flux creep [26,27].
The (1 − t) 3/2 law is actually an approximation of the (1 − t 2 ) 3/2 law as t → 1, because (1−t 2 ) ≈ (1−t)(1+t) ≈ 2(1−t). Alsoaccount, one have F p = j c B = n p f 2 p (u 0 /f p )d/a 2 0 ,
where u 0 is the maximum distortion of the flux line lattice, d is the range of the pinning force, typically of the order of xi, and a 0 the flux line lattice constant. These two strong models yield j c (B) characteristics of j c ∝ B −1 and j c ∝ B −0.5 , respectively. Due to the large densities of the pins n p and the small elementary interaction forces f p , these two models are not representative for most real pinning systems [28]. For randomly distributed weak pinning centers, the macroscopic pinning force F p can be estimated using the basic concept of collective pinning [29], which has been proved to be very successful in most real pinning systems,
F p = j c B = W V c = W R 2 c L c(2)
with the correlation volume V c = R 2 c L c , with the correlation lengths R c perpendicular to the field direction and L c along the vortex line, and the pinning parameter W = n p < f 2 p >.
R c and L c depend on the applied magnetic field, the dimension of the flux line lattice (3D or 2D), and the elasticity or plasticity of the flux line lattice. For the 3D elastic flux line lattice, it has been derived by Blatter et al. [17] that j c is field-independent when the applied magnetic field is lower than the crossover field B sb (single vortex pinning)
B sb = β sb j sv j 0 B c2(3)
where β sb ≈ 5 is a constant,
j 0 = 4B c /3 √ 6µ 0 λ the depairing current, B c = Φ 0 /2 √ 2πλξ the thermodynamic critical field, B c2 = µ 0 Φ 0 /
When the applied magnetic field is larger than B lb = β lb B c2 (j sv /j 0 )[ln(κ 2 j sv /j 0 )] 2/3 (where β lb is a constant ≈ 2), this large bundle pinning regime is governed by a power law j c (B) ∝ B −3 . The 2D elastic flux line lattice shows single pancake pinning at low magnetic fields with field-independent j c , then a 2D collective-pinning region at higher fields with j c ∝ 1/B.
For fields higher than a crossover field B 3D b , three-dimensional pinning is predicted. From the characteristics of the j c (B) curves shown in Fig. 2, it is expected that the 3D elastic pinning model (single vortex pinning followed by small bundle pinning, then large bundle pinning) may the dominant pinning mechanism in MgB 2 . We therefore use The crossover field B sb we have obtained between single vortex pinning and small bundle pinning as a function of temperature is shown in Fig. 5 as open circles. We now compare the experimental data with theoretical predictions to get some insight into the pinning mechanism in MgB 2 . Using λ ∝ (1 − t 4 ) −1/2 and ξ ∝ [(1 + t 2 )/(1 − t 2 )] 1/2 , Griessen et al.
have found that for δl-pinning the critical current density in the single vortex pinning regime j sv ∝ (1 − t 2 ) 5/2 (1 + t 2 ) −1/2 , while for δT c -pinning j sv ∝ (1 − t 2 ) 7/6 (1 + t 2 ) 5/6 . Inserting all these expressions into Eq.(3), we have
B sb = B sb (0) 1 − t 2 1 + t 2 2/3(5)
for δT c -pinning, and
B sb = B sb (0) 1 − t 2 1 + t 2 2(6)
for δl-pinning. The lines corresponding to Eqs. (5) and (6) are indicated as δT c -pinning and δl-pinning respectively in Fig. 5. The central result of this Letter is the remarkably good agreement found between B sb and the corresponding δT c -pinning line in the figure. In sharp contrast, the δl-pinning line shown in Fig. 5 is in total disagreement with the experimental data.
Having derived the crossover fields B sb and B th , we now reconstruct the B-T phase diagram shown in Fig. 3. The final B-T phase diagram is shown in Fig. 6. The vortex solid region is divided into three smaller regions. Dingle vortex pinning governs the region below (1) to the experimental data (see Fig. 3). B th is the crossover field to thermal dominant region (see Fig. 5).
Again the B c2 (T ) line is taken from Takano's data using resistive measurements.
Fig. 1
1shows the magnetization hysteresis loops of the MgB 2 sample every 2 K in the 14-36 K range. The results at lower temperatures, which have large flux jumping[23], are not shown here. The symmetric magnetization hysteresis loops with respect to the magnetic field indicate the dominance of bulk current up to temperatures near T c , rather than surface shielding current. Therefore, the bulk pinning is dominant in this sample, while the surface pinning is negligible. As a comparison, we show in the inset ofFig. 1 the magnetization hysteresis loop of a pressed MgB 2 sample at 5 K. This sample is fabricated by pressing the MgB 2 powder into a pellet without sintering. The loop is highly asymmetric, showing a large reversible magnetization resulting from the surface current, and surface pinning plays an important role in this sample. No flux jumping is observed down to T=5 K, indicating that the grains in the sample are decoupled. The surface pinning effect has also been observed by Takano et al. [13] in their powder sample and bulk sample sintered at low temperature. From these M(H) loops, we can calculate the critical current density using j c = ∆M/a(1− a/3b), with a, b the width and length of the sample perpendicular to the applied magnetic field, respectively. The resultant j c (B) curves at various temperatures are shown in a double logarithmic plot in
plotted in the figure is the B c2 (T ) data taken from Takano's work (the dashed line is just guide to the eye), showing that B irr is well below B c2 . Therefore, giant flux creep also plays an important role in the new superconductor MgB 2 .j c (B) characteristics very similar to those shown in Fig. 2 have been observed by other groups [3,14,15]. Based on different physical assumptions on summation of the elementary pinning force f p to obtain the macroscopic pinning force F p , different pinning models yield different j c (B) characteristics. The simplest model is the direct summation of f p to have F p = j c B = n p f p , where n p is the density of pinning centers in the sample. This strong pinning model neglects the influence of the flux line lattice. When the influence is taken into
0
02πξ 2 the upper critical field and j sv the critical current density in the single vortex pinning regime. When the applied field is larger than B sb (small bundle pinning), j c (B) follows a exponential law j c (B) ≈ j c (
Eq.(4) to fit the j c (B) curves, with fitting parameters j c (0) and B 0 . The fitting results for different temperatures are shown as solid lines in Fig. 2. At intermediate fields, Eq.(4) fits the experimental data very well, while deviations from the fitting curves can be observed at both low and high fields. A clearer plot is shown in Fig. 4, where the j c (B) curve at 24 K is shown in a double-logarithmic plot of − log[j/j(B = 0)] versus the applied field, which clearly shows a straight line at intermediate magnetic fields. The deviation at low fields is denoted as B sb , indicating the crossover from the single vortex pinning regime to the small bundle pinning regime. The point of deviation at high fields was first considered as the crossover field from small bundle pinning to large bundle pinning. However, when we fit the j c (B) data at high fields to the power law j c (B) ∝ B −n , n is found to be as large as 20 rather than the theoretically predicted value of 3, making it unlikely that the system changes to the large bundle pinning regime. As the high field deviation is very close to the irreversibility line, which results from giant flux creep, it is likely that the deviation at high field may result from large thermal fluctuations, which lead to the rapid decrease in j c , and therefore is denoted as B th , indicating thermal fluctuations. Other j c (B) expressions have also been tested for the fitting, such as j c (B) ∝ 1/[1 + (B/B 0 ) n ] and j c (B) ∝ exp[−(B/B 0 )], but both yield poor fitting results. The inset of Fig. 4 shows the j c (B) of the pressed sample at 5 K in a double-logarithmic plot, which indicates that when the surface pinning is important, the exponential drop in j c (B) [see Eq.(4)] no longer applies, but a power law j c (B) ∝ B −1.2 is obvious.
B
sb , between B sb and B th , small bundle pinning becomes dominant, while between B th and B irr , thermal fluctuations are more important. Large flux bundle pinning is not observed in MgB 2 , but may be concealed by the thermal fluctuation effects. In summary, we have found strong evidence for δT c -pinning, i.e. pinning via the spatial fluctuations in the transition temperature, in the new superconductor MgB 2 , while δl-pinning i.e., pinning via the spatial fluctuations of the charge carrier mean free path, is not observed. The B-T phase diagram of the MgB 2 sample has been derived, showing that at low fields below B sb the system is dominated by single vortex pinning and changes to smaller bundle pinning when B > B sb . When B > B th , this region in the vortex solid area is dominated by thermal fluctuations. The irreversibility line may result from the giant flux creep effect. The authors would like to thank the Australian Research Council for financial support.FIGURES FIG. 1. Magnetization hysteresis loops of the MgB 2 bulk sample taken every 2 K in the 14-36 K range. Results at lower temperatures are not shown because of large flux jumping. Inset shows the magnetization hysteresis loop of a pressed MgB 2 sample at 5 K, showing large reversible magnetization from the surface current. FIG. 2. Critical current density j c calculated from the Bean critical state model, indicated by different symbols for different temperatures. Solid lines are fitting curves using Eq.(4). FIG. 3. B-T phase diagram of the new superconductor MgB 2 . Open circles represent the irreversibility line obtained from Fig. 2, and the solid line is a fit to B irr = 5.2(1 − t 2 ) 3/2 . Solid circles represent the B c2 (T ) line from Takano's data using resistive measurements. The dashed line is just a guide to the eye. FIG. 4. Critical current density at 24 K in a double-logarithmic plot of − log[j/j(B = 0)] versus the applied field. The solid line is a fit using Eq.(4). B sb indicates the crossover field from single vortex pinning to small bundle pinning, while B th is the crossover field to the thermal fluctuations dominated regime. Inset shows the j c (B) curve of the pressed MgB 2 sample with a large contribution from the surface current, which shows a B −1.2 behavior. FIG. 5. Temperature dependence of the crossover field B sb . The δT c -pinning line corresponds to Eq.(5), which is in agreement with the experimental data, while The δl-pinning line corresponds to Eq.(6), which is not in agreement with the experimental data. FIG. 6. B-T phase diagram of the new superconductor MgB 2 . B sb (T ) is the fitting curve of Eq.(3) to the experimental data (see Fig. 5). B irr (T ) is the fitting curve of Eq.
Fig.1 by Qin et al.
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| {'fraction_non_alphanumeric': 0.07101874858820872, 'fraction_numerical': 0.03758753105940818, 'mean_word_length': 3.5556698909240585, 'pattern_counts': {'":': 0, '<': 1, '<?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': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The magnetic field dependent critical current density j c (B) of a MgB 2 bulk sample has been obtained by means of magnetization hysteresis measurements. The j c (B) curves at different temperatures demonstrate a crossover from single vortex pinning to small-bundle vortex pinning, when the field is larger than the crossover field B sb . The temperature dependence of the crossover field B sb (T ) is in agreement with a model of randomly distributed weak pinning centers via the spatial fluctuations of the transition temperature (δT c -pinning), while pinning due to the mean free path fluctuations (δl-pinning) is not observed. 74.70.Ad, 74.25.Ha, 74.25.Dw Typeset using REVT E X', 'arxivid': 'cond-mat/0108172', 'author': ['M J Qin \nInstitute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia\n', 'X L Wang \nInstitute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia\n', 'H K Liu \nInstitute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia\n', 'S X Dou \nInstitute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia\n'], 'authoraffiliation': ['Institute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia', 'Institute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia', 'Institute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia', 'Institute for Superconducting and Electronic Materials\nUniversity of Wollongong\n2522WollongongNSWAustralia'], 'corpusid': 18291923, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7610, 'n_tokens_neox': 6594, 'n_words': 3977, 'pdfsha': '2377afc2cd8c72facf9f57d56b3c1105660dd0e1', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/0108172v1.pdf'], 'title': ['Evidence for Transition Temperature Fluctuation Induced Pinning in MgB 2 Superconductor', 'Evidence for Transition Temperature Fluctuation Induced Pinning in MgB 2 Superconductor'], 'venue': []} |
arxiv |
A note on the partition bound for one-way classical communication complexity
21 Feb 2023 February 22, 2023
Srinivasan Arunachalam
João F Doriguello
Rahul Jain
A note on the partition bound for one-way classical communication complexity
21 Feb 2023 February 22, 2023
We present a linear program for the one-way version of the partition bound (denoted prt 1 ε (f )). We show that it characterizes one-way randomized communication complexity R
Introduction
The two-party communication model was introduced by Yao in 1979 [Yao79] and offers a simple, yet rich model useful in numerous areas in theoretical computer science. One of the main tasks in communication complexity is to prove non-trivial lower bounds, either general or to specific problems. To this end, several general methods were proposed in the settings of randomized and quantum communication, the first ones being the fooling set [Yao79], (approximate) rank [MS82,Kra96,BdeW01], discrepancy [CG88,BNS89], and rectangle/corruption [Yao83,BFS86,Raz92,KN97,Kla03] bounds. Since then several new and stronger lower bounds were discovered. Jain, Klauck, and Nayak [JKN08] proposed the subdistribution bound, a relaxation of the rectangle/corruption bounds. Klauck [Kla07] introduced the smooth-discrepancy lower bound, while Linial and Shraibman [LS07] introduced the factorization norm (γ 2 ) lower bound, which also holds for quantum protocols. Sherstov [She08] later showed that both bounds coincide. All these bounds were later subsumed by the partition bound from Jain and Klauck [JK10], who also introduced relaxed versions of their partition bound named smooth-rectangle and smooth-discrepancy bounds. This left the field with one unified general lower bound method for randomized communication complexity. Laplante, Lerays, and Roland [LLR12] introduced the quantum partition bound (a.k.a. the efficiency bound). The efficiency bound subsumes the factorization norm when shared entanglement is allowed, and coincides with the partition bound when shared randomness is allowed.
In this note we present a linear program for the one-way version of the partition bound and show that it lower bounds the one-way randomized communication complexity and is tight up to an additive log log term. More specifically, for ε ∈ (0, 1/2) and a partial function f : X × Y → Z, let prt 1 ε (f ) be the optimal value of the linear program defining our one-way partition bound and let R 1 ε (f ) be the one-way randomized communication complexity with shared randomness for f with worst-case error at most ε. Then our main result is the following.
Theorem 1. For δ, ε ∈ (0, 1/2), R 1 ε (f ) ≥ log prt 1 ε (f ) and R 1 ε+δ (f ) ≤ log prt 1 ε (f ) + log log(1/δ).
It is known that the rectangle bound is tight for one-way randomized communication complexity up to an additive O(log(1/δ))-term [JKN08]. Our result shows that the one-way partition bound is tight up to an additive log log(1/δ)-term. We mention that [LLR12] had also introduced a one-way version of their efficiency bound, but through a different linear program than ours.
The one-way partition bound
Consider a partial function f :
X × Y → Z and let f −1 ⊆ X × Y be the set where f (·) is defined. Define f −1 (z) := {(x, y) ∈ f −1 : f (x, y) = z}.
In a two-party one-way communication protocol P, Alice is given x ∈ X and sends a message to Bob with input y ∈ Y. Upon receiving Alice's message, Bob produces the output of the protocol P(x, y). We shall always assume shared randomness between the two players. Let err x,y (P, f ) := Pr[P(x, y) = f (x, y)] be the error of the protocol on input (x, y) ∈ f −1 and err(P,
f ) := max (x,y)∈f −1 {err x,y (P, f )} be the error with worst- case (x, y) ∈ f −1 . Let R 1
ε (f ) be the classical one-way communication complexity with worst-case error at most ε, R 1 ε (f ) := min P {bits communicated by P : err(P, f ) ≤ ε}. For ease of notation we do not include the superscript "pub" to signal that shared randomness is allowed.
In a one-way zero-communication protocol with abort, P ⊥ : X × Y → Z ∪ {⊥}, Alice and Bob are given x ∈ X and y ∈ Y, respectively. Using shared randomness R ∈ {0, 1} * , but without communicating to each other, Alice outputs z a ∈ {⊢, ⊥} and Bob outputs z b ∈ Z. If z a =⊥, the protocol's output, denoted by P ⊥ (x, y), is ⊥, otherwise it is z b . Let err ⊥
x,y (P ⊥ , f ) := Pr R [P ⊥ (x, y) = f (x, y)|P ⊥ (x, y) =⊥] be the error of the protocol on input (x, y) ∈ f −1 given it does not abort and err(P ⊥ , f ) := max (x,y)∈f −1 {err ⊥ x,y (P ⊥ , f )} be the error with worst-case (x, y) ∈ f −1 . Let eff 1 (P ⊥ , (x, y)) := Pr R [P ⊥ (x, y) =⊥] be the non-abort probability of P ⊥ for (x, y) ∈ X × Y. We require such probability eff 1 (P ⊥ , (x, y)) to be the same for all (x, y) ∈ X × Y, denoted eff 1 (P ⊥ ). Define eff 1 ε (f ) := max P ⊥ {eff 1 (P ⊥ ) : err(P ⊥ , f ) ≤ ε}, the one-way zero-communication efficiency with worst-case error at most ε [LLR12].
We now present our linear program for the one-way partition bound.
Definition 2 (One-way partition bound). Given ε ∈ (0, 1/2), the one-way ε-partition bound of f , denoted by prt 1 ε (f ), is the optimal value of the following linear program.
Primal min A⊆X w A ,
∀x ∈ X :
A⊆X :x∈A
w A = 1, ∀A ⊆ X , y ∈ Y : z∈Z w A,y,z = w A ,
∀(x, y) ∈ f −1 :
A⊆X :x∈A w A,y,f (x,y) ≥ 1 − ε, ∀A ⊆ X , (y, z) ∈ Y × Z : w A ≥ 0, w A,y,z ≥ 0. Dual max (x,y)∈f −1 (1 − ε)µ x,y − x∈X λ x ,
∀A ⊆ X , (y, z) ∈ Y × Z :
x∈A:(x,y)∈f −1 (z) µ x,y ≤ λ A,y ,
∀A ⊆ X :
y∈Y λ A,y ≤ 1 + x∈A λ x ,
∀A ⊆ X , (x, y) ∈ X × Y : µ x,y ≥ 0, λ A,y ≥ 0, λ x ∈ R.
In order to prove Theorem 1, we start by showing that optimal one-way zero-communication protocols are equivalent to optimal solutions to our partition-bound linear program.
Lemma 3. For all ε ∈ (0, 1/2), prt 1 ε (f ) = 1/eff 1 ε (f ).
Proof. We first show that prt 1 ε (f ) ≥ 1/eff 1 ε (f ). Consider an optimal solution for the primal of prt 1 ε (f ) with weights w A and w A,y,z for all A ⊆ X , (y, z) ∈ Y × Z. We define a one-way zero-communication protocol as follows: using public coins, Alice chooses A ⊆ X with probability w A / A ′ ⊆X w A ′ and does not abort if and only if x ∈ A. Bob, on the other hand, outputs z b = z with probability w A,y,z /w A . Therefore, for all x ∈ X , the probability that Alice does not abort is
Pr[x ∈ A] = A⊆X :x∈A w A A ′ ⊆X w A ′ = 1 prt 1 ε (f )
, using that A⊆X :x∈A w A = 1. Moreover, for all (x, y) ∈ f −1 ,
Pr[z b = f (x, y)|z a =⊥] = A⊆X Pr[Alice chooses A] Pr[z b = f (x, y), z a =⊥ |Alice chooses A] Pr[z a =⊥] = A⊆X :x∈A Pr[Alice chooses A] Pr[z b = f (x, y)|Alice chooses A] Pr[z a =⊥] = A⊆X :x∈A w A A ′ ⊆X w A ′ w A,y,f (x,y) w A 1 A ′ ⊆X w A ′ = A⊆X :x∈A w A,y,f (x,y) ≥ 1 − ε.
Hence, our one-way zero-communication protocol has non-abort probability 1/prt 1 ε (f ) and worstcase error at most ε. Thus eff 1 ε (f ) ≥ 1/prt 1 ε (f ). On the other direction, we now prove that prt 1 ε (f ) ≤ 1/eff 1 ε (f ). Consider a one-way zerocommunication protocol P ⊥ with public randomness R, worst-case error at most ε, and nonabort probability eff 1 ε (f ) for all (x, y) ∈ X × Y. Given a public coin r with probability p(r), let X r := {x ∈ X |z a (x, r) =⊥} be the set of inputs for which Alice does not abort. Then, for A ⊆ X and (y, z) ∈ Y × Z, define the weights
w ′ A := 1 eff 1 ε (f ) r:A=Xr p(r), w ′ A,y,z := 1 eff 1 ε (f ) r:A=Xr,z b (y,r)=z p(r).
Clearly, ∀A ⊆ X :
A⊆X :x∈A w ′ A = 1 eff 1 ε (f ) r:za(x,r) =⊥ p(r) = 1 eff 1 ε (f ) Pr R [P ⊥ (x, y) =⊥] = 1.
Also, regardless of Alice's output, Bob must output something with probability 1, meaning that ∀A ⊆ X , y ∈ Y :
z∈Z w ′ A,y,z = w ′ A .
Finally, since for all (x, y) ∈ f −1 Bob outputs the correct answer with probability at least 1 − ε given that Alice does not abort, ∀(x, y) ∈ f −1 : 1 − ε ≤ Pr[P ⊥ (x, y) = f (x, y)|P ⊥ (x, y) =⊥] = Pr[P ⊥ (x, y) = f (x, y), P ⊥ (x, y) =⊥] Pr[P ⊥ (x, y) =⊥] = A⊆X :x∈A w ′ A,y,f (x,y) .
Thus, the weights w ′ A and w ′ A,y,z are a feasible solution for prt 1 ε (f ) and therefore
prt 1 ε (f ) ≤ A⊆X w ′ A = 1 eff 1 ε (f ) r p(r) = 1 eff 1 ε (f ) .
In the next two lemmas, we relate one-way randomized and one-way zero-communication protocols, i.e., we show that an optimal one-way randomized communication protocol can simulate a one-way zero-communication protocol and vice-versa. This was already observed in [LLR12].
Acknowledgements. The research of JFD and RJ is supported by the National Research Foundation, Singapore and A*STAR under the CQT Bridging Grant and the Quantum Engineering Programme Award number NRF2021-QEP2-02-P05.
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arxiv |
SINGULARITIES OF SOLUTIONS OF HAMILTON-JACOBI EQUATIONS
6 Jan 2021
Piermarco Cannarsa
Wei Cheng
SINGULARITIES OF SOLUTIONS OF HAMILTON-JACOBI EQUATIONS
6 Jan 2021
This is a survey paper on the quantitative analysis of the propagation of singularities for the viscosity solutions to Hamilton-Jacobi equations in the past decades. We also review further applications of the theory to various fields such as Riemannian geometry, Hamiltonian dynamical systems and partial differential equations.
INTRODUCTION
This is a survey paper concerning the progress made for the singularities of the solutions to Hamilton-Jacobi equations in the past decades. We begin with a quote from the paper [KS16] by Khanin and Sobolevski: The evolutionary Hamilton-Jacobi equation
+ ( , , ∇ ) = 0
(HJ) appears in diverse mathematical models ranging from analytical mechanics to combinatorics, condensed matter, turbulence, and cosmology ⋯. In many of these applications the objects of interest are described by singularities of solutions, which inevitably appear for generic initial data after a finite time due to the nonlinearity of (HJ). Therefore one of the central issues both for theory and applications is to understand the behavior of the system after singularities form.
The notion of viscosity solutions, introduced in the seminal papers [CL83,CEL84], provides the right class of generalized solutions to study existence, uniqueness, and stability issues for problem (HJ). An overview of the main features of this theory can be found in the monographs [BCD97] for first order equations and [FS06] for second order equations.
It is well known that Hamilton-Jacobi equations have no global smooth solutions in general, because solutions may develop singularities due to crossing or focusing of characteristics. The persistence of singularities, i.e, once a singularity is created, it will propagate forward in time up to +∞, affords an evidence of irreversibility for equation (HJ), while the compactness after the evolution of the associated Lax-Oleinik semi-group gives another one ( [ACN16b,ACN16a]).
The expected maximal regularity for solutions of (HJ) is the local semiconcavity of ( , ⋅) for > 0. Indeed, semiconcave functions were used to study well-posedness for (HJ) before the theory of viscosity solution was developed ([Dou61, Kru75,Kry87]). Nowadays, the notion of semiconcavity has been widely used in many mathematical fields, such as [Hru78, CF91, FM00, Rif00,Rif02] in control theory and sensitivity analysis, [Roc82,CM06] in nonsmooth and variational analysis, [Pet07] in metric geometry. Good references on semiconcave functions include the monographs [CS04,Vil09].
To our knowledge, the first paper dealing with the singularities of viscosity solutions of (HJ) is the paper by the first author and Soner ( [CS87]). Thanks to the discovery of semiconcave functions in the study of viscosity solutions of (HJ) ( [CS89]), some propagation results for general semiconcave functions were obtained in [ACS93]. The propagation of singularities of semiconcave functions along Lipschitz arcs was firstly studied in [AC99] and then extend to solutions of Hamilton-Jacobi equations ( [AC02]).
It is the first time in [AC02] the authors introduced the important notion of generalized characteristics for Hamilton-Jacobi equation (HJ), which is a keystone for the further progress later. In one-dimensional case, the idea of generalized characteristics also comes from earlier work by Dafermos [Daf77] on Burgers equation. The readers can also refer to Arnold's book [Arn90] on the shock wave singularities and perestroikas of Maxwell sets and the references therein.
A Lipschitz curve ∶ [0, ] → Ω, (0) = 0 ∈ Sing ( ), is called a generalized characteristic with respect to ( , ) from 0 if the following differential inclusion is satisfieḋ ( ) ∈ co ( , ( ), + ( ( ))),
. ., ∈ [0, ].
(1.1)
It was proved in [AC02] that there exists a generalized characteristic from any initial singular point 0 propagating the singularities if 0 ∉ co ( 0 , + ( 0 )). Using the approximation method introduced by Yu ( [Yu06]), the first author and Yu showed the existence of singular characteristics (see Definition 3.8) which has more regularity information. More precisely, any such a singular characteristic satisfies the condition lim where is a smooth manifold, is a Tonelli Hamiltonian and 0 is the Mañé's critical value. Weak KAM theory bridges Mather theory ( [Mat91,Mat93,Mn92]) from Hamiltonian dynamical systems to the theory of viscosity solutions of (1.2). Any weak KAM solution of (1.2) is exact the common fixed point of the associated negative type Lax-Oleinik semi-group { } >0 , = for all > 0.
For any (Lipschitz and semiconcave) weak KAM solution of (1.2), an intrinsic method was developed in the paper [CC17]. Using the positive Lax-Oleinik semi-group {̆ } >0 , one can obtain an intrinsic singular characteristic propagating singularities from any singular initial point, or general cut point of ( [CCF17]).
Although singular characteristics satisfy (1.1), the convex hull in the differential inclusion (1.1) is an obvious obstacle to establish uniqueness and stability. The only well-understood system with such well-posedness properties is the system with Hamiltonians quadratic in the momentum variable. A typical example is the Hamiltonian = 1 2 | | 2 , where differential inclusion (1.1) becomes the generalized gradient systeṁ ( ) ∈ + ( ( )),
∈ [0, ].
Inspired by earlier works [Bog02,Bog06,Str13], Khanin and Sobolevski essentially proved the existence of singular characteristics satisfying (1.1) without convex hull under some extra conditions on the initial data ( [KS16]). These kinds of singular characteristics are called strict singular characteristics (or broken characteristics in [Str13]). The fact that singular characteristics satisfy more restrictive dynamics than (1.1) might help to obtain some kind of uniqueness result. Indeed, in the recent work [CC20], we solved such a well-posedness problem in ℝ 2 for non-critical initial data.
When we pursue applications of this theory, global propagation results for solutions of Hamilton-Jacobi equations turn out to be necessary. Global propagation for the closure of the singular set was obtained by Albano in [Alb16]. For the propagation of genuine singularities, a global result for a Cauchy-Dirichlet problem with quadratic Hamiltonian was obtained in [CMS15] using an energy method. More global results for weak KAM solutions and Dirichlet problem using intrinsic method can be found in [CC17, CCF17, CCMW19, CCF19].
One important application of the global propagation result is the homotopy equivalence between the complement of Aubry set and the singular set of any weak KAM solution, and the local contractibility of the singular set ( [CCF17,CCF19]). An earlier result for such homotopy equivalence for the distance function on Riemannian manifolds was obtained in [ACNS13] based on invariance properties of the generalized gradients flow. Moreover, global propagation result in [CCF19] can also be applied to some basic problem in Riemmanian geometry such as the analysis of the set of points which can be joined by at least two minimizing geodesics. There are also some applications of this theory to Hamiltonian dynamical systems, mainly in the frame of Mather theory and weak KAM theory ([CCZ14, CC15, CCC19, Zha20]). Above evidence suggests that the story of singularities will continue and further applications to various topics will appear in the near future.
The paper is organized as follows. In section 2, we introduce some necessary materials on Hamilton-Jacobi equations and semiconcavity. In section 3 and 4, we will review the progress in local and global propagation of singularities for various kinds of problem. In section 5, we will concentrate on the setting of the weak KAM theory, especially the applications to the topological and dynamical applications. There is also a short concluding remark in section 6. We also provide a new proof of the Lipschitz regularity for the intrinsic singular characteristics in the appendix, which looks quite natural and intuitive comparing to the original one in [CC17].
Acknowledgements. Piermarco Cannarsa was supported in part by the National Group for Mathematical Analysis, Probability and Applications (GNAMPA) of the Italian Istituto Nazionale di Alta Matematica "Francesco Severi" and by Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Wei Cheng is partly supported by National Natural Science Foundation of China (Grant No. 11871267, 11631006 and 11790272). The second author also thanks to Jiahui Hong for helpful discussion on some part of the appendix.
PRELIMINARIES
Let Ω ⊂ ℝ be a convex open set. We recall that a function ∶ Ω → ℝ is semiconcave (with linear modulus) if there exists a constant > 0 such that
( ) + (1 − ) ( ) − ( + (1 − ) ) ⩽ 2 (1 − )| − | 2 (2.1)
for any , ∈ Ω and ∈ [0, 1].
For any continuous function ∶ Ω ⊂ ℝ → ℝ and any ∈ Ω, the closed convex sets
− ( ) = ∈ ℝ ∶ lim inf → ( ) − ( ) − ⟨ , − ⟩ | − | ⩾ 0 , + ( ) = ∈ ℝ ∶ lim sup → ( ) − ( ) − ⟨ , − ⟩ | − | ⩽ 0 .
are called the subdifferential and superdifferential of at , respectively. The following statement characterizes semiconcavity (with linear modulus) for a continuous function by using superdifferentials.
Proposition 2.1. Let ∶ Ω → ℝ be a continuous function. If there exists a constant > 0 such that, for any ∈ Ω, there exists ∈ ℝ such that
( ) ⩽ ( ) + ⟨ , − ⟩ + 2 | − | 2 , ∀ ∈ Ω, (2.2)
then is semiconcave with constant and ∈ + ( ). Conversely, if is semiconcave in Ω with constant , then (2.2) holds for any ∈ Ω and ∈ + ( ).
Let ∶ Ω → ℝ be locally Lipschitz. We recall that a vector ∈ ℝ is called a reachable (or limiting) gradient of at if there exists a sequence { } ⊂ Ω ⧵ { } such that is differentiable at for each ∈ ℕ, and
lim →∞ = and lim →∞ ( ) = .
The set of all reachable gradients of at is denoted by * ( ).
Proposition 2.2. Let ∶ Ω ⊂ ℝ → ℝ be a continuous semiconcave function and let ∈ Ω. Then the following properties hold. To study the rectifiability of the singular set Sing ( ) of a semiconcave function , we need some concepts from geometric measure theory. Definition 2.5. A continuous real-valued function on Ω is called a viscosity solution of (2.3) if for every ∈ Ω and ∈ 1 (Ω, ℝ) (a) − has a local maximum at implies ( , , ( ), ( )) ⩽ 0, or is a viscosity subsolution of (2.3); (b) − has a local minimum at implies ( , , ( ), ( )) ⩾ 0, or is a viscosity supersolution of (2.3).
The relation between continuous viscosity solution and its semiconcavity is , considered as a function of loc (Ω, ) and is a countably −1 -rectifiable set. Apart from earlier contributions for distance functions as in [Erd45], to our knowledge the first general results about the rectifiability of the singular sets of concave functions are due to Zajíček [Zaj78,Zaj79] and Veselý [Ves86,Ves87]. Similar properties were later extended to semiconcave functions with general modulus in [AAC92].
To obtain a fine description of Sing ( ) for a semiconcave function on Ω, it is convenient to introduce a hierarchy of subsets of Sing ( ) according to the dimension of the superdifferential. The magnitude of a point ∈ Ω (with respect to u) is the integer ( ) = dim + ( ). Given an integer ∈ {0, … , } we set
Sing ( ) = { ∈ Ω ∶ ( ) = }. Proposition 3.1 ([CS04]). If is semiconcave in Ω, then the set Sing ( ) is countably ( − )- rectifiable for any integer ∈ {0, … , }. In particular, Sing ( ) is countably ( − 1)-rectifiable.
Now, we turn to the analyze of rectifiability of Sing ( ), with the value function to the classical one free endpoint problem from calculus of variation, i.e.,
( , ) = inf ∈ , 0 ( (0)) + ∫ 0 ( , ( ),̇ ( )) , ( , ) ∈ (0, ) × ℝ , (CV , )
where is a Tonelli Lagrangian of class +1 ( ⩾ 1) and 0 is of class +1 , and , is the set of all absolutely continuous curves ∶ [0, ] → ℝ such that ( ) = ∈ ℝ .
We have already seen in Proposition 3.1 that Sing ( ) is countably ( − 1)-rectifiable. Recall that, under the assumption on and 0 ,
Sing ( ) = Sing ( ) ∪ Conj ( ),
where Conj ( ) is the set of conjugate points of problem (CV , ) (see [CS04,Page 155] or [CMS97] for the definition). So, we only need to analyze the rectifiability of Conj ( ). By a Sard type argument ( [Fle69]) one has +1∕ (Conj ( )) = 0. However, the above estimate does not imply the rectifiability of Conj ( ) even if 0 is of class ∞ .
Proposition 3.2 ([CMS97]). Under previous assumption, (a) Sing ( ) = Sing ( ) ∪ Conj ( ). (b) Conj ( ) is countably -rectifiable, and so is Sing ( ). (c) −1+2∕ (Sing ( ) ⧵ Conj ( )) = 0. (d) If and 0 is of class ∞ , then dim (Sing ( ) ⧵ Conj ( )) ⩽ − 1.
If the initial datum 0 has weaker regularity than 2 , then Sing ( ) can fail to be countably -rectifiable, see [CS04, Example 6.6.13]. Notice that, in the mentioned example, is of class ∞ . For the further progress along this line, see [Pig02,Men04].
Generalized characteristics.
Let Ω ⊂ ℝ be open and let ∶ Ω → ℝ be a Lipschitz and semiconcavity viscosity solution of the Hamilton-Jacobi equation
( , ( ), ( )) = 0, ∈ Ω. (3.1)
The notion of generalized characteristics with respect to ( , ) plays a central rôle in the study the phenomenon that the singularities propagates along a Lipschitz curve from an initial point 0 ∈ Sing ( ). 3.2.1. Propagation of singularities for general semiconcave functions. Before dealing with viscosity solutions of (3.1), we begin with a result concerning propagation of singularities for semiconcave functions with linear modulus.
Proposition 3.3 ([AC99]). Let ∶ Ω → ℝ be a semiconcave function. If 0 ∈ Sing ( ), or + ( 0 ) ⧵ * ( 0 ) ≠ ∅, (3.2)
then, there exists a Lipschitz singular arc ∶ [0, ] → Ω with (0) = 0 such thaṫ + (0) exists,
+ (0) ≠ 0 and inf ∈[0, ] diam ( + ( ( ))) > 0.
We note that condition (3.2) is equivalent to the existence of two vectors 0 ∈ + ( 0 ) ⧵ * ( 0 ) and ∈ ℝ ⧵ {0} such that
⟨ − 0 , ⟩ ⩾ 0, ∀ ∈ + ( 0 ). (3.3)
We will see later the importance of condition (3.2) which was initially pointed out in [ACS93].
The key idea of the proof of Proposition 3.3 is to construct a function
( ) = ( ) − ( 0 ) − ⟨ 0 − , − 0 ⟩ − 1 2 | − 0 | 2 , ∈ Ω.
Being strictly concave for small > 0, has a unique maximizer in a small neighborhood of 0 in Ω. The curve ↦ is exactly the local singular arc constructed in Proposition 3.3. It is rather surprising that a similar idea also works with the intrinsic singular characteristics, for the study of which the term 1 2 | − 0 | 2 will be replaced by the fundamental solution.
3.2.2. Generalized characteristics. Applying the basic idea from [AC99] to the viscosity solutions of (3.1), Albano and the first author introduced the notion of generalized characteristic in [AC02]. Suppose ∶ Ω × ℝ × ℝ → ℝ is a continuous function satisfying the following conditions:
(A1) ↦ ( , , ) is convex; (A2) for any ∈ Ω and ∈ ℝ the function ( , , ⋅) is uniformly quasi-convex, or the 0-level set { ∈ ℝ ∶ ( , , ) = 0} contains no straight line; (A3) for any 1 , 2 ∈ Ω, 1 , 2 ∈ ℝ, ∈ ℝ | ( 1 , 1 , ) − ( 2 , 2 , )| ⩽ 0 (| 1 − 2 | + | 1 − 2 |) for some constant 0 > 0; (A4)
is differentiable with respect to and, for any 1 , 2 ∈ Ω, 1 , 2 ∈ ℝ, 1 , 2 ∈ ℝ
| ( 1 , 1 , 1 ) − ( 2 , 2 , 2 )| ⩽ 1 (| 1 − 2 | + | 1 − 2 | + | 1 − 2 |) for some constant 1 > 0. Proposition 3.4 ([AC02]). Suppose satisfies (A1)-(A4).
Let be a locally semiconcave solution of (3.1) and let 0 ∈ Sing ( ) be such that 0 ∉ co ( 0 , ( 0 ), + ( 0 )). (1) is a generalized characteristic with respect to ( , ) from 0 , that is,
̇ ( ) ∈ co ( ), + ( ( )) a.e. ∈ [0, ], (0) = 0 . (3.5) (2) is an injection. (3) ( ) ∈ Sing ( ) for all ∈ [0, ]. (4)̇ + (0) exists anḋ + (0) = ( 0 , ( 0 ), 0 ) where 0 = arg min ∈ + ( 0 ) ( 0 , ( 0 ), ).
The proof of Proposition 3.4 uses the result in Proposition 3.3 together with an Euler segment approximation method. Moreover, one can also derive the useful energy estimate
( ( ), ( ( )), ( )) ⩽ 1 2 ( 0 , ( 0 ), 0 ), ∈ [0, ], where ∶ [0, ] → ℝ is defined by (0) = 0 and ( ) − 0 = [ ( ) − 0 + ( 0 , ( 0 ), 0 )], ∀ ∈ (0, ].
This kind of energy estimate can be used to deduce global propagation results.
3.2.
3. An approximation method and singular characteristics. Needless to say, the proof of Proposition 3.3 and Proposition 3.4 utilises techniques from nonsmooth analysis and control theory. A simpler method was introduced in [Yu06] and [CY09]. The following approximation lemma, proved in [CY09], will be frequently used in what follows.
Lemma 3.5. Given a semiconcave function on Ω, we assume there are positive constants , = 0, 1, 2, such that | ( )| ⩽ 0 for all ∈ Ω, | ( )| ⩽ 1 for almost all ∈ Ω, and has semiconcavity constant 2 . Let 0 ∈ Ω and let be an open subset of Ω such that
0 ∈ ⊂ ⊂ Ω. Then for any ∈ + ( 0 ) there is a sequence { } ⩾1 ⊂ ∞ ( ) such that (a) | ( )| ⩽ 0 , | ( )| ⩽ 1 , 2 ( ) ⩽ 2 for all ∈ , (b) lim →∞ = uniformly in and lim →∞ ( 0 ) = .
The following result can be regarded as a refinement of Proposition 3.4.
Proposition 3.6 ([CY09]). Suppose is semiconcave function on Ω and is a function of class 1 satisfying (A1) and (A2') for any ∈ Ω, ∈ ℝ and ∈ ℝ, the -level set { ∈ ℝ ∶ ( , , ) = } contains no straight line. Let 0 ∈ Sing ( ) and 0 = arg min ∈ + ( 0 ) ( 0 , ( 0 ), ). Then, there exists a Lipschitz arc
∶ [0, ] → Ω such that: (i) is a generalized characteristic for ( , ) starting at 0 ; (ii) ( ) ∈ Sing ( ) for all ∈ [0, ]; (iii)̇ + (0) exists anḋ + (0) = ( 0 , ( 0 ), 0 ); (iv) lim →0 + ess sup ∈[0, ] |̇ ( ) −̇ + (0)| = 0.
Remark 3.7. For what follows we need further details related to Proposition 3.6. -The semiconcave function is not required to be a solution of (3.1). But, if Ω is bounded, being Lipschitz, must be a subsolution of (3.1) with Hamiltonian − for some ∈ ℝ. -Observe that, in general, a generalized characteristic may well be a constant arc. But, for solutions of (3.1), it was proved in [AC02] that singularities propagate along genuine shocks (injective generalized characteristics) under assumption (3.4). If is a solution of (3.1), as a corollary, one can show that the generalized characteristics in Proposition 3.6 propagates singularities under the more natural condition 0 ∉ ( 0 , ( 0 ), + ( 0 )). -For the generalized characteristic , constructed in Proposition 3.6, the right-continuity oḟ at 0 is important for further applications. Later, we will call a singular generalized characteristic satisfying properties (i)-(iv) in Proposition 3.6 a singular characteristic.
Owing to Lemma 3.5, there is a sequence of smooth functions { } enjoying properties (a) and (b) in the lemma for = 0 . It is easy to see that, for every ⩾ 1, the Cauchy problem
̇ = ( , ( ), ( )), (0) = 0 ,
has a 1 solution ∶ [0, ] → Ω. Without loss of generality, we can assume that uniformly converges to on [0, ] as → ∞. A standard argument (see, for instance, [Yu06]) shows that is a generalized characteristic for ( , ) starting at 0 .
3.3. Strict singular characteristics. The rôle of the convex hull in the definition of generalized characteristic is quite mysterious. This is a big obstacle for us to reveal more information about the propagation of singularities and related Hamiltonian dynamics. The next notion gets rid of such a convexity operator. The existence of strict singular characteristics for equation (HJ) was proved in [KS16] (see also the appendix of [CC20]), where additional regularity properties of such curves were established, including the right-differentiability of for every . However, the intrinsic nature of the strict singular characteristics is still unclear. One of the most important issues of the theory is to establish the uniqueness of solutions to (3.6). We describe below a partial answer to such a fundamental problem, following the paper [CC20].
Hereafter in this section we assume = 2. Given a semiconcave solution of ( , ( )) = 0, ∈ Ω, (HJ loc )
we denote by Lip 0 (0, ; Ω) the set of Lipschitz arcs satisfying properties (ii), (iii), and (iv) of Definition 3.8 for all ∈ [0, ].
Theorem 3.9 ([CC20]). Let be a semiconcave solution of (HJ loc ) and let 0 ∈ Sing ( ) be such that 0 ∉ ( 0 , + ( 0 )). Let ∈ Lip 0 (0, ; Ω) ( = 1, 2) be such that (0) = 0 . Then, there exists ∈ (0, ] such that there exists a unique bi-Lipschitz homeomorphism For strict singular characteristics, uniqueness holds without reparameterization.
∶ [0, ] → [0, ( )] ⊂ [0, ] satisfying 1 ( ) = 2 ( ( )) for all ∈ [0, ].
Theorem 3.11 ( [CC20]). Let be a semiconcave solution of (HJ loc ) and let 0 ∈ Sing ( ) be such that 0 ∉ ( 0 , + ( 0 )). Let ∶ [0, ] → Ω ( = 1, 2) be strict singular characteristics with initial point 0 . Then there exists ∈ (0, ] such that 1 ( ) = 2 ( ) for all ∈ [0, ]. Theorem 3.9 and Theorem 3.11 establish a connection between the absence of critical points and uniqueness of strict singular characteristics. In this direction, we also have the following global result.
GLOBAL PROPAGATION OF SINGULARITIES
In this section, we will discuss the global behavior of the propagation of singularities along generalized characteristics. 4.1. Propagating structure of the 1 singular support. A typical problem is the following evolutionary Hamilton-Jacobi equation The proof of Proposition 4.2 is based on an improvement of some classic results when 0 is of class 2 (see, for instance, Chapter 6 of [CS04]). In fact, even if 0 is just continuous, one can show that if ( , ) ∉ sing supp 1 ( ), then the associated optimal curve ending at must satisfies the property that ( , ( )) ∉ sing supp 1 ( ) for ∈ (0, ]. Now, suppose there exists ( , ( )) ∉ sing supp 1 ( ) for some ∈ ( 0 , ). Then there exists a tubular neighborhood
( , ) + ( , ,( , )⊂ ℝ +1 of {( , ( )) ∶ ∈ [ 0 , )},
where is the optimal curve such that ( ) = ( ). Moreover, ∩ sing supp 1 ( ) = ∅. On the open set , and are essentially identified because both solve the same ordinary differential equation (4.2) (by the claim above) and satisfy the same endpoint condition. This leads to a contradiction and it follows that the 1 singular support must propagate to ( , ( )) along the generalized characteristic .
Remark 4.3. We should emphasize that the proof of Proposition 4.2 is based on an intrinsic approach, i.e., the argument just uses the analysis of the associated characteristics system.
Global propagation of genuine singularities.
It is quite natural to ask the question if the singularities of the viscosity solution of (4.1) can propagation along the associated generalized characteristic for all > 0. In general, the answer is negative (see, for instance, Example 5.6.7 in [CS04]). If Ω = ℝ , is of class 2 , −1 ⩽ 2 ⩽ for some > 0, and equation (4.3) admits a concave solution, then it was showed in [AC00] that if ( 0 , 0 ) ∈ Sing ( ) then there exists a Lipschitz arc ( , ( )), ∈ [ 0 , +∞), with ( 0 ) = 0 , such that ( , ( )) ∈ Sing ( ) for all ∈ [ 0 , +∞). We remark that is concave in [0, ] × ℝ if and only if 0 is concave. So, this result is very special.
Generalized gradients. Let
⊂ ℝ be closed and denote by the distance function from . It is well known that = satisfies the eikonal equation If Ω is a bounded open subset of ℝ , it was shown in [ACNS13] that the generalized gradient flow given by (4.5) propagates singularities for all > 0. This is also true for the case of Riemannian manifolds. A significant application of this global propagation result to geometry is that the singular set of has the same homotopy type as Ω. Further deep extension of this topological result to the weak KAM context will be discussed later. We will also discuss more general Dirichlet problem in Section 5.3.2. for all ( , ), ( , ) ∈ such that > ⩾ 0.
| ( )| = 1, ∈ ℝ ⧵ , ( ) = 0, ∈ .
Proposition 4.4 ([CMS15]). Let ∶ → ℝ be a Lipschitz continuous function satisfying (4.7)
and let be a viscosity solution of (4.6). Given ( 0 , 0 ) ∈ Sing ( ), let be the generalized characteristic determined by (4.2) such that ( 0 ) = 0 . Then, there exists ∈ (0, +∞] such that ( , ( )) ∈ Sing ( ) for all ∈ [ 0 , 0 + ) and lim →( 0 + ) − ∈ whenever < +∞.
The proof of the above result relies on two main ideas that are converted in two technical results, respectively. The first one is a sharp semiconcavity estimate for a suitable transform of the solution in [ACNS13]. The second one is an inequality established showing that the full Hamiltonian associated with (4.6), that is,
( , ) = + ( ),
decreases along a selection of the superdifferential of , evaluated at any point of a suitable arc.
Remark 4.5. We remark that if Ω = ℝ , Proposition 4.4 directly leads to a global propagation result. For general case, the statement ensures that the singularities will have global propagation or hit the boundary (see also Section 5.3.2).
WEAK KAM ASPECTS OF SINGULARITIES
In this section, we will discuss the problem of propagation of singularities in the frame of weak KAM theory ( 2) can be regarded as the value function of some basic problem in the calculus of variation or optimal control. For any , ∈ and > 0, we denote by Γ , the set of all absolutely continuous curves ∶ [0, ] → such that (0) = and ( ) = . We define the fundamental solution of (5.2) as
( , ) = inf ∈Γ , ∫ 0 ( ,̇ ) , , ∈ , > 0. (5.3)
Recall that for any Tonelli Lagrangian, the function ( , ) ↦ ( , ) is locally semiconcave and semiconvex for small > 0. Moreover, the function ↦ ( , ) is convex with constant ∕ for small (see, for instance, [CC17, Proposition B.8]). In symplectic geometry, ( , ) is also known as generating function. The following result is known for generating functions in symplectic geometry (see, for instance, [MS17,Chapter 9]). The readers can compare Proposition B.8 in [CC17] (see also [CF14,Theorem 4.2] for Cauchy problems) and the following lemma for fundamental solutions of Hamilton-Jacobi equations, with two analogous concepts of convexity radius and injectivity radius from Riemannian geometry.
Lemma 5.1. For any > 0 there exists > 0 such that the function ( , ) ↦ ( , ) is of class 2 in the cone
( , ) ∶= ( , ) ∈ ℝ × ℝ ∶ 0 < < , | − | < .
Proof. Fix ∈ ℝ and let ∈ ℝ . For > 0 consider the Hamiltonian system Observe that Φ is of class 2 . The associated variational equation is
′ ( , ) = ( , ) ( , ) + ( , ) ( , ), (0, ) = 0; ′ ( , ) = − ( , ) ( , ) − ( , ) ( , ), (0, ) = . (5.5)
Consequently, we have ′ (0, ) = ( , ). Since is a Tonelli Hamiltonian, we conclude that for any > 0 there exists ( ) > 0 such that if | | ⩽ then ( , ) > ( ) . Moreover, by the Lipschitz dependence of the solution of (5.4) and (5.5) with respect to initial data, we obtain
( , ) = ∫ 0 ′ ( , ) = ∫ 0 ′ (0, ) + ∫ 0 ( ′ ( , ) − ′ (0, )) ⩾ ( ( ) ) − ( ) 2 ⋅ 2 ⩾ ( ) − ( ) 2 .
We conclude that for any > 0 there exist ( ), ( ) > 0 such that and fix 0 ∈ (0, 0 ). Set 0 = ( ,̇ 0 , (0)). We want to show that Φ ∶ (0, 0 ) × ℝ → (0, 0 ) × ℝ is locally invertible at ( 0 , 0 ) with a 2 inverse. For this we observe that
| | ⩽ ⇒ ( , ) ⩾ ( ) 2 , ∀| | ⩽ , 0 ⩽ ⩽ ( ).Φ( , ) = 1 ′ ( , ) 0
( , ) and (5.7) implies that det Φ( , ) > 0, ( , ) ∈ (0, 0 ) × (0, 0 ). Then the conclusion follows from the inverse mapping theorem.
We now claim that We say such an absolutely continuous curve is a ( , )-calibrated curve, or a -calibrated curve for short, if the equality holds in the inequality above. A curve ∶ (−∞, 0] → ℝ is called a -calibrated curve if it is -calibrated on each compact sub-interval of (−∞, 0]. In this case, we also say that is a backward calibrated curve (with respect to ). Recall that a continuous function on is called a weak KAM solution of (5.1) if = for all > 0. The following result explains the relation between the set of all reachable gradients and the set of all backward calibrated curves from (see, e.g., [CS04] or [Rif08] for the proof).
Proposition 5.4. Let ∶ → ℝ be a weak KAM solution of (5.1) and let ∈ . Then ∈ * ( ) if and only if there exists a unique 2 curve ∶ (−∞, 0] → with (0) = and = ( ,̇ (0)), which is a backward calibrated curve with respect to . A confirmative result that no isolated singular point exists for a weak KAM solution of (5.10) was proved in [CCZ14] for mechanical systems using a topological argument.
5.3. Intrinsic singular characteristics.
Characteristics of weak KAM solution.
In this section, we suppose is a Lipschitz semiconcave weak KAM solution of (5.1) on = ℝ . In [CC17], another kind of singular curves for is constructed as follows. First, it is shown that there exists 0 > 0 such that for any ( , ) ∈ ℝ + × ℝ and any maximizer for the function (⋅) − ( , ⋅), we have that | − | ⩽ 0 . Then, taking = 0 + 1, one shows that, for some 0 > 0 and any ∈ (0, 0 ], there exists a unique , ∈ ( , ) such that ( ) = ( , ) − ( , , ). ). Let ∈ ℝ and let be the curve defined in (5.13). Then, the following holds:
(1) is Lipschitz continuous;
(2) if ∈ Sing ( ), then ( ) ∈ Sing ( ) for all ∈ [0, 0 ];
(3)̇ + (0) exists anḋ
+ (0) = ( 0 , 0 )
where 0 = arg min{ ( 0 , ) ∶ ∈ + ( )}.
Hereafter, we refer to the arc defined in (5.13) as the intrinsic characteristic from . Notice that 0 is independent of the initial point. Thus, when ∈ Sing ( ), Proposition (5.5) yields global propagation of singularities.
The reader can compare to the idea of the proof-that we outline below-to the argument used to deduce the propagation of the 1 singular support in Section 4.1. Suppose ∈ Sing ( ) but , ∉ Sing ( ) (0 < ⩽ 0 ). Applying Fermat's rule, we have that ( , , ) = ( , ). Invoking Proposition 5.4 and the differentiability property of the fundamental solution for small time, we conclude that there exist two minimal curves. One is the backward calibrated curve satisfying
| ( )| ⩽ 0 , | ( )| ⩽ 1 , 2 ( ) ⩽ 2 , ∈ Ω.
Take any sequence of ∞ -functions { } such that
| ( )| ⩽ 0 , | ( )| ⩽ 1 , 2 ( ) ⩽ 2 , ∀ ∈ Ω, (5.18)
converging uniformly to on Ω as → ∞ (for instance, the sequence given by Lemma 3.5). As was observed above, the sequence of curves
( ) = arg max ( ) − ( , }, ∈ (0, 0 ] , = 0. (5.19)
is well defined for some 0 > 0. So,
3 − 2 | ( ) − ( )| 2 ⩽ 2‖ − ‖ ∞ .
Recall that 0 is chosen that that 2 − 3 ∕ < 0 for ∈ (0, 0 ]. Recall that the family { } is equi-Lipschitz by (5.16). This implies converges to uniformly on [0, 0 ].
Remark 5.8. The method used here is closely related to the Lasry-Lions regularization from convex analysis ( [Att84,AA93]) and PDE ( [LL86]). In a weak KAM context, this method was also widely used as an interaction of the positive-negative Lax-Oleinik operators ( [Ber07,Ber10,Ber12,FZ10]). The relation between Lasry-Lions regularization and generalized characteristics was also studied in [CC16,CCZ18]. This method was applied to minimal homoclinic orbits with respect to the Aubry set ([CC15]). (5.20)
where Ω ⊂ ℝ is a bounded Lipschitz domain, is a Tonelli Hamiltonian, and is the boundary datum. For any , ∈ Ω and any < , we define the set of admissible arcs from to as Let be the value function of the following problem:
( ) = inf ∈ Ω ( ) + Φ Ω ( , ) , ∈ Ω, (5.21)
where ∶ Ω → ℝ is a continuous function satisfying the compatibility condition
( ) − ( ) ⩽ Φ Ω ( , ), ∀ , ∈ Ω. (5.22)
Observe that the function given by (5.21) is the value function of an optimal exit problem (see, for instance, [BCD97]) and a viscosity solution of (5.20). The following result can be regarded as an extension of Proposition 4.4. To exclude the case that the singularities hit the boundary we need more conditions on Ω. We shall suppose the following, where we denote Ω by Γ: (G1) there exists ∈ [0, 1) such that ( 1 ) − ( 2 ) ⩽ Φ Ω ( 2 , 1 ), ∀ 1 , 2 ∈ Γ; (G2) there exists ∈ 1,1 (Γ ) for some > 0 such that = | Γ and
⟨∇ ( ), − ⟩ ⩽̆ | − | 2 ∀ , ∈ Γ (5.23)
for somĕ > 0, where Γ denotes the -neighborhood of Γ.
Proposition 5.10. Let Ω ⊂ ℝ be a bounded domain with 2 boundary, let be a Tonelli Lagrangian satisfying ⩾ > 0 and let satisfy (G1),(G2). If 0 ∈ Cut ( ), then there exists a generalized characteristic ∶ [0, +∞) → Ω starting from (0) = 0 such that ( ) ∈ Sing ( ) for all ∈ [0, +∞).
Remark 5.11. We note that the energy condition Ω ( ) < 0 (which is implicitly assumed even in the above proposition as a consequence of the hypothesis ⩾ > 0) ensures that any optimal curve touches the boundary in finite time in the associated optimal exit time problem. On the other hand, the case of Ω ( ) = 0 is still open, especially the analysis of the Aubry set on the boundary. For a state constrained problem, weak KAM aspects of the boundary behaviour of solutions were studied in [CCMW20]. 5.4.1. Aubry set and cut locus. Let be compact and be a weak KAM solution of (5.1). We define the projected Aubry set ( ) of as the subset of such that ∈ ( ) if there exists a -calibrated curve ∶ (−∞, +∞) → passing though . We also define the cut locus of , denoted by Cut ( ), as the set of points ∈ where no backward -calibrated curve ending at can be extended to a -calibrated curve beyond . In general we have the following inclusions:
Sing ( ) ⊂ Cut ( ) ⊂ ⧵ ( ), Sing ( ) ⊂ Cut ( ) ⊂ Sing ( ).Sing ( ) = Sing ( ) ∩ (0, ) × ⊂ (0, ) × ⧵ ( ) is a homotopy equivalence.
Notice that we just assume the solution of (5.24) to be uniformly continuous without any extra conditions on the initial data. So, there are a lot of technical points one needs to clear in order to deal with arbitrary initial conditions (see [Fat20]). 5.4.3. Applications to Riemmanian geometry. Now, suppose ( , ) is a complete Riemannian manifold, and is the distance function to a closed subset ⊂ . We denote by Sing * ( ) the set of points in ⧵ where is not differentiable.
Proposition 5.16 ([CCF19]
). If is a closed subset of a complete Riemannian manifold ( , ), then Sing * ( ) is locally contractible.
In classical Riemmanian geometry, for any ∈ one denotes by Cut ( , ) ( ) the cut locus with respect to . It is well-known that, when is compact, such a cut locus Cut The difficulty for all these studies is an unavoidable dichotomy for cut points: the mixture of points with two different segments and conjugate points.
We now proceed to explain how to distinguish the study of these two sets by using the above methods. We will begin with another consequence of Proposition 5.16, for which we need the following definition: for a complete Riemannian manifold ( , ), we define ( , ) = ( × ) ⧵ ( , ). The set ( , ) contains a neighborhood of the diagonal Δ ⊂ × . In fact, we have ( , ) = Sing * ( Δ ), the set of singularities in ( × ) ⧵ Δ of the distance function of points in × to the closed subset Δ . Therefore, Proposition 5.16 implies:
Proposition 5.17 ([CCF19]). For every complete Riemannian manifold ( , ), the set
( , ) ⊂ × ⧵ Δ
is locally contractible. In particular, the set ( , ) is locally path connected.
For a closed subset ⊂ , we define its Aubry set * ( ) as the set of points ∈ ⧵ such that there exists a curve ∶ [0, +∞) → parameterized by arc-length such that ( ( )) = and = ( 0 ) for some 0 > 0. We remark that if is a bounded connected component of ⧵ , then ∩ * ( ) = ∅, and Sing * ( ) ∩ ⊂ is a homotopy equivalence (see also [Lie04] and Section 4.2.2). As for unbounded components, see also [CP01] for the Euclidean case.
Proposition 5.19 ([CCF19]). For every compact connected Riemannian manifold ( , ), the inclusion ( , ) ⊂ × ⧵ Δ is a homotopy equivalence. Therefore the set ( , ) is path connected and even locally contractible.
CONCLUDING REMARKS
The study of singularities of solutions to HJ equation has made remarkable progress in the past decades. Many results that seemed impossible have been obtained, and connections with other domains have been established. Nevertheless, many interesting problems remain open. Some open problems were proposed in [CC18].
In [CC20], the uniqueness of strict singular characteristic on = ℝ 2 is proved when the initial point is not a critical point. However, the uniqueness issue is still open for higher dimensional manifolds. Recalling some results in [CCC19], assuming uniqueness for generalized characteristics, one can bridge the Aubry set (Mather set) and the invariant set of the associated semi-flow of generalized characteristics. Recently, relations between propagation of singularities and global dynamics of lower dimensional Hamiltonian systems have also been pointed out in [Zha20]. More concrete applications to problems from Hamiltonian dynamical systems in the scheme of Mather theory and weak KAM theory are expected, including applications to the study of Burgers turbulence as noted in [KS16].
[Zha20]
Jianlu Zhang. Global behaviors of weak KAM solutions for exact symplectic twist maps. J. Differential Equations, 269 (7)
→0 + ess sup ∈[0, ] |̇ ( ) −̇ + (0)| = 0. From late 1990's, Fathi established weak KAM theory mainly for the stationary Hamilton-Jacobi equation ([Fat97b, Fat97a, Fat98b, Fat98a, Fat, Fat20]) ( , ( )) = 0, ∈ , (1.2)
(a) + ( ) is a nonempty closed convex set in ℝ and * ( ) ⊂ + ( ), where + ( ) denotes the topological boundary of + ( ). (b) The set-valued function ⇉ + ( ) is upper semicontinuous. (c) + ( ) is a singleton if and only if − ( ) ≠ ∅. If + ( ) is a singleton, then is differentiable at . Moreover, if + ( ) is a singleton for every point in Ω, then ∈ 1 (Ω). (d) + ( ) = co * ( ).(e) If is both semiconcave and semiconvex in Ω, then ∈ 1,1 (Ω).
Definition 2. 3 .
3Let ∶ Ω → ℝ be a semiconcave function. ∈ Ω is called a singular point of if + ( ) is not a singleton. The set of all singular points of is denoted by Sing ( ).
Definition 2 . 4 .
24Let ∈ {0, 1, ⋯ , } and let ⊂ ℝ . (1) is called a -rectifiable set if there exists a Lipschitz continuous function ∶ ℝ → ℝ such that ⊂ (ℝ ). (2) is called a countably -rectifiable set if it is the union of a countable family of -rectifiable sets. (3) is called a countably -rectifiable set if there exists a countably -rectifiable set ⊂ ℝ such that ( ⧵ ) = 0, where stands for -dimensional Hausdorff (outer) measure. Let Ω ⊂ ℝ be an open set and let be a continuous real-valued function on Ω × ℝ × ℝ . Let us again consider the general nonlinear first order equation ( , , unknown ∶ Ω → ℝ.
Proposition 2 . 6 .
26For ∶ Ω → ℝ semiconcave and ∈ (Ω × ℝ × ℝ , ℝ), (i) if u is a viscosity solution of ( , ( ), ( )) = 0 in Ω, then ( , ( ), ) = 0, ∀ ∈ Ω, ∈ * ( ); (ii) if ( , , ⋅) is convex, then ( , ( ), ( )) = 0, . ., ⟺ is a viscosity solution of ( , ( ), ( )) = 0; 3. LOCAL PROPAGATION OF SINGULARITIES 3.1. Rectifiability of Sing ( ) for semiconcave functions and viscosity solutions. For a semiconcave function on an open subset Ω ⊂ ℝ , is a function of bounded variation (see, for instance, [EG92]). The singular set Sing ( ) coincides with the jump set
Then, there exists a Lipschitz arc ∶ [0, ] → Ω satisfying the following.
Definition 3. 8 .
8A Lipschitz curve ∶ [0, ] → Ω is called a strict singular characteristic for ( , ) starting at 0 ∈ Sing ( ) if: (i) denoting by ( ) the minimal energy selection of + ( ( )), i.e., ) ( ) ∈ Sing ( ) for all ∈ [0, ]; (iii)̇ + (0) exists anḋ + (0) = ( 0 , ( 0 ), (0)); (iv) lim →0 + ess sup ∈[0, ] |̇ ( ) −̇ + (0)| = 0.
Corollary 3.10 ([CC20]). Let be a strict singular characteristic starting from 0 and let be any singular characteristic as in Proposition 3.4. If 0 ∉ ( 0 , + ( 0 )), then there exists > 0 and a bi-Lipschitz homeomorphism ∶ [0, ] → [0, ( )] such that ( ( )) = ( ) ∀ ∈ [0, ].
Corollary 3 .
312 ([CC20]). Let be a semiconcave solution of (HJ loc ) and let 0 ∈ Sing ( ). Let ∶ [0, ] → Ω ( = 1, 2) be strict singular characteristics with initial point 0 such that 0 ∉ ( ( ), + ( ( ))) for all ∈ [0, ]. Then 1 ( ) = 2 ( ) for all ∈ [0, ].
, but related, problem is the study of the propagation of the closure of the singular set of , i.e, the 1 singular support of .Definition 4.1. Let be a viscosity solution of (4.1). We say that ( , ) is not in the 1 singular support of , denoted by ( , ) ∉ sing supp 1 ( ), if there exists a neighborhood ⊂ (0, ) × ℝ of ( 0 , 0 ), such that ∈ 1 ( ). In other words, sing supp 1 ( ) is the complement of the largest open set on which is of class 1 .Consider the system of generalized characteristics with respect to (4.1), that is, ̇ ( ) ∈ co ( , ( ), + ( ( ))), ∈ [ 0 , ) ( 0 ) = 0 .(4.2)Proposition 4.2 ([Alb14]). Suppose ( , , ) is a Tonelli Lagrangian with the associated Hamiltonian ( , , ) and 0 is continuous. Let ( 0 , 0 ) ∈ sing supp 1 ( ), then ( , ( )) ∈ sing supp 1 ( ) ∀ ∈ [ 0 , ) , where is solution of (4.2).
4. 2 . 1 .
21Concave initial data. For any open subset Ω ⊂ ℝ , consider the following Hamilton-Jacobi equation ( , ) + ( ( , )) = 0, ( ,
let = ℝ ⧵ Ω where Ω is an open domain in ℝ .In this case, the system of generalized characteristics becomes the generalized gradient system: also the case for the Hamiltonian of mechanical system which has the form ( , ) = 1 2 ⟨ ( ) , ⟩ + ( ), where is a smooth function and ( ) is a positive definite symmetric × real matrix smoothly depending on .
4.2. 3 .
3Mechanical systems. Now, suppose ( ) = 1 2 ⟨ , ⟩ with a positive definite × real matrix. Consider the following Cauchy-Dirichlet problem ( , ) + ( ( , )) = 0, ( , ) ∈ (0, +∞) × Ω =∶ ( , ) = ( , ), ( , ) ∈ . (4.6) Moreover, assume satisfies the following compatibility condition ( , ) − ( , ) ⩽ ( − ) − − (4.7)
Weak KAM aspects of Hamilton-Jacobi equations. Suppose is a smooth manifold without boundary and (resp. * ) is the tangent (cotangent) bundle of . Let ∶ → ℝ be a Tonelli Lagrangian, i.e., is of class 2 , and ( , ⋅) is strictly convex for all ∈ and uniformly superlinear. Let ∶ * → ℝ be the associated Tonelli Hamiltonian. We consider the stationary Hamilton-Jacobi equation( , ( )(5.1) we always suppose 0 on the right side equals Mañé's critical value. The solution of equation (5.
the solution of (5.4) by ( ( , ), ( , )). Define Φ ∶ (0, ∞) × ℝ → (0, ∞) × ℝ as Φ( , ) = ( , ( , )).
|
′ ( , ) − ′ (0, )| ⩽ ( ) , ∀ ∈ [0, 1], ∀| | ⩽ (5.6) with (⋅, ⋅) > 0 nondecreasing for all variables. So
Let > 0 .
0Then there exists by Proposition B.8 in[CC17] such that ( , ⋅) is of class 1,1 in ( , ) for 0 < ⩽ . For any 0 < ⩽ , ∈ ( , ) there exists a unique minimizer , ∈ Γ , for ( , ). Notice | ( , ,̇ , )| ⩽ ( ) for some constant ( ) > 0.
from the fact that (⋅, 0 ) and 0 (⋅) are both solutions of the Cauchy problem ( ( ),̇ ( )) = ( ( ),̇ ( )), ∈ [0, ] (0) = ,̇ (0) =̇ 0 , (0). Consequently, ( 0 , 0 ) = 0 ( 0 ) = . Recalling that ( , ) = − ( , ( ), ( , ( ),̇ , ( ))) and ( , ) = ( , ( ),̇ , ( )) one completes the proof. Whenever (5.2) has a unique solution, such a solution satisfies the Lax-Oleinik formula. More precisely, for any ∶ → ℝ, any > 0 and any ∈ we define ( , ) = ( ) is the (unique) viscosity solution of (5.2). Similarly,̆ ( ) gives the representation of the viscosity solution of (5.2) when replacing by − . We call { } >0 and {̆ } >0 the negative and positive type Lax-Oleinik operators, respectively. Both of them are continuous semigroups on suitable function spaces of initial data. Definition 5.2. Let be a continuous function on . We say is -dominated if ( ( )) − ( ( )) ⩽ ∫ ( ( ),̇ ( )) , for all absolutely continuous curves ∶ [ , ] → ℝ ( < ), with ( ) = and ( ) = .
Proposition 5.3 ([Fat97b,FM07]). There exists a Lipschitz semiconcave viscosity solution of (5.1). Moreover, such a solution is a common fixed point of the semigroup { }, i.e., = for all ⩾ 0.
5. 2 .
2Local propagation. In the study of singularities of weak KAM solutions, the first issue to address is the possible existence of isolated singular points. A typical family of Hamilton-Jacobi equations on the -torus is( , + ( )) = ( ), ∈ , (5.10) where ( ) is Mather's -function evaluated at ∈ ℝ ([Mat91]). For given ∈ ℝ , ( ) is exactly Mañé's critical value for the Hamiltonian ( , + ). Recall the function is convex and superlinear. Usually, the level set Λ = arg min ∈ℝ ( ) has a flat part. Observe that, for the one-dimensional pendulum system, there exist isolated singular points of a weak KAM solution if is contained in the relative interior of Λ. A criterion to ensure the non-existence of isolated singular points is * ∉ Λ, or ( * )
(⋅) − ( , ⋅) is concave with constant 1 − 2 ∕ < 0 for 0 < ⩽ 0 . For any fixed ∈ ℝ define
.
Then ( , ) is positive definite and ( , 3.3 in [CC17] we have that {̇ , } (0, 0 ] is an equi-Lipschitz family. Hence, | ( ( ), ( , ( ))) + ( ( ), ( , ( ))) 2 ( , ( ))| ⩽ 4 ,where 4 at most depends on Lip ( ). Therefore, we conclude that |̇ ( )
Theorem 5. 6 .
6Let be a Lipschitz and semiconcave solution of (5.17). Let ( ) be the intrinsic singular characteristic defined on [0, 0 ] starting from a given point ∈ Sing ( ), and let ( ) be the curve defined in (5.19). Then { } converges to uniformly on [0, ] and is Lipschitz continuous on [0, 0 ].Remark 5.7. Since the sequence of functions (⋅)− ( , ⋅) converges to (⋅)− ( , ⋅) uniformly as → ∞ and the family is equi-Lipschitz, then it is straightforward to see that the (unique) maximizer of (⋅) − ( , ⋅) converges to the maximizer of (⋅) − ( , ⋅) uniformly with respect to . However, we give a detailed proof of this fact below, in order to establish a precise estimate of the convergence rate.Proof. Let = ( , ( ),̇ , ( )), then = ( , ( )) ∈ + ( ( )). By the semiconcavity of and the convexity of the fundamental solution we deduce that 0 ⩽ [ ( ( )) − ( , ( ))] − [ ( ( )) − ( , ( ))] = [ ( ( )) − ( ( ))] − [ ( , ( )) − ( , ( ))] ⩽ [ ( ( )) − ( ( ))] + [ ( ( )) − ( ( ))] + [ ( ( )) − ( ( ))] − ⟨ , ( ) − ( )⟩ + 3 | ( ) − ( )| 2 ⩽ [ ( ( )) − ( ( ))] + [ ( ( )) − ( ( ))] + 2 − 3 | ( ) − ( )| 2 .
5. 3 . 2 .
32Dirichlet problem. The proof of Proposition 5.5 actually affords a method to handle various kind of problems for propagation of singularities if the solution can be represented in the form of an inf-convolution. For example, in[CCMW19], a global result for the Dirichlet problem was obtained using the above intrinsic approach.Consider the Dirichlet boundary-value problem for a first-order Hamilton-Jacobi equation ( , ( )) = 0, ∈ Ω, | Ω = .
Γ
, , (Ω) = { ∈ 1,1 ([ , ]; ℝ ) ∶ ( ) ∈ Ω, ∀ ∈ [ , ]; ( ) = ; ( ) = }. For any , ∈ Ω and > 0, we define the fundamental solution Ω ( , ) relative to Ω, Mañé's potential Φ Ω ( , ) relative to Ω, and critical value Ω ( ) relative to Ω by Ω ( , ) ∶= inf ∈Γ 0, , (Ω) ∫ 0 ( ( ),̇ ( )) , Φ Ω ( , ) ∶= inf >0 Ω ( , ), Ω ( ) ∶= − inf >0, ∈Ω 1 Ω ( , ).
Proposition 5. 9
9([CCMW19]). Suppose the energy condition Ω ( ) < 0. Let 0 ∈ Cut ( ). Then, the following alternative holds: (a) either there exists a generalized characteristic ∶ [0, +∞) → Ω starting from (0) = 0 such that ( ) ∈ Sing ( ) for all ∈ [0, +∞), (b) or there exist > 0 and a generalized characteristic ∶ [0, ) → Ω starting from (0) = 0 such that ( ) ∈ Sing ( ) for all ∈ [0, ), and a sequence of positive real numbers { } such that lim →∞ = , and lim →∞ Ω ( ( )) = 0.
5. 4 .
4Topology of Sing ( ) and Cut ( ). Recall the homotopy equivalence between a bounded open subset Ω ⊂ ℝ and the singular set of the distance function Ω discussed in Section 4.2.2 is based on a global propagation result for the generalized gradient flow. It is quite natural to use the global result in the last section to study the similar problem in the weak KAM setting.
Using the construction in the last section, one can obtain a continuous homotopy ∶ × [0, ] → , > 0, with the following properties: (a) for all ∈ we have ( , 0) = ; (b) if ( , ) ∉ Sing ( ) for some > 0 and ∈ , then the curve ↦ ( , ) is -calibrating on [0, ]; (c) if there exists a -calibrated curve ∶ [0, ] → with (0) = , then ↦ ( , ) = ( ) for every ∈ [0, min{ , }]. Proposition 5.12 ([CCF17]). The inclusions Sing ( ) ⊂ Cut ( ) ⊂ ( ⧵ ( )) ∩ Sing ( ) ⊂ ⧵ ( ) are all homotopy equivalences. As a consequence, for every connected component of ⧵ ( ) the three intersections Sing ( ) ∩ , Cut ( ) ∩ , and Sing ( ) ∩ are path connected. Similar to the homotopy constructed above, for any open subset ⊂ ⧵ ( ), we can also construct a local homotopy by ( , ) = ( , ( )), where is continuous interpolation of the cut time function ∶ → [0, +∞] (the supremum of the time ⩾ 0 such that there exists a -calibrated curve ∶ [0, ] → , with (0) = ) and the local exit function ∶ → [0, +∞] defined by ( ) = sup{ ∈ [0, +∞) ∶ ( , ) ∈ , for all ∈ [0, ]}. Notice < on an open subset of , and is upper semicontinuous, is lower semicontinuous.
Proposition 5 .
513 ([CCF17]). The spaces Sing ( ) and Cut ( ) are locally contractible. 5.4.2. Singular set on noncompact manifolds. Let 0 < ⩽ ∞ and suppose is a noncompact manifold and (resp. ) is a Tonelli Lagrangian (resp. Hamiltonian). We will review some topological results for the singular set of a uniformly continuous viscosity solution of+ ( , ) = 0 on (0, ) × , (5.24)which were obtained in[CCF19], together with their applications to Riemannian geometry.Proposition 5.14 ([CCF19]). Let ∶ * → ℝ be a Tonelli Hamiltonian. If is a continuous viscosity solution of the evolutionary Hamilton-Jacobi equation (5.24), then the set Sing ( ) is locally contractible in (0, ) × . To formulate the global homotopy equivalence result, we need extend the notion of Aubry set of as follows: let ∶ (0, ) × → ℝ, with ∈ (0, +∞], be a viscosity solution of the evolutionary Hamilton-Jacobi equation (5.24). The Aubry set ( ) of is the set of points ( , ) ∈ (0, ) × for which we can find a curve ∶ [0, ) → , with ( ) = and ( , ( )) − ( , ( )) = ∫ ( ( ),̇ ( )) , for every < ∈ [0, ).
( , ) ( ) is a deformation retract of ⧵ { }, therefore it is locally contractible. On the other hand, very little was known up to now about the set ( , ) = {( , ) ∈ × ∶ there exists a unique minimal -geodesic between and }. As Marcel Berger wrote in [Ber03, Page 284]
Proposition 5 .
518 ([CCF19]). If C is a closed subset of the complete Riemannian manifold ( , ), then the inclusion Sing * ( ) ⊂ ⧵ ∪ * ( ) is a homotopy equivalence.
Proposition 5.15 ([CCF19]). Let ∶ * → ℝ be a Tonelli Hamiltonian. Assume that the uniformly continuous function ∶ [0, ) × → ℝ, with ∈ (0, +∞], is a viscosity solution of the evolutionary Hamilton-Jacobi equation (5.24). Then the inclusion
:5730-5753, 2020. DIPARTIMENTO DI MATEMATICA, UNIVERSITÀ DI ROMA "TOR VERGATA", VIA DELLA RICERCA SCIEN-TIFICA 1, 00133 ROMA, ITALY Email address: [email protected] DEPARTMENT OF MATHEMATICS, NANJING UNIVERSITY, NANJING 210093, CHINA Email address: [email protected]
, ∶ (−∞, ] → ℝ such that , ( ) = , and
It follows that , and , coincide on [0, ] since both of them are extremal curves for the action functional in (5.3) and satisfy the same endpoint conditions at , . This leads to a contradiction since we suppose , ∉ Sing ( ). This argument shows the stronger result that any in the cut locus of , Cut ( ), is the initial point of a singular arc which remains singular at all times. We will emphasise this point in the next section. Now, we want to give a new proof of the Lipschitz continuity of intrinsic characteristics, by a reasoning that seems more natural than the method we used in[CC17]. This proof is based on the combination of the approximation argument used in[CY09]and regularity of the fundamental solution (Lemma 5.1).First, suppose ∈ 2 (Ω) with Ω a bounded open subset of ℝ . Fix ∈ Ω. By following the approach in[CC17], we have that there exists 0 > 0 such that for any ∈ (0, 0 ] the strictly concave function (⋅) − ( , ⋅) attains its maximum at a unique point ( ). In other words, the curve satisfies the equation Thus, by differentiating (5.14) with respect to we obtain 2 ( , ( )) − 2 ( ( )) ̇ ( ) = ( ( ), ( , ( ))) + ( ( ), ( , ( ))) ( , ( )).Notice that 2 ( , ) − 2 ( ) is invertible since 2 ( , ) − 2 ( ) > 2 − 1 > 3 , 0 < ⩽ 0 .
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arxiv |
Some observations about the MOLSCAT
17 Jul 2015
M K Sharma
School of Studies in Physics
Jiwaji University
474 011Gwalior, (M.P.)India
Monika Sharma s:[email protected]
School of Studies in Physics
Jiwaji University
474 011Gwalior, (M.P.)India
Suresh Chandra
Physics Department
Lovely Professional University
144411 (Phagwara, Punjab)India
Some observations about the MOLSCAT
121117 Jul 2015ISM: MoleculesCollisional transitionsMOLSCAT
For calculation of cross sections for collisional transitions between rotational levels in a molecule, a computer code, MOLSCAT has been developed byHutson & Green (1994). For the transitions between rotational levels in H 2 CS due to collisions with He atom, we have calculated cross sections under the CS approximation. In the MOLSCAT, there is provision to input more than one values of total energies. Here, for example, we are interested in the cross sections for total energy 11 cm −1 . The calculations have been done for the single energy 11 cm −We have found that the cross sections for 11 cm −1 , in general, differ from one another in all the 9 calculations. The reason for the difference in the results appears that the MOLSCAT uses the intermediate data of calculations for one energy, in the calculations for other energies. Under such circumstances, the possible suggestion can be to run the MOLSCAT for a single energy at a time.
Introduction
In most of the cosmic objects, spectral lines of molecules are formed under non-thermal conditions and for analysis of spectrum, the radiative and collisional transition probabilities for the transitions between rotational levels of the molecule are essentially required.
For the calculation of scattering cross sections for collisional transitions, the first requirement is to calculate the interaction potential between the molecule of interest and the collision partner. Being the most abundant in the interstellar medium, the hydrogen molecule H 2 is taken as the colliding partner. Because of two hydrogen atoms, the H 2 has two species, called the ortho (parallel spins) and para (anti-parallel spins). Thus, one needs to consider the collisions between the molecule of interest and para-H 2 (in the J = 0 state) or ortho-H 2 (in the J = 1 state).
The molecule of interest may be a linear molecule (diatomic molecule), symmetric top molecule or asymmetric top molecule. Large number of molecules found in the cosmic objects belong to the category of asymmetric top molecules. Treatment of an asymmetric top molecule is rather complicated, as there is no preferential direction. The molecule of our interest, the thioformaldehyde H 2 CS, is asymmetric top molecule.
Often, for simplification of calculations, the molecular hydrogen (collision partner) is taken as structure-less, and has been replaced by the He atom, as both the H 2 and He have two protons and two electrons, and the interaction depends on the charges. In the present discussion, we have considered the collision between the H 2 CS and He.
In section 2, the interaction potential between the H 2 CS and He has been calculated. Section 3 has been devoted for the calculations for collisional cross sections with the help of MOLSCAT. In the last section 4, we have discussed about the results and the conclusions have been drawn. The H 2 CS is a planar molecule with electric dipole moment along the axis having the lowest moment of inertia.
Interaction potential
Then, we have included the He atom whose positions have been expressed in terms of the spherical polar coordinates (R, θ, φ) with origin at the center-of-mass of H 2 CS. For the interaction between H 2 CS and He, we have used the Coupled Cluster with Single and Double and perturbative Triple CCSD(T) method and cc-pVTZ basis set. In order to account for the Basis Set Superposition Errors (BSSE) (Sharma et al., 2014a, b;, we have done three sets of calculations:
(i) energy E 1 of H 2 CS + He.
(ii) energy E 2 of H 2 CS while He is present as a ghost atom.
(iii) energy E 3 of He while all the atoms of H 2 CS are present as ghost atoms.
There are 106 basis functions, 204 primitive gaussians, 119 cartesian basis functions. The interaction potential V (R, θ, φ) between the H 2 CS and He is then The interaction potential V (R, θ, φ) has been calculated for R = 2.25 (0.25) 5.25Å,
V (R, θ, φ) = E 2 (R, θ, φ) + E 3 (R, θ, φ) − E 1 (R, θ, φ)θ = 0 • (15 • ) 180 • and φ = 0 • (15 • ) 90 • .
The calculated potential has been fitted in terms of the spherical harmonics with the help of the expression:
V (R, θ, φ) = lm v lm (R) (1 + δ m0 ) [Y lm (θ, φ) + (−1) m Y l−m (θ, φ)]
where the azimuthal quantum number l has been allowed to vary for the integer values form 0 to 5. For a given value of l, the magnetic quantum number m could assume even integer values from 0 to l. The values of the expansion coefficients v lm (R) as a function of R are given in Table 1.
For the present investigation, the accuracy of interaction potential does not matter.
However, an interaction potential is required. This interaction potential has been used as input in the computer code MOLSCAT. When the interaction potential is not appropriate for the MOLSCAT, the MOLSCAT does not converge and no output is produced. For example, in the calculations of Green (1980) and Palma & Green (1987), the BSSE were not considered. When the BSSE are considered, the potential would definitely be different.
The MOLSCAT had given results for that potential and would give for new potential also.
Calculations with MOLSCAT
The where the basis set with JMAX = 14 is used. In the MOLSCAT, there is a provision to input more than one values of total energies. In the input file, NNRG is the number of total energies included in the input file. The calculations have been done for the single energy 11 cm −1 and for eight combinations, (11, 12), (12, 11), (10, 11), (11, 10), (11, 12, 13), (9, 10, 11), (10, 11, 12), and (9, 10, 11, 12, 13) cm −1 , as given in Table 2. In Table 2, column 3 gives the number of energies (NNRG) given in the input file. The energies and their sequence are given in column 4.
The cross sections for different sets of energies are denoted by C1, C2, . . . , C9. In C1, the MOLSCAT is run for the single energy 11 cm −1 . In C2 and C3, and in C4 and C5, and in C7, two energies are before the 11 cm −1 . In C8, one energy before and one energy after the 11 cm −1 have been taken. In C9, two energies before and two energies after the 11 cm −1 are taken. One may consider other combinations also. We assume that these combinations are sufficient for our investigation. All the parameters (except NNRG and ENERGY) in the input file for all the combinations are the same.
The cross sections have been calculated with the help of MOLSCAT. In Table 3, we have given the cross sections for 11 cm −1 . Up to 11 cm −1 in H 2 CS, there are four para levels (0 00 , 1 01 , 2 02 , 3 03 ) and two ortho levels (1 11 , 1 10 ). The ortho and para species of H 2 CS behave as they are two distinct molecules, as there are no transitions between them.
Thus, there are 12 (excitations + deexcitations) transitions between the para levels and 2 (excitations + deexcitation) transitions between the ortho levels. The cross sections show random values.
In order to understand the range of variation in nine combinations, for each transition,
we have chosen the maximum cross-section C max and the minimum cross-section C min , and have calculated the percent variation P of C max relative to C min as
P = C max − C min C min × 100
For example, for the transition 0 00 → 1 01 , we have C max = 23.611 and C min = 14.809.
The value of P for each transition is given in the last column of Table 3.
Discussion and conclusions
First, we have to state that we have no comment on the the papers where CS and CC approximations have been used. The only point to be discussed here is that we have found different cross sections for the same energy (11 cm −1 ) when different combinations, which depend on the direction also, having 11 cm −1 are considered.
Except one pair (between 3 03 and 2 02 ), for other pairs, the cross sections for C1 and C5 are equal. Other values of cross sections show random variation. It is obvious that the value of P is the same for excitation and deexcitation between a pair of levels. It supports the detailed equilibrium where the rate coefficients for collisional excitation and deexcitation are proportional to each other. The value of P for ortho transitions is 88.5.
It is quite high and shows that the cross sections can vary up to almost a factor of 2. It may be because the separation between the levels is very small.
To some extent, the value of P is found to decrease with the increase of separation between the levels. However, it is not the case for all the transitions. In absence of any trend shown by the cross sections, it is difficult to draw any legitimate conclusions.
However, it may be suggested that one should calculate the cross sections for a single value of energy (NNRG = 1) with the help of the MOLSCAT. It would avoid the possibility of using intermediate data for the calculations for one energy into the calculations for other energy.
One may still ask if doing the calculations with MOLSCAT for a single value of energy (NNRG = 1) is sufficient. We do not find ourselves qualified to make any comment about it. Probably some one who knows more details about the MOLSCAT may answer about it. Since there is no substitute for the MOLSCAT, one has to depend on the MOLSCAT.
For calculation of cross sections for collisional transitions between rotational levels with the help of MOLSCAT, one requires the interaction potential between the molecule of interest and the collision partner. For calculation of the interaction potential between H 2 CS and He, as the first step, the geometry of the H 2 CS molecule has been optimized with the help of GAUSSIAN 2003 and the coordinates of atoms in the H 2 CS are obtained.
MOLSCAT has provision to do calculations under the Infinite Order Sudden (IOS) approximation, Coupled States (CS) approximation, and Close Coupling (CC) approach.For scattering in an asymmetric top molecule, these three approaches can be invoked by choosing the value of ITYPE as 106, 26 and 6, respectively, in the input file for the MOLSCAT. In the IOS approximation, the energies of rotational levels in the molecule are neglected in comparison to the energy of the collision partner. Therefore, it is valid for high energies of collision partner. Consequently, the the scientists prefer to use the CS approximation which is valid for all energies of collision partner.Though the CC approximation is better than the CS approximation, but it is too expensive from the computation point of view. A calculations in the CC approximation takes many times more computer time as compared to that in the CS approximation. Ina large number of calculations, the CS and CC approximations have been used. Some of the papers where such calculations have been done are: Cernicharo et al. (2011), Daniel et al. (2014; 2015), Dubernet et al. (2006), Dumouchel et al. (2010), Faure & Josselin (2008), Faure et al. (2012), Flower & Lique (2015), Gotoum et al. (2011), Machin & Roueff (2006, 2007), Pottage et al. (2004), Rabli & Flower (2010a, b; 2011), Sarrasin et al. (2010), Troscompt et al. (2009), Wernli et al. (2006, 2007a, b), Wiesenfeld & Faure (2013), Wiesenfeld et al. (2011).In the present work, for example, we are interested in the cross sections for total energy 11 cm −1 . The calculations have been done under the CS approximation (ITYPE = 26)
Table 1 :
1H 2 CS-He interaction potential in cm −1 .R (Å)
v 00
v 10
v 20
v 22
v 30
v 32
2.25
52115.61 -34010.48 47297.66 -6818.03 -22737.70 9602.23
2.50
23789.73 -15582.89 21426.39 -3027.45 -10256.35 4016.18
2.75
10490.22
-6937.04
9373.59 -1364.29
-4506.79 1729.56
3.00
4431.47
-2931.71
3928.66
-598.34
-1905.54
715.10
3.25
1787.34
-1177.20
1579.96
-254.33
-779.49
280.10
3.50
681.02
-449.18
609.07
-105.13
-310.85
103.56
3.75
237.12
-160.60
222.77
-42.20
-121.20
35.63
4.00
67.18
-50.97
74.92
-16.17
-45.83
10.70
4.25
6.66
-11.77
21.12
-5.66
-16.41
2.10
4.50
-11.84
0.65
3.10
-1.58
-5.26
-0.46
4.75
-15.19
3.48
-2.03
-0.14
-1.26
-0.89
5.00
-13.69
3.32
-2.88
0.30
0.02
-0.74
5.25
-11.08
2.55
-2.53
0.34
0.35
-0.51
R (Å)
v 40
v 42
v 44
v 50
v 52
v 54
2.25
2142.69 -19887.50
1152.16 6949.83 12372.54 -2420.97
2.50
1259.07
-8181.70
413.10 2480.66
4807.46
-851.89
2.75
377.35
-3557.91
167.63 1134.88
2029.72
-349.71
3.00
56.16
-1523.69
67.94
525.16
848.60
-144.78
3.25
-22.72
-635.48
26.52
234.40
345.14
-58.10
3.50
-28.50
-259.23
9.84
100.89
137.07
-22.45
3.75
-20.23
-104.16
3.43
42.73
53.74
-8.42
4.00
-12.32
-41.10
1.09
18.05
20.84
-3.02
4.25
-6.80
-15.59
0.24
7.44
7.80
-0.97
4.50
-3.42
-5.47
-0.03
2.85
2.71
-0.24
4.75
-1.60
-1.65
-0.08
1.00
0.81
0.01
5.00
-0.65
-0.30
-0.06
0.29
0.16
0.05
5.25
-0.26
0.09
-0.05
0.06
-0.03
0.06
Table 2 :
2Parameters No. Combination NNRG ENERGY (cm −1 )1
C1
1
11
2
C2
2
11, 12
3
C3
2
12, 11
4
C4
2
10, 11
5
C5
2
11, 10
6
C6
3
11, 12, 13
7
C7
3
9, 10, 11
8
C8
3
10, 11, 12
9
C9
5
9, 10, 11, 12, 13
the sequence of energies have been reversed. In C6, two energies are after the 11 cm −1
Table 3 :
3Cross sections for various transitions in H 2 CS for total energy 11 cm −1 01 → 3 03 12.348 11.737 11.836 11.772 12.348 9.631 12.575 11.673 13.476 39.9 1 11 → 1 10 38.799 40.058 73.123 39.529 38.799 40.875 43.286 40.311 45.526 88.5 1 10 → 1 11 40.759 42.082 76.818 41.526 40.759 42.940 45.473 42.347 47.02 → 1 01 18.919 18.847 25.536 21.389 18.919 18.367 23.192 21.100 21.220 39.0 2 02 → 3 03 38.157 36.981 29.993 35.978 31.286 33.465 32.927 42.974 30.286 43.3 03 → 1 01 12.627 12.002 12.104 12.037 12.627 9.848 12.859 11.936 13.781 39.9 3 03 → 2 02 49.919 48.381 39.238 47.069 40.931 43.781 43.077 56.222 39.623 43.3in 9
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| {'fraction_non_alphanumeric': 0.06951033927132755, 'fraction_numerical': 0.0940828932056217, 'mean_word_length': 3.529292519764849, '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': 1, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'For calculation of cross sections for collisional transitions between rotational levels in a molecule, a computer code, MOLSCAT has been developed byHutson & Green (1994). For the transitions between rotational levels in H 2 CS due to collisions with He atom, we have calculated cross sections under the CS approximation. In the MOLSCAT, there is provision to input more than one values of total energies. Here, for example, we are interested in the cross sections for total energy 11 cm −1 . The calculations have been done for the single energy 11 cm −We have found that the cross sections for 11 cm −1 , in general, differ from one another in all the 9 calculations. The reason for the difference in the results appears that the MOLSCAT uses the intermediate data of calculations for one energy, in the calculations for other energies. Under such circumstances, the possible suggestion can be to run the MOLSCAT for a single energy at a time.', 'arxivid': '1507.04882', 'author': ['M K Sharma \nSchool of Studies in Physics\nJiwaji University\n474 011Gwalior, (M.P.)India\n', 'Monika Sharma s:[email protected] \nSchool of Studies in Physics\nJiwaji University\n474 011Gwalior, (M.P.)India\n', 'Suresh Chandra \nPhysics Department\nLovely Professional University\n144411 (Phagwara, Punjab)India\n'], 'authoraffiliation': ['School of Studies in Physics\nJiwaji University\n474 011Gwalior, (M.P.)India', 'School of Studies in Physics\nJiwaji University\n474 011Gwalior, (M.P.)India', 'Physics Department\nLovely Professional University\n144411 (Phagwara, Punjab)India'], 'corpusid': 118135256, 'doi': '10.1088/0253-6102/64/6/731', 'github_urls': [], 'n_tokens_mistral': 8840, 'n_tokens_neox': 6955, 'n_words': 3735, 'pdfsha': 'b4c90e4e09d790d9a2f88f358741cf049f2f1d8e', 'pdfurls': ['https://arxiv.org/pdf/1507.04882v1.pdf'], 'title': ['Some observations about the MOLSCAT', 'Some observations about the MOLSCAT'], 'venue': []} |
arxiv |
Divergences of the scalar sector of quadratic gravity
14 Mar 2022
Enriqueálvarez [email protected]
Departamento de Física Teórica and Instituto de Física Teórica
IFT-UAM/CSIC
Universidad Autónoma de Madrid
28049Cantoblanco, MadridSpain
Jesús Anero [email protected]
Departamento de Física Teórica and Instituto de Física Teórica
IFT-UAM/CSIC
Universidad Autónoma de Madrid
28049Cantoblanco, MadridSpain
Divergences of the scalar sector of quadratic gravity
14 Mar 2022
The divergences coming from a particular sector of gravitational fluctuations around a generic background in general theories of quadratic gravity are analyzed. They can be summarized in a particular type of scalar model, whose properties are analyzed.
1 Introduction.
In this paper we are going to analyze a particular sector of quantum graviton fluctuations around an arbitrary background when the action is given by general theories quadratic in spacetime curvature. In the particular case where the background metric coincides with the Minkowski oneḡ µν = η µν , the decomposition of the graviton fluctuation in terms of spin components can be expressed in terms of the good old Barnes-Rivers projectors [10] [9]. Let us briefly recall the formalism.
Let us define two projectors in momentum space
P (0)β α = kαk β k 2 ≡ ω α β = 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 P (1)β α = δ β α − kαk β k 2 ≡ θ α β = 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 (1)
It should be pointed out that these operators are non-local in position space where 1 k 2 stands for 2 −1 . As it is well-known, the metric h µν transforms in the euclidean setting under the representation 10 ≡ of SO(4), so the spin content and corresponding projectors are given by
s=2 : h T ij ≡ h ij − 1 3 hδ ij (P 2 ) ρσ µν ≡ 1 2 θ ρ µ θ σ ν + θ σ µ θ ρ ν − 1 3 θ µν θ ρσ s=1 : h 0i (P 1 ) ρσ µν ≡ 1 2 θ ρ µ ω σ ν + θ σ µ ω ρ ν + θ ρ ν ω σ µ + θ σ ν ω ρ µ s=0 : h 00 (P w 0 ) ρσ µν ≡ ω µν ω ρσ s=0 : h ≡ δ ij h ij (P s 0 ) ρσ µν ≡ 1 3 θ µν θ ρσ(2)
these particular projectors are complete in the symmetrized direct product
Sym (T x ⊗ T x )(3)
where T x is the tangent space at the point x ∈ M of the space-time manifold.
It is convenient to define another projector P 0 ≡ P w 0 + P s 0 (4) and the non-differential projectors are
I ρσ µν ≡ 1 2 δ ρ µ δ σ ν + δ σ µ δ ρ ν T ρσ µν ≡ 1 4 η µν η ρσ(5)
then we can write a closure relation for these projectors. To be specific, (P 2 ) ρσ µν + (P 1 ) ρσ µν + (P 0 ) ρσ µν = I ρσ µν (6) These projectors are not enough for some tasks, because they do not form a basis of the space of four-index tensors of the type of interest. We then need to add a new independent operator
P × 0 ρσ µν = 1 √ 3 (ω µν θ ρσ + θ µν ω ρσ )(7)
that can be identified with the mixing of the two spin 0 components, h and h 00 . It is clear that this new operator cannot be orthogonal to the other four, since closure implies that the only operator orthogonal to the set that closes is the null operator. This decomposition has been generalized for arbitrary background metric by York [20] by means of his conformally invariant orthogonal decomposition of the graviton fluctuation which reads
h µν ≡ h T T µν + h L µν + h tr µν(8)
where the longitudinal component is given by
h L µν ≡∇ µ ξ ν +∇ ν ξ µ − 2 nḡ µν ∇ λ ξ λ ≡ (Lξ) µν(9)
and
h tr µν ≡ 1 n hḡ µν(10)
Besides, in [20] it is demonstrated how to uniquely determine ξ µ from the knowledge of h µν . The transversality condition
∇ µ h T T µν = 0(11)
leads a covariant equations for the vector field ξ µ
Dξ µ = −∇ ν (Lξ) µν(12)
defining Λ, the projector into the traceless piece of the graviton fluctuation, this defines a partial differential equation for ξ, namely
Dξ = −∇ · Λh(13)
which uniquely determines ξ in terms of h µν unless there are in the background metric conformally Killing vectors, for which Lξ = 0. The different spin contributions are mutually orthogonal. It is clear that through a gauge transformation the spin one piece can be completely eliminated (although this is not usually the most convenient way of fixing the gauge). For example, the usual harmonic (de Donder) background field gaugē
∇ µ h µν = 1 2∇ ν h(14)
reads 2ξ µ +R µλ ξ λ = 0 (15) in case the spin one part is eliminated, the residual term proportional to∇ξ can be absorbed in the spin 0 scalar φ. This vector can in turn be decomposed as
ξ = ξ T + ξ L(16)
(where∇.ξ T =∇.ξ = 0). The scalar component of the graviton fluctuation comprises then two parts: h and∇.ξ L . In this paper we are going to consider the full scalar piece denoted by φḡ µν just by simplicity. Previous related works are to be found in [19] as well as in [12]. The renormalizability of the model has been exploited in [14] to build particle physics models. For its relationship with more general setup of quantum gravity cf. [7]. Related scalar models have been studied in [15]. The existence of runaway solutions in higher order theories has been first pointed out by Dirac [13] in his classic paper, and in the context of quadratic gravity, in [17] [9]. In this paper we shall use the usual second order formalism all the way; some points to the much more general first order approach are to be found in [1]. Finally, some particular models have enhanced Dif f × W eyl symmetry [2].
Quadratic gravity
The general lagrangian quadratic in curvature is a combination of monomials in R 2 , R 2 µν and R 2 µνρσ . The most general diff invariant low energy action is then
S = d(vol) λ − 1 2κ 2 R + α 1 R 2 + α 2 R µν R µν + α 3 R µνρσ R µνρσ(17)
where the diff invariant measure is given by
d(vol) ≡ |g|d n x(18)
In n = 4 dimensions the Gauss-Bonnet formula permits to discard the Riemann squared term; but in higher dimensions this is not so, so we prefer to keep the whole set of operators. Let us analyze them in turn. In the appendix we have collected the expansion of the different monomials up to quadratic order in the perturbartions, which is enough to compute the one loop divergences.
3 The scalar sector of quadratic gravity.
We are to restrict now the graviton fluctuations to the scalar sector as explained in the introduction, i.e h µν = φḡ µν In the case of the R squared action
S = d 4 x √ −gR 2(19)
the EoM is
−2R µνR + 1 2ḡ µνR 2 + 2∇ µ∇νR − 2ḡ µν2R = 0 (20) which trace yields n − 4 2R 2 − 2(n − 1)2R = 0(21)
with the ansatz h µν = φḡ µν , the expansion (106) reduces to
S φ R 2 = d n x √ḡ φ (n − 1) 222 + (−n 2 + 7n − 6) 2R2 + n 2 − 10n + 24 8R 2 φ(22)
we can write
S φ R 2 = (n − 1) 2 d n x √ḡ φF φ(23)
where the explicit expression for this operator is then
F =2 2 +D µν∇ µ∇ν +P (24) whereD µν = (−n 2 + 7n − 6) 2(n − 1) 2ḡ µνR P = n 2 − 10n + 24 8(n − 1) 2R 2 (25)
The form of the usual four-dimensional one-loop counterterm is tabulated in [11], namely,
∆S = − 1 (4π) 2 2 d n x |ḡ|tr 1 180 2R µνρσR µνρσ − 2R µνR µν + 5R 2 I + + 1 48D 2 + 1 24D µνD µν + 1 12DR − 1 6D µνR µν −P + 1 6 W µν W µν (26) in our case ∆S φ R 2 = − 1 (4π) 2 2 d n x (n − 1) 2 90R 2 µνρσ − (n − 1) 2 90R 2 µν + + 3n 4 − 54n 3 + 196n 2 + 424n − 1424 576R 2(27)
on-shell, using (21)
∆S φ R 2 = − 1 (4π) 2 2 d n x (n − 1) 2 90R 2 µνρσ − (n − 1) 2 90R 2 µν + + (3n 4 − 54n 3 + 196n 2 + 424n − 1424)(n − 1) 144(n − 4)2R(28)
We continue with the next monomial, Ricci squared
S = d 4 x √ −gR 2 µν (29) the EoM is −2R µλR λ ν + 1 2ḡ µνR 2 αβ −2R µν + 2∇ λ∇νR λ µ −ḡ µν∇α∇βR αβ = 0 (30) which trace is n − 4 2R 2 αβ −2R − (n − 2)∇ α∇βR αβ = 0 (31)
with the approximation h µν = φḡ µν , in this case (108) reduces to
S φ R 2 µν = d n x √ḡ φ n 4 (n − 1)2 2 +R2 + (−n 2 + 6n − 8) 4R µν∇ µ∇ν + + n 2 − 10n + 24 8R 2 µν φ (32) again, we can write S φ R 2 µν = n 4 (n − 1) d n x √ḡ φF φ(33)
where the explicit expression for this operator is then
F =2 2 +D µν∇ µ∇ν +P (34) whereD µν = 1 n(n − 1) 4ḡ µνR + −n 2 + 6n − 8 R µν P = n 2 − 10n + 24 2n(n − 1)R 2 µν(35)
using the one-loop counterterm (26), we obtain
∆S φ R 2 µν = − 1 (4π) 2 2 d n x n(n − 1) 360R 2 µνρσ + −109n 4 + 1388n 3 − 4504n 2 + 2400n + 960 1440n(n − 1)R 2 µν + + 5n 4 − 64n 3 + 8n 2 − 96n + 192 576n(1 − n)R 2(36)
on-shell, using (31)
∆S φ R 2 µν = − 1 (4π) 2 2 d n x n(n − 1) 360R 2 µνρσ + −109n 4 + 1388n 3 − 4504n 2 + 2400n + 960 720n(n − 1)(n − 4)2R + + (−109n 4 + 1388n 3 − 4504n 2 + 2400n + 960)(n − 2) 720n(n − 1)∇ µ∇νR µν + + 5n 4 − 64n 3 + 8n 2 − 96n + 192 576n(1 − n)R 2(37)
And finally, the action for Riemann squared
S = d 4 x √ −gR 2 µνρσ (38) the EoM is −2R µαβλR αβλ ν + 1 2ḡ µνR 2 αβρσ − 4∇ ρ∇σR µρνσ = 0 (39) which trace is n − 4 2R 2 αβρσ − 4∇ ρ∇σR ρσ = 0(40)
with approximation h µν = φḡ µν , in this case (110) reduces to
S φ R 2 µνρσ = d n x √ḡ φ (n − 1)2 2 +R2 + (4 − n)R µν∇ µ∇ν + n 2 − 10n + 24 8R 2 µνρσ φ(41)
one time more, we can write
S φ R 2 µνρσ = (n − 1) d n x √ḡ φF φ(42)
the explicit expression for this operator is then
F =2 2 +D µν∇ µ∇ν +P (43) whereD µν = 1 (n − 1) ḡ µνR + (4 − n)R µν P = n 2 − 10n + 24 8(n − 1)R 2 µνρσ(44)
and like the previous case, with (26), we obtain
∆S φ R 2 µνρσ = − 1 (4π) 2 2 d n x −45n 2 + 454n − 1084 360R 2 µνρσ + 71n 2 − 412n + 476 360(n − 1)R 2 µν + + 2n 2 + 5n + 38 72(n − 1)R 2 (45) on-shell, using (40) ∆S φ R 2 µνρσ = − 1 (4π) 2 2 d n x −45n 2 + 454n − 1084 45(n − 4)∇ ρ∇σR ρσ + 71n 2 − 412n + 476 360(n − 1)R 2 µν + + 2n 2 + 5n + 38 72(n − 1)R 2(46)
This results shows clearly that the truncation made on the quantum fluctuations is not self-consistent; it is known [19] that the full model is renormalizable; so that there will be mixing between different spins at the quantum level. We pal to study this in the future.
Four dimensional check.
Note that in the physical dimension n = 4
S φ R 2 µνρσ = S φ R 2 µν = 1 3 S φ R 2 = d n x √ḡ φ 32 2 +R2 φ (47) obviously S φ R 2 − 4S φ R 2 µν + S φ R 2 µνρσ = 0(48)
and the same with the counterterms
∆S φ R 2 µνρσ = ∆S φ R 2 µν = 1 3 ∆S φ R 2 = − 1 (4π) 2 2 d n x 1 30R 2 µνρσ − 1 30R 2 µν + 5 12R 2 (49) again ∆S φ R 2 − 4∆S φ R 2 µν + ∆S φ R 2 µνρσ = 0 (50)
which is consistent with the Gauss-Bonnet theorem. 4 Simple scalar model.
As we have just seen, the scalar sector of quadratic gravity reduces to a scalar lagrangian of a particular type. Let us examine a particular scalar model which is closely related to it, namely
S = d n xφ 2 2 + (M 2 − m 2 )2 − M 2 m 2 φ(51)
we can write
S 2 = d n xφF φ(52)
the explicit expression for this operator is then
F =2 2 +D µν∇ µ∇ν +P (53) whereD µν =ḡ µν (M 2 − m 2 ) P = −M 2 m 2(54)
the one-loop counterterm, with (26), reads
∆S M m = − 1 (4π) 2 2 d n x 1 180 2R 2 µνρσ − 2R 2 µν + 5R 2 + (n − 2) 12 (M 2 − m 2 )R + + n(n + 2) 48 (M 2 − m 2 ) 2 + M 2 m 2(55)
but the same action can be written like
S = d n x √ḡ φ −2 − M 2 −2 + m 2 φ (56) the counterterm of operator −2 − M 2 is ∆S M = − 1 (4π) 2 2 d n x 1 360 2R 2 µνρσ − 2R 2 µν + 5R 2 + 1 6 M 2R + 1 2 M 4
and the corresponding to −2 + m 2 is
∆S m = − 1 (4π) 2 2 d n x 1 360 2R 2 µνρσ − 2R 2 µν + 5R 2 − 1 6 m 2R + 1 2 m 4
if and only if n = 4 the multiplicative anomaly [18] cancels
∆S M m = ∆S M + ∆S m(57)
This result is interesting insofar as the vanishing of the product anomaly in this case seems at variance with a general theorem in [5]. Let us elaborate. The product anomaly is conventionally defined [18] as
a C (A, B) ≡ log det (AB) − log det A − log det B(58)
and it is known to be non vanishing in general when determinants are defined through the zeta functions associated to the corresponding operators [3] and to be given by the Wodzicki residue [6]. Our result indicates that in our case the anomaly vanishes a C = 0 (59)
In fact, in [5] the anomaly a C was computed for the operators
A ≡ −2 + m 2 1 B ≡ −2 + m 2 2(60)
in the case m 2 = 0, with the result in terms of the digamma function
a C (A, B) = V n (4π) n/2 (−1) n/2 2 n 2 ! m n 1 (Ψ(1) − Ψ(n/2)) = 0(61)
We do not understand the reason for this discrepancy.
5 Some comments on unitarity of the effective action.
• In perturbation theory the Kallen-Lehmann spectral theorem [8] (for a scalar theory, to simplify things) asserts that the exact propagator in Minkowski space can be expressed as
Ω |T φ(x)φ(y)| Ω = d 4 p (2π) 4 e ip(x−y) ∞ 0 ρ(s) p 2 − s + i(62)
where the spectral function ρ(s) ≥ 0. In fact the free Feynman propagator
∆(p) = ∞ 0 ρ(s) p 2 − s + i(63)
can be easily obtained out of the euclidean one through a Wick rotation
∆ E (p) ≡ −1 p 2 e + m 2 −→ ∆ F (p) ≡ 1 p 2 − m 2 + i(64)
This sheds light on what is wrong with ghosts and/or tachyons. Ghosts have got the wrong sign for residues at the pole; tachyons instead have the pole located at spacelike momenta.
We would like to get a similar clear understanding from the heat kernel point of view. Let us concentrate in situation at the origin of spatial coordinates, id est, the point x = y = z = 0. The well defined onedimensional heat kernel
K(τ ) ≡ tr e −Oτ(65)
corresponding to the positive operator
O ≡ − d 2 dt 2 + m 2(66)
with plane wave eigenfunctions e iωt (67) reads, properly normalized,
K(τ ) = 1 √ 4πτ e − t 2 4τ −m 2 τ(68)
Now it is clear what happens when we want to compute with −2 (this is just the simplest ghost). Then we should start with
K(τ ) ≡ tr e +Oτ(69)
and this does not exist. There is a solution of the heat equation, namely
K gh (τ ) = 1 √ 4πτ e + t 2 4τ −m 2 τ(70)
which however does not reduce to a Dirac's delta when τ → 0.
lim →0 + 1 √ π e − x 2 2 = δ(x) = lim →0 + 1 √ π e x 2 2(71)
This is then an ultraviolet problem.
Something similar happens when m 2 < 0 (the simplest tachyon).
K tq ≡ 1 √ 4πτ e − t 2 4τ +m 2 τ(72)
The problem there does not lie in the non existence of the small proper time limit, but rather is that the propagator does not exist, because the integral
∞ 0 dτ K tq (τ )(73)
does not converge. This is an infrared problem.
• Let us now comment on Wick's rotation. The path integral is defined by
e iW = Dφe iS(74)
Assume a massive scalar in physical Minkowski space
S ≡ 1 2 dx 0 d 3 x φ −2 − m 2 φ(75)
Perform now the analytic continuation to Euclidean space. It is determined by the need for the mass term to be negative definite, in order for the path integral to formally converge. This implies
x 0 = −ix 4(76)
It follows that the relationship between the differential operators is
simply 2 → −2 E (77)
which is negative definite. This means that
S −→ − i 2 dx 4 d 3 x φ 2 E − m 2 φ(78)
The Euclidean effective action is then proportional to
dx 4 dx log det O E → i dx 0 dx log det O E ≡ iS ef f(79)
in conclusion, this means that the Minkowskian effective action is related to the euclidean determinant through
S ef f = dx 0 dx log det O E(80)
so that an imaginary piece in the euclidean determinant means an imaginary part in the Minkowskian effective action.
• The ζ-function can be recovered from the heat kernel through
ζ KG (s) = 1 Γ(s) ∞ 0 dτ τ 1−s tr K(τ ; x, y) = 1 Γ(s) ∞ 0 dτ τ 1−s 1 (4πτ ) n 2 e −m 2 τ = = (m 2 ) n 2 −s (4π) n 2 Γ s − n 2 Γ(s)(81)
which yields an imaginary part in the effective action.
The heat kernel of the d'Alembertian squared.
Let is return to the model (51)
S = d n xφ 2 2 + (M 2 − m 2 )2 − M 2 m 2 φ(87)
and consider it as defined in flat space. The characteristic equation reads
k 4 + (m 2 − M 2 )k 2 − M 2 m 2 = 0 (88)
whose solutions are
k 2 = M 2 − m 2 ± (M 2 − m 2 ) 2 + 4M 2 m 2 2 = M 2 −m 2(89)
This means that there are runaway classical solutions as long as m 2 = 0. It is to be expected that this carries in a ghostly form to the quantum regime. Let us try to obtain an exact solution to the heat equation at least in flat space.
∂ ∂τ K(τ, x) = 2 2 + (M 2 − m 2 )2 − M 2 m 2 K(τ, x)(90)
we can take a Fourier transform
K(τ, x) = 1 (2π) n+1 F (ω, k)e iωτ +i k· x d n kdω(91)
this implies
F (ω, k) = f (k)δ(−iω + k 4 + (M 2 − m 2 )k 2 + M 2 m 2 )(92)
and
K(τ, x) = 1 (2π) n e [k 4 +(M 2 −m 2 )k 2 −M 2 m 2 ]τ+i k· x f (k) (93)
the boundary condition at τ = 0 does imply
f (k) = 1(94)
then we can calculate the integral
K(τ, x) = 1 (2π) n d n ke [k 4 +(M 2 −m 2 )k 2 −M 2 m 2 ]τ+i k· x = = 1 (2π) n π n−1 2 Γ n+1 2 1 −1 dµ ∞ 0 dkk n−1 e [k 4 +(M 2 −m 2 )k 2 −M 2 m 2 ]τ+ikxµ = = 1 (2π) n x 2π n−1 2 Γ n+1 2 ∞ 0 dkk n−2 e [k 4 +(M 2 −m 2 )k 2 −M 2 m 2 ]τ sin kx (95)
First of all, it is clear that the integral does not converge unless the coefficient of |k| 4 is negative, which we will assume from now on. Then, in the particular case M = m there is an explicit solution
K(τ, x) = 1 (2π) n π n−1 2 Γ n+1 2 τ − n+2 4 12 e −M 4 τ − x 2 Γ n + 2 4 p F q n + 2 4 , 5 4 , 3 2 , 7 4 , x 4 256τ + +6 √ τ Γ n 4 p F q n 4 , 1 2 , 3 4 , 5 4 , x 4 256τ (96)
where the modified hypergeometric function is defines as
p F q ({a 1 . . . a p } , {b 1 . . . b q } , z) ≡ (a 1 ) k . . . (a p ) k (b 1 ) k . . . (b q ) k z k k!(97)
and the Pochhammer's symbol is defined as
(a) n ≡ Γ(a + n) Γ(a)(98)
Let us come back to the issue of the sign of the 2 2 term (or the 2 one for that matter). In [16] the scaling of determinants when the operator in question is multiplied by a constant factor λ has been studied. The corresponding zeta function reads
ζ λ (s) = n (λλ n ) −s = λ −s ζ(s)(99)
It follows that dζ λ (s)
ds = −λ −s ζ(s) log λ + λ −s ζ (s)(100)
For the determinant itself
log det ∆ ≡ −ζ (0)(101)
In the basic case when ∆ = 2
log det(λ2) = ζ(0) log λ + log det(2)(102)
so that when λ = −1 log det(−2) = iπζ(0) + log det(2)
which carries over an imaginary part in the effective action, clearly demonstrating violation of unitarity.
Conclusions
In this paper we have discussed the divergent structure of the scalar spin 0 sector of the graviton fluctuations in theories of gravitation quadratic in curvature. All those contributions have a similar structure, which we summarized in a simplified scalar model, which we studied in some detail. Although the general lagrangian is known to be renormalizable, owing to Stelle's [19] pioneering work, there is mixing between different spins, and the scalar sector is not. It shares however, many of the properties of the general lagrangian, like runaway solutions and the related presence of ghosts.
We were finally interested in the manifestation of non-unitarity in our language. It is a nontrivial problem, because the heat kernel is defined as an analytic continuation from the riemannian solution. In the analysis of the scalar model we have found that the product anomaly vanishes in this particular case, which seems to challenge some theorems on the contrary.
We were able, nevertheless to discover some imaginary contributions to the effective action for ghosts, although the quantum manifestation of the classical runaway solutions still eludes us in the general case. This is still work in progress.
Acknowledgements
We acknowledge partial financial support by the Spanish MINECO through the Centro de excelencia Severo Ochoa Program under Grant CEX2020-001007-S funded by MCIN/AEI/10.13039/501100011033 All authors acknowledge the European Union's Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 860881-HIDDeN and also byGrant PID2019-108892RB-I00 funded by MCIN/AEI/ 10.13039/501100011033 and by "ERDF A way of making Europe" A Expansion of the general action to quadratic order in quantum fluctuations.
In this section we present the expansion of the different monomials up to quadratic order in the perturbations
g µν =ḡ µν + h µν(104)
Let us examine the different monomials in order. First, we begin with the R squared action
S = d 4 x √ −gR 2(105)
the action up to quadratic order in the perturbations yields
S R 2 = d n x √ḡ h µν ∇ µ∇ν∇ρ∇σ −ḡ µν2∇ρ∇σ −ḡ ρσ∇µ∇ν2 +ḡ µνḡρσ2 2 + +R 2 ḡ µν∇ρ∇σ +ḡ ρσ∇µ∇ν + δ µρ,νσ2 −ḡ µνḡρσ2 − − 1 4 ḡ νσ∇µ∇ρ +ḡ µσ∇ν∇ρ +ḡ νρ∇µ∇σ +ḡ µρ∇ν∇σ − − 1 4 ḡ νσ∇ρ∇µ +ḡ µσ∇ρ∇ν +ḡ νρ∇σ∇µ +ḡ µρ∇σ∇ν + +R µν ḡ ρσ2 −∇ ρ∇σ +R ρσ ḡ µν2 −∇ µ∇ν −R 2 4 δ µρ,νσ − 1 2ḡ µνḡρσ + +R 2 3 4 ḡ νσRµρ +ḡ µσRνρ +ḡ νρRµσ +ḡ µρRνσ −ḡ µνRρσ −ḡ ρσRµν − − 1 2 R µσρν +R νσρµ +R µνRρσ h ρσ(106)
Let us snow turn to the Ricci squared monomial
S = d 4 x √ −gR 2 µν(107)
the action up to quadratic order in the perturbations is
S R 2 µν = d n x √ḡ h µν 1 2∇ µ∇ν∇ρ∇σ − 1 4ḡ µν2∇ρ∇σ − 1 4ḡ ρσ∇µ∇ν2 − − 1 16 ḡ µρ∇ν∇σ2 +ḡ νρ∇µ∇σ2 +ḡ µσ∇ν∇ρ2 +ḡ νσ∇µ∇ρ2 − −
tachyonic case the proper time integral clearly diverges. We can define it by analytic continuation from the real Klein-Gordon case ζ tachyon (s
νσ∇µ∇ρ +R µσ∇ν∇ρ +R νρ∇µ∇σ +R µρ∇ν∇σ + + 1 16 R νσ∇ρ∇µ +R µσ∇ρ∇ν +R νρ∇σ∇µ +R µρ∇σ∇ν + And finally we take the Riemann squared action the action up to quadratic order in the perturbations reads µσRντ ρλ +ḡ νσRµτ ρλ +ḡ µρRντ σλ +ḡ νρRµτ σλ + + 5 ḡ µσRνλρτ +ḡ νσRµλρτ +ḡ µρRνλστ +ḡ νρRµλστ − − ḡ µσRνρλτ +ḡ νσRµρλτ +ḡ µρRνσλτ +ḡ νρRµσλτ ∇ λ∇τ +∇ τ∇λ − µνRρλστ +ḡ µνRσλρτ +ḡ ρσRνλµτ +ḡ ρσRµλντ ∇ λ∇τ +∇ τ∇λ + νσ∇µ∇ρ +R µσ∇ν∇ρ +R νρ∇µ∇σ +R µρ∇ν∇σ + νσ∇ρ∇µ +R µσ∇ρ∇ν +R νρ∇σ∇µ +R µρ∇σ∇ν −1
16
ḡ µρ∇σ∇ν2 +ḡ νρ∇σ∇µ2 +ḡ µσ∇ρ∇ν2 +ḡ νσ∇ρ∇µ2 −
+
1
4ḡ
µνḡρσ2
2 +
1
4
δ µρ,νσ2
2 +
+
1
16
ḡ µσRνρ2 +ḡ νσRµρ2 +ḡ µρRνσ2 +ḡ νρRµσ2 +
+
1
16
R +
3
8
R µρνσ2 +R νρµσ2 +
1
2
δ µρ,νσRλτ∇
λ∇τ +
+
1
8
ḡ ρσRµλ∇ν∇
λ +ḡ ρσRνλ∇µ∇
λ +ḡ µνRρλ∇σ∇
λ +ḡ µνRσλ∇ρ∇
λ +
+
1
8
ḡ ρσRµλ∇
λ∇
ν +ḡ ρσRνλ∇
λ∇
µ +ḡ µνRρλ∇
λ∇
σ +ḡ µνRσλ∇
λ∇
ρ −
−
3
32
ḡ νσRµλ∇ρ∇
λ +ḡ µσRνλ∇ρ∇
λ +ḡ νρRµλ∇σ∇
λ +ḡ µρRνλ∇σ∇
λ +
+ḡ σνRρλ∇µ∇
λ +ḡ σµRρλ∇ν∇
λ +ḡ ρνRσλ∇µ∇
λ +ḡ ρµRσλ∇ν∇
λ −
−
3
32
ḡ νσRµλ∇
λ∇
ρ +ḡ µσRνλ∇
λ∇
ρ +ḡ νρRµλ∇
λ∇
σ +ḡ µρRνλ∇
λ∇
σ +
+ḡ σνRρλ∇
λ∇
µ +ḡ σµRρλ∇
λ∇
ν +ḡ ρνRσλ∇
λ∇
µ +ḡ ρµRσλ∇
λ∇
ν −
−
1
8
R µλνσ∇ρ∇
λ +R νλµσ∇ρ∇
λ +R µλνρ∇σ∇
λ +R νλµρ∇σ∇
λ −
−
1
8
R µλνσ∇
λ∇
ρ +R νλµσ∇
λ∇
ρ +R µλνρ∇
λ∇
σ +R νλµρ∇
λ∇
σ −
−
1
4ḡ
µνḡρσRλτ∇
λ∇τ −
1
4
δ µρ,νσR
2
αβ +
1
8ḡ
µνḡρσR
2
αβ −
−
1
2
ḡ µνR
λ
ρR σλ +ḡ ρσR
λ
µR νλ +
1
8
R µρRνσ +R νρRµσ +
+
5
16
ḡ νσR
λ
µR ρλ +ḡ µσR
λ
νR ρλ +ḡ νρR
λ
µR σλ +ḡ µρR
λ
νR σλ −
−
1
4
R λ
µR νρσλ +R λ
νR µρσλ +R λ
µR νσρλ +R λ
νR µσρλ −
−
1
8
R µλντR
τ λ
σρ +R νλµτR
τ λ
σρ +R µλντR
τ λ
ρσ +R νλµτR
τ λ
ρσ
h ρσ
(108)
S = d 4 x
√
−gR 2
µναβ
(109)
S R 2
µνρσ = d n x
√ḡ
h µν ∇ µ∇ν∇ρ∇σ + δ µρ,νσ2
2 −
−
1
4
ḡ µρ∇ν∇σ2 +ḡ νρ∇µ∇σ2 +ḡ µσ∇ν∇ρ2 +ḡ νσ∇µ∇ρ2 −
−
1
4
ḡ µρ∇σ∇ν2 +ḡ νρ∇σ∇µ2 +ḡ µσ∇ρ∇ν2 +ḡ νσ∇ρ∇µ2 +
+ R µρνσ2 +R νρµσ2 +
+
1
8
− 3 ḡ −
1
4
ḡ + δ µρ,νσRλτ∇
λ∇τ −
−
1
8
ḡ νσRµλ∇
λ∇
ρ +ḡ µσRνλ∇
λ∇
ρ +ḡ νρRµλ∇
λ∇
σ +ḡ µρRνλ∇
λ∇
σ +
+ḡ σνRρλ∇
λ∇
µ +ḡ σµRρλ∇
λ∇
ν +ḡ ρνRσλ∇
λ∇
µ +ḡ ρµRσλ∇
λ∇
ν −
−
1
8
ḡ νσRµλ∇
λ∇
ρ +ḡ µσRνλ∇
λ∇
ρ +ḡ νρRµλ∇
λ∇
σ +ḡ µρRνλ∇
λ∇
σ +
+ḡ σνRρλ∇
λ∇
µ +ḡ σµRρλ∇
λ∇
ν +ḡ ρνRσλ∇
λ∇
µ +ḡ ρµRσλ∇
λ∇
ν +
+
1
2
R +
1
2
R −
1
4
ḡ µσRνρ2 +ḡ νσRµρ2 +ḡ µρRνσ2 +ḡ νρRµσ2 +
+
1
8ḡ
µνḡρσ −
1
4
δ µρ,νσ R 2
αβλτ +
+
3
4
ḡ µσRναβλR
αβλ
ρ
+ḡ νσRµαβλR
αβλ
ρ
+ḡ µρRναβλR
αβλ
σ
+ḡ νρRµαβλR
αβλ
σ
−
−
1
2
ḡ µνRραβλR
αβλ
σ
+ḡ ρσRµαβλR
αβλ
ν
+
+
1
4
R µλR
λ
σνρ +R νλR
λ
σµρ +R µλR
λ
ρνσ +R νλR
λ
ρµσ +
+
1
4
R µρλτ +R µλρτ +R ρµλτ +R ρλµτ R λ τ
σν +
+
1
4
R νσλτ +R νλστ +R σνλτ +R σλντ R λ τ
ρµ −
−
1
4
ḡ νρRµλR
λ
σ +ḡ µρRνλR
λ
σ +ḡ νσRµλR
λ
ρ +ḡ µσRνλR
λ
ρ
h ρσ
(110)
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R 2 /Higgs inflation and the hierarchy problem. P G Ferreira, C T Hill, J Noller, G G Ross, arXiv:2108.06095hep-phP. G. Ferreira, C. T. Hill, J. Noller and G. G. Ross, "R 2 /Higgs inflation and the hierarchy problem," [arXiv:2108.06095 [hep-ph]].
Agravity. A Salvio, A Strumia, 10.1007/JHEP06(2014)080arXiv:1403.4226[hep-ph]].Easof02JHEP. 0680A. Salvio and A. Strumia, "Agravity," JHEP 06 (2014), 080 doi:10.1007/JHEP06(2014)080 [arXiv:1403.4226 [hep-ph]]. E as of 02 Feb 2022
The Lee-Wick standard model. B Grinstein, D O'connell, M B Wise, 10.1103/PhysRevD.77.025012arXiv:0704.1845Phys. Rev. D. 7725012hep-phB. Grinstein, D. O'Connell and M. B. Wise, "The Lee- Wick standard model," Phys. Rev. D 77, 025012 (2008) doi:10.1103/PhysRevD.77.025012 [arXiv:0704.1845 [hep-ph]].
Zeta function regularization of path integrals in curved spacetime. S W Hawking, Commun.Math. Phys. 55Hawking, S.W. "Zeta function regularization of path integrals in curved spacetime." Commun.Math. Phys. 55, 133-148 (1977).
. 10.1007/BF01626516https://doi.org/10.1007/BF01626516
Living with ghosts. S W Hawking, T Hertog, 10.1103/PhysRevD.65.103515arXiv:hep-th/0107088Phys. Rev. D. 65103515hep-thS. W. Hawking and T. Hertog, "Living with ghosts," Phys. Rev. D 65 (2002), 103515 doi:10.1103/PhysRevD.65.103515 [arXiv:hep-th/0107088 [hep-th]].
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Renormalization of Higher Derivative Quantum Gravity. K S Stelle, 10.1103/PhysRevD.16.953doi:10.1007/BF00760427Gen. Rel. Grav. 16Phys. Rev. DK. S. Stelle, "Renormalization of Higher Derivative Quantum Gravity," Phys. Rev. D 16 (1977), 953-969 doi:10.1103/PhysRevD.16.953 "Classical Gravity with Higher Derivatives," Gen. Rel. Grav. 9 (1978), 353-371 doi:10.1007/BF00760427
Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity. J W York, Jr , 10.1063/1.1666338J. Math. Phys. 14J. W. York, Jr., "Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial value problem of general relativity," J. Math. Phys. 14 (1973), 456-464 doi:10.1063/1.1666338
| {'fraction_non_alphanumeric': 0.09299079397786567, 'fraction_numerical': 0.08501445711057197, 'mean_word_length': 3.0922072623419012, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, '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': 'The divergences coming from a particular sector of gravitational fluctuations around a generic background in general theories of quadratic gravity are analyzed. They can be summarized in a particular type of scalar model, whose properties are analyzed.', 'arxivid': '2202.04536', 'author': ['Enriqueálvarez [email protected] \nDepartamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain\n', 'Jesús Anero [email protected] \nDepartamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain\n', 'Enriqueálvarez [email protected] \nDepartamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain\n', 'Jesús Anero [email protected] \nDepartamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain\n'], 'authoraffiliation': ['Departamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain', 'Departamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain', 'Departamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain', 'Departamento de Física Teórica and Instituto de Física Teórica\nIFT-UAM/CSIC\nUniversidad Autónoma de Madrid\n28049Cantoblanco, MadridSpain'], 'corpusid': 246680178, 'doi': '10.1088/1361-6382/ac7cb6', 'github_urls': [], 'n_tokens_mistral': 15912, 'n_tokens_neox': 12431, 'n_words': 5857, 'pdfsha': 'eed2669968c978a94281112fc1af176096402e76', 'pdfurls': ['https://export.arxiv.org/pdf/2202.04536v2.pdf'], 'title': ['Divergences of the scalar sector of quadratic gravity', 'Divergences of the scalar sector of quadratic gravity', 'Divergences of the scalar sector of quadratic gravity', 'Divergences of the scalar sector of quadratic gravity'], 'venue': []} |
arxiv |
Simultaneous expansion and rotation of shear-free universes in modified gravity
17 Feb 2012
Amare Abebe
Rituparno Goswami
Peter K S Dunsby
South African Astronomical Observatory
7925ObservatorySouth Africa
Astrophysics, Cosmology and Gravity Centre (ACGC)
University of Cape Town
7701RondeboschSouth Africa
Simultaneous expansion and rotation of shear-free universes in modified gravity
17 Feb 2012cosmologyshear-freeexpansionrotationperturbations PACS: 0425Nx
We show in a fully covariant way that, there exist a class of f (R) models for which a shear-free, almost FLRW universe can expand and rotate at the same time .
INTRODUCTION
The action of a generalized fourth-order gravity is given by (for more details, see [1] and references therein):
A = 1 2 d 4 x √ −g [ f (R) + 2L m ] ,(1)
where L m represents the matter contribution, and the generalized field equations read
G ab =T m ab + T R ab ≡ T ab ,(2)whereT m ab = T m ab f ′ , T R ab = 1 f ′ 1 2 ( f − R f ′ )g ab + ∇ b ∇ a f ′ − g ab ∇ c ∇ c f ′ .(3)
Here we have defined f ′ ≡ d f (R)/dR and T m ab = µ m u a u b + p m h ab + q m a u b + q m b u a + π m ab , where µ m , p m , q m and π m ab denote the standard matter density, pressure, heat flux and anisotropic stress respectively.
The total thermodynamics of the matter-curvature composite is then given by
µ ≡ µ m f ′ + µ R , p ≡ p m f ′ + p R , q a ≡ q m a f ′ + q R a , π ab ≡ π m ab f ′ + π R ab ,(4)
where µ R , etc. are thermodynamical quantities of the curvature fluid defined in the next section. The covariant derivative of a timelike vector u a can be decomposed into basic parts as
∇ a u b = −A a u b + 1 3 h ab Θ + σ ab + ε abc ω c ,(5)
where A a =u a is the acceleration, Θ =∇ a u a is the expansion, σ ab =∇ a u b is the shear tensor and ω a = ε abc∇ b u c is the vorticity vector. For the Weyl curvature tensor one has
E ab = C abcd u c u d = E ab , H ab = 1 2 ε acd C cd be u e = H ab ,(6)
giving a covariant description of tidal forces and gravitational radiation respectively.
LINEARIZED FIELD EQUATIONS
We consider the background to be Friedmann-Lemȃitre-Robertson-Walker (FLRW), where the Hubble scale sets the characteristic scale of the perturbations. In the perturbed spacetime the standard matter is considered to be a perfect fluid with the energy momentum tensor given by:
T m ab = (µ m + p m )u a u b + p m g ab .(7)
with p m = wµ m and the heat flux (q m a ) and the anisotropic stress (π m ab ) vanishing in the perturbed spacetime. In addition, since we consider shear-free perturbations, the shear tensor σ ab vanishes identically.
For the curvature fluid the linearized thermodynamic quantities are given by
µ R = 1 f ′ 1 2 (R f ′ − f ) − Θ f ′′Ṙ + f ′′∇2 R , p R = 1 f ′ 1 2 ( f − R f ′ ) + f ′′R + f ′′′Ṙ2 + 2 3 Θ f ′′Ṙ − f ′′∇2 R , q R a = − 1 f ′ f ′′′Ṙ∇ a R + f ′′∇ aṘ − 1 3 f ′′ Θ∇ a R , π R ab = 1 f ′ f ′′∇ a∇b R .(8)
With the conditions above, the propagation and constraint equations can be given bẏ
Θ −∇ a A a = − 1 3 Θ 2 − 1 2 (µ + 3p) ,(9)(ω a ) . − 1 2 ε abc∇ b A c = − 2 3 Θω a ,(10)E ab − ε cd a∇ c H b d = −ΘE ab − 1 2π ab R − 1 2∇ a q b R − 1 6 Θπ ab R ,(11)H ab + ε cd a∇ c E b d = −ΘH ab + 1 2 ε cd a∇ c π b d R ,(12)µ m = −(µ m + p m )Θ, (13) µ +∇ a q R a = −(µ + p)Θ;(14)(C 0 ) ab := E ab −∇ a A b − 1 2 π ab R = 0 ,(15)(C 1 ) a :=∇ a Θ − 3 2 ε abc∇ b ω c − 3 2 q a R = 0 ,(16)(C 2 ) :=∇ a ω a = 0 ,(17)(C 3 ) ab := H ab +∇ a ω b = 0 ,(18)(C 4 ) a :=∇ a p m + (µ m + p m )A a = 0 ,(19)(C 5 ) a :=∇ b E ab + 1 2∇ b π ab R − 1 3∇ a µ + 1 3 Θq a R = 0 ,(20)(C 6 ) a :=∇ b H ab + (µ + p)ω a + 1 2 ε abc∇ b q R c = 0 .(21)
The conditions σ ab = 0 and q a m = 0 give the two new constraints (C 0 ) ab and (C 4 ) a respectively. Substituting (C 0 ) bd into (C 5 ) b and using (C 4 ) b we obtain the constraint
w w + 1∇ d∇ b∇d φ + 1 3∇ b µ −∇ d π R bd − 1 3 Θq R b = 0,(22)
where φ ≡ ln µ m . To check the spatial consistency of the above constraint on any initial hypersurface we take the curl of (22) to obtain
ω a wΘ 3 +Ṙ f ′′ 3 f ′ R + 2(1 + w)µ m Θ 3 f ′ + Ṙ f ′′ f ′ + wΘ ∇ 2 ω a = 0 ,(23)
whereR = 2 µ − 1 3 Θ 2 . Now defining the expansion, acceleration, jerk and snap parameters by the following relations
Θ = 3ȧ a , q = −ä ȧ a 2 , j = a 2 a 3 d 3 a dt 3 , s = a 3 a 4 d 4 a dt 4 ,(24) ACKNOWLEDGMENTSThe authors thank the organisers of the Spanish Relativity Meeting (ERE2011) and the National Research Foundation (South Africa) for financial support.and usinġwherewe can rewrite (23) asSpatial consistency requires the vanishing of either Θ or the terms in the curly brackets. For temporal consistency differentiate (27) w.r.t. time to getIt follows that for the new constraints to be spatially and temporally consistent we must have either Θω a = 0 or the expression in the curly brackets must vanish. It is easy to see from (28) that if the 3-curvature vanishes, then Θω a = 0 for vacuum universes (µ m = 0). This implies that a shear-free, spatially flat vacuum universe in any f (R) theory can rotate and expand simultaneously in the linearized regime.In the non -vacuum case, there exists at least one non-trivial case which does violate the Ellis condition. For a flat Milne universe, the Friedmann equation is given byand has the following general solution:Considering the particular solution (C 1 = 0 = C 2 ), and comparing it with Eqn.(28), for the corresponding flat Milne universe in R n gravity, we obtainComparing solutions (32) and the particular solution of (31) (with n = 3(1 + w)/2) we find that w = 1 if µ m = 0. In other words, for a stiff fluid in R 3 gravity, there exists a flat Milne-universe solution which can rotate and expand simultaneously at the level of linearized perturbation theory.DISCUSSION AND CONCLUSIONIn this work we showed that if the 3-curvature vanishes, then the result of[2]can always be avoided for vacuum universes. We also demonstrated there is at least one physically realistic non-vacuum case in which both rotation and expansion are simultaneously possible. This suggests that there are situations where linearized fourth-order gravity shares properties with Newtonian theory not valid in General Relativity.
. A Abebe, R Goswami, P K S Dunsby, Phys. Rev. D. 84124027A. Abebe, R. Goswami, P. K. S. Dunsby, Phys. Rev. D 84 124027 (2011).
. G F R Ellis, J. Math. Phys. 8G. F.R. Ellis, J. Math. Phys. 8, 1171-1194 (1967);
. J M Senovilla, C F Sopuerta, P Szekeres, Gen. Rel. Grav. 30J. M. M Senovilla, C. F. Sopuerta and P. Szekeres, Gen. Rel. Grav. 30, 389-411, (1998);
. Phys Rev D. 401804Phys Rev D 40 1804 (1989);
. A M Nzioki, R Goswami, P K S Dunsby, G F R Ellis, Phys.Rev.D. 84124028A. M. Nzioki, R. Goswami, P. K. S. Dunsby and G. F. R. Ellis, Phys.Rev.D.84.124028 (2011).
| {'fraction_non_alphanumeric': 0.08676886441434738, 'fraction_numerical': 0.042714689685965174, 'mean_word_length': 3.050632911392405, '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': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We show in a fully covariant way that, there exist a class of f (R) models for which a shear-free, almost FLRW universe can expand and rotate at the same time .', 'arxivid': '1202.3864', 'author': ['Amare Abebe ', 'Rituparno Goswami ', 'Peter K S Dunsby \nSouth African Astronomical Observatory\n7925ObservatorySouth Africa\n', '\nAstrophysics, Cosmology and Gravity Centre (ACGC)\nUniversity of Cape Town\n7701RondeboschSouth Africa\n'], 'authoraffiliation': ['South African Astronomical Observatory\n7925ObservatorySouth Africa', 'Astrophysics, Cosmology and Gravity Centre (ACGC)\nUniversity of Cape Town\n7701RondeboschSouth Africa'], 'corpusid': 118425112, 'doi': '10.1063/1.4734421', 'github_urls': [], 'n_tokens_mistral': 2732, 'n_tokens_neox': 2317, 'n_words': 1397, 'pdfsha': 'c03ab902bfe20dc49b0c0526c9e5690783247af6', 'pdfurls': ['https://arxiv.org/pdf/1202.3864v1.pdf'], 'title': ['Simultaneous expansion and rotation of shear-free universes in modified gravity', 'Simultaneous expansion and rotation of shear-free universes in modified gravity'], 'venue': []} |
arxiv |
29 Jan 1999
David Auckly
CONTROL OF NONLINEAR UNDERACTUATED SYSTEMS
Department of Mathematics
Department of Mathematics
Kansas State University
66506-2602ManhattanKSUSA
Lev Kapitanski
Department of Mechanical and Nuclear Engineering
Kansas State University
66506-2602ManhattanKSUSA
Warren White
Kansas State University
66506-5106ManhattanKSUSA
29 Jan 19998/26/98
In this paper we introduce a new method to design control laws for non-linear underactuated systems. Our method will often work in situations where standard control techniques fail. Even when standard techniques apply, we believe that our approach will achieve better performance. This is because we produce an infinite dimensional family of control laws, whereas most control techniques only produce a finite dimensional family of control laws. We will describe the problem and our solution invariantly, using differential geometry and in local coordinates. We hope that this paper will be useful for both mathematicians and engineers. We include an abstract example of a system which is open-loop unstable and cannot be stabilized using any linear control law, and demonstrate that our method produces a stabilizing control law. We also apply our method to the inverted pendulum cart to compare it with some standard control techniques.An important problem in control theory is how to modify a system of ordinary differential equations so that solutions to the equations satisfy some property. In order to
describe the class of differential equations that we consider, let Q denote the configuration space. The configuration space is a finite dimensional manifold that represents, for example, every possible position of a mechanical system. Let g ∈ Γ(T * Q ⊗ T * Q) be a metric. This is the mass matrix in a mechanical system. Let c, f : T Q → T Q be fiber-preserving maps. These maps are not assumed to be linear, but we do assume that c is odd, i.e., c(−X) = −c(X). In a mechanical system c will represent the inherent dissipation and f will represent the input forces. Finally, let V : Q → R. This will represent the potential energy of a mechanical system. The differential equation that we consider in this paper is ∇γγ + c(γ) + grad γ V = f (γ).
(1)
Let P ∈ Γ(T * Q ⊗ T Q) be a g-orthogonal projection. We consider the situation where a constaint P (f ) = 0 is imposed. A system is called underactuated if P = 0. A mechanical system is underactuated if we require that control forces are zero in certain directions. Many problems in chemical, electrical and mechanical engineering may be formulated as follows: find a function f with P (f ) = 0 so that solutions to Equation (1) have some specified property. For example, solutions with initial conditions in some region will remain close to some path, or as a different example, some point in T Q will be an asymptotically stable equilibrium.
In local coordinates x = (x 1 , . . . , x n ) Equation (1) reads
x k + Γ k ij (x)ẋ iẋj + c k (x,ẋ) + g ik ∂V ∂x i = f k (x,ẋ),(2)
where Γ k ij (x) are the Christoffel symbols (of the Levi-Civita connection) associated to the metric g, [Hicks, 1965], and g ik is the inverse matrix to g kj . In Equation (2) and throughout the rest of this paper we are using the summation convention, that repeated indices are summed from 1 to n. As we have mentioned, we assume that c k (x, −ẋ) = −c k (x,ẋ) and the projection of f k to some specified set of directions must be zero.
The main question addressed in this paper is how to find a function, f , with P (f ) = 0, so that solutions to Equation (1) satisfy some preassigned conditions. Our approach to this question is to find functions g, c, V and f so that solutions to Equation (1) are automatically solutions to ∇γγ + c(γ) + grad γ V = 0. This will clearly be the case if
f (X) ≡ ∇ X X − ∇ X X + grad γ V − grad γ V + c(X) − c(X),(3)
for every vector field X. The condition P (f ) = 0 then becomes a system of nonlinear partial differential equations for g, c, and V . Notice that constant multiples of g, c, and V satisfy P (f ) = 0 even when P has full rank. Thus, one would expect many solutions when P does not have full rank. Separating P (f ) = 0 into terms which are quadratic in the velocity, independent of the velocity or odd functions of the velocity gives
P (∇ X X − ∇ X X) = 0, (4.1) P (grad γ V − grad γ V ) = 0, (4.2) P (c(X) − c(X)) = 0. (4.3)
We will look for solutions to these matching equations with g non-degenerate so that
g(X, Y ) = g(λX, Y ) with λ ∈ Γ(T * Q ⊗ T Q)
. It is clear that λ has to be g self-adjoint, i.e., g(λX, Y ) = g(X, λY ). We will derive a linear system of partial differential equations for λ which must be satisfied if g is to solve Equation (4.1).
To derive the equations for λ we will use the relation between the connection, Lie bracket (commutator) and metric. In fact, we only derive equations for λ Im P .
It is known, [Hicks], that the covariant derivative ∇ compatible with the metric g is determined uniquely as a bilinear operator which associates to any pair of vectors X and Y a third vector ∇ X Y so that the following equations are satisfied:
X(g(Y, Z)) = g(∇ X Y, Z) + g(Y, ∇ X Z),(5)∇ X Y − ∇ Y X = [X, Y ],(6)
where [X, Y ] is the Lie bracket (commutator) of X and Y . Using the above properties of the covariant derivative we get:
2g(∇ X Y, Z) =Xg(Y, Z) + Y g(Z, X) − Zg(X, Y ) + g([X, Y ], Z) + g([Z, X], Y ) − g([Y, Z], X).(7)
This may be solved for ∇ X Y since g is non-degenerate.
Proposition 1. If g = gλ and g satisfies the matching condition P (∇ X X − ∇ X X) = 0, then λ satisfies:
∇gλ Im P ⊗2 = 0,(8)
or, equivalently,
g(∇ Z λP X, P X) − g(λP X, ∇ Z P X) = 0.(9)
Proof. We begin by polarizing the matching equation
P (∇ X Y − ∇ X Y ) = 1 2 P (∇ X Y + ∇ Y X − ∇ X Y − ∇ Y X) = 1 2 P (∇ X+Y (X + Y ) − ∇ X+Y (X + Y )) −P (∇ X X − ∇ X X) − P (∇ Y Y − ∇ Y Y ) = 0.
The first line is true because the covariant derivatives are torsion free (Equation (6)). The second line is true because the covariant derivatives are bilinear (Equation (7)). Now,
0 = 2g(P (∇ λP X Z − ∇ λP X Z), X) = 2g(∇ λP X Z − ∇ λP X Z, P X)
= 2g(∇ λP X Z, P X) − 2 g( ∇ λP X Z, λP X)
= λP Xg(Z, P X) + Zg(P X, λP X) − P Xg(λP X, Z) + g([λP X, Z], P X) + g([P X, λP X], Z) − g([Z, P X], λP X)
− λP X g(Z, λP X) − Z g(λP X, λP X) + λP X g(λP X, Z)
− g([λP X, Z], λP X) − g([λP X, λP X], Z) + g([Z, λP X], λP X)
= λP Xg(Z, P X) − P Xg(λP X, Z) + g([P X, λP X], Z)
− g([Z, P X], λP X) + g([Z, λP X], P X)
= g(∇ λP X Z, P X) + g(Z, ∇ λP X P X) − g(λP X, ∇ P X Z) − g(Z, ∇ P X λP X) + g([Z, λP X], P X) − g([P X, λP X], Z) − g([Z, P X], λP X) = g(∇ Z (λP X), P X) − g(∇ Z P X, λP X) = g((∇ Z λ)(P X), (P X)) = (∇ Z gλ)(P X, P X). 4
The covariant derivative is extended to all tensors by requiring every reasonable product rule to hold. For example, Z(λ(X)) = ∇ Z (λ(X)) = (∇ Z λ)(X) + λ(∇ Z X) and X(g(Y, Z)) = ∇ X (g(Y, Z)) = (∇ X g)(Y, Z) + g(∇ X Y, Z) + g(Y, ∇ X Z). We used these two product rules in the last two lines and Equations (5), (6) and (7) in the previous lines.
Now that we have equations for λ Im P , we will derive equations for g in terms of λ Im P .
Proposition 2. If g = gλ, then
λP X g(Z, Z) + 2 g([Z, λP X], Z) = 2Zg(P X, Z) − 2g(P X, ∇ Z Z)(10)
Proof.
λP X g(Z, Z) + 2 g([Z, λP X], Z)
= 2 g( ∇ λP X Z, Z) + 2 g([Z, λP X], Z) = 2 g( ∇ Z λP X, Z) = 2 Zg(P X, Z) − 2g(P X, ∇ Z Z).
The matching equation for the potential energy may be expressed as a linear partial differential equation using λ.
Proposition 3. If the matching Equations (4) are satisfied and g = gλ, then λP X( V ) = P X(V ).
Proof. λP X( V ) = (d V )(λP X) = g( grad γ V , λP X) = g( grad γ V , P X)
= g(P grad γ V , X) = g(P grad γ V, X) = P X(V ).
By solving the matching equations, we find every control law that will result in dynamical behavior equivalent to a system of the form:
∇γγ + c(γ) + grad γ V = 0.
This is a very large collection of vector fields on T Q. Thus far, we have shown that every solution to the matching equations may be found by first solving one system of 5 linear partial differential equations and then solving a different system of linear partial differential equations.
The previous three propositions suggest a natural approach for solving the matching equations. First, solve Equation (9) for λP X, then solve Equation (10) for g, and Equation (11) for V . Finally solve the algebraic Equation (4.3) for c. Every solution to the matching equations can be found in this way. What is not clear is whether every set of functions generated in this way is a solution to the matching equations. The problem is that once we have a solution to the λ-equation, (Equation (9)), and then a solution to Equation (10), it is not clear that λ can be extended from Im P so that the condition g = gλ holds. We will address this question for general P in a future paper. We will next show that g = gλ holds if it is true at the initial conditions, provided that the rank of P is one.
Proposition 4. Let X be a non-zero vector field in an open set U , generating Im P , and let λP X be a non-vanishing solution of the λ Equation (9) in U . Let Σ be a codimension one non-characteristic hypersurface in U . Assume that g is defined, non-degenerate, and satisfies gλP X = gP X on Σ, and V is defined on Σ. Then there is a unique solution to the matching Equations (4.1) and (4.2), and a unique extension of λ away from Im P in a neighborhood of Σ. Furthermore, gλ = g everywhere in the neighborhood.
Proof. Notice that Σ is non-characteristic for the g-Equation (10) if it is non-characteristic for the V -Equation (11), and visa versa. The functions g and V may be found in a neighborhood of Σ using the method of characteristics. The main content of this proposition is that gP X = gλP X everywhere in the neighborhood. After polarization, the g-Equation (10) reads:
λP X g(λP X, Z) + g(λP X, [Z, λP X]) = Zg(λP X, P X) + λP Xg(P X, Z) − g(∇ Z λP X, P X) − g(∇ λP X Z, P X) = λP X g(P X, Z) + g(P X, [Z, λP X]) + g(λP X, ∇ Z P X) − g(∇ Z λP X, P X).
This last line follows after applying the product rule to the first term of the previous line and the torsion-free condition (6) to the last term of the previous line. In view of the λ-Equation (9), it is clear that gλP X = gP X is a solution of this equation. This solution is unique in a neighborhood of Σ. Finally, since g is non-degenerate on Σ, it is non-degenerate in a neighborhood of Σ, so we may extend λ to the full tangent space by λ = ( g) −1 g.
6
EXAMPLE
In order to demonstrate and clarify our method, we will apply it to an inverted pendulum cart as an example.
y Figure 1 7 The (mass) metric for the cart system depicted in Figure 1 is: g = (M + m)dy 2 + 2mℓ cos(θ)dθdy + (mℓ 2 + I)dθ 2 , and the potential energy is:
V = mgℓ cos(θ).
Here M is the mass of the base of the cart, m is the mass of the pendulum, ℓ is the length from hinge to the center of mass of the pendulum,
I
is the moment of inertial about the center of mass, and g is the acceleration due to gravity.
By a change of length, time and mass scales, the metric and the potential energy may be transformed into
g = dx 2 + 2b cos(θ)dθdx + dθ 2 , V = cos(θ),(12)where b = mℓ(M + m) − 1 2 (mℓ 2 + I) − 1 2 ∈ (0, 1) is a dimensionless parameter.
The problem is to find the force applied to the base of the cart as a function of the state of the cart so that the origin (θ = 0, x = 0) will be an asymptotically stable equilibrium of the resulting dynamical system.
In this application, we may only directly apply force in the x direction, so P is projection onto the ∂ ∂θ direction, i.e.,
P = (b cos(θ) dx + dθ) ⊗ ∂ ∂θ .
Projecting system (1) onto the coordinate directions gives:
P (∇γγ + c(γ) + grad γ V ) = 0, g ∂ ∂x , ∇γγ + c(γ) + grad γ V = u ≡ g ∂ ∂x , f .(13)
We will assume that there is no inherent dissapation: c ≡ 0. Then, in local coordinates, the equations read:θ
+ b cos(θ)ẍ − sin(θ) = 0 b cos(θ)θ +ẍ − b sin(θ)θ 2 = u.(14)
8 Equation (3) expresses our solution to this problem in terms of functions, g ij , V and c i , i, j = 1 or 2, which satisfy the Equation (4). In local coordinates, these matching equations read:
( g 22 − b cos(θ) g 12 ) ∂ g 11 ∂θ + (b cos(θ) g 11 − g 12 ) 2 ∂ g 12 ∂θ − ∂ g 11 ∂x = 0, ( g 22 − b cos(θ) g 12 ) 2 ∂ g 12 ∂x − ∂ g 22 ∂θ + (b cos(θ) g 11 − g 12 ) ∂ g 22 ∂x = 0, ( g 22 − b cos(θ) g 12 ) ∂ g 11 ∂x + (b cos(θ) g 11 − g 12 ) ∂ g 22 ∂θ = 0 ( g 22 − b cos(θ) g 12 ) ∂ V ∂θ + (b cos(θ) g 11 − g 12 ) ∂ V ∂x = − sin(θ)( g 11 g 22 − ( g 12 ) 2 ),
and c 1 + b cos(θ) c 2 = 0.
We have slightly simplified these equations by multiplying by the determinant of g. (The terms on the right hand side of each equation would be non-zero if the cart was on a hill, and the hinge was rusty. Therefore, multiplying by the determinant of g would not be a great simplification.) These equations are messy, but the same functions may be found by solving the equations in Propositions 1, 2, and 3 without ever writing out the matching equations. Take X = ∂ ∂θ in Proposition 1 (so P X = ∂ ∂θ ) and let
λ ∂ ∂θ = σ ∂ ∂θ + µ ∂ ∂x .
Using the bilinearity of the covariant derivative and the product rule, the λ-Equation (9) becomes:
g ∂ ∂θ , ∂ ∂θ ∂σ ∂θ + g ∂ ∂x , ∂ ∂θ ∂µ ∂θ + g ∇ ∂ ∂θ ∂ ∂x , ∂ ∂θ µ − g ∂ ∂x , ∇ ∂ ∂θ ∂ ∂θ µ = 0 and g ∂ ∂θ , ∂ ∂θ ∂σ ∂x + g ∂ ∂x , ∂ ∂θ ∂µ ∂x + g ∇ ∂ ∂x ∂ ∂x , ∂ ∂θ µ − g ∂ ∂x , ∇ ∂ ∂x ∂ ∂θ µ = 0
Using Equation (7), it follows that:
∂σ ∂θ + b cos(θ) ∂µ ∂θ + b sin(θ)µ = 0, ∂σ ∂x + b cos(θ) ∂µ ∂x = 0.
By the Frobenius Theorem, or equivalently checking that mixed derivatives are equal, we see that this system has a solution if and only if b sin(θ) ∂µ ∂x = 0. The second equation then implies that σ only depends on θ, so the first equation reduces to an ordinary differential equation. The general solution to this system of equations is, therefore,
σ = σ(θ), µ = µ 0 cos(θ) − 1 b cos(θ) ∂σ ∂θ sec 2 θ dθ.
For the rest of this example, we will use the particular solution:
σ = σ 0 , µ = µ 0 cos(θ).(15)
It is easiest to find g using the basis ∂ ∂θ , λ ∂ ∂θ . By the relation gλ = g, we already know g λ( ∂ ∂θ ), ∂ ∂θ and g λ ∂ ∂θ λ ∂ ∂θ , so we only need to find g ∂ ∂θ , ∂ ∂θ . The Lie bracket,
∂ ∂θ , λ( ∂ ∂θ ) = −µ 0 sin(θ) ∂ ∂x = σ 0 tan(θ) ∂ ∂θ − tan(θ)λ ∂ ∂θ
may be plugged into the g-Equation (10) to get:
σ 0 ∂ g 11 ∂θ + µ 0 cos(θ) ∂ g 11 ∂x + 2σ 0 tan(θ) g 11 − 2 tan(θ) = 0.
The surface θ = 0 is non-characteristic, so the above equation may be solved with initial data g 11 (0, x) = h(x).
The flow of the vector field λP X is given by:
θ = σ 0 ,ẋ = µ 0 cos(θ), or θ = σ 0 t, x = a + µ 0 σ 0 sin(σ 0 t),
using θ = 0 at t = 0. With this flow, the g equation becomes:
d g 11 dt + 2σ 0 tan(σ 0 t) g 11 − 2 tan(σ 0 t) = 0, so g 11 = 1 σ 0 + h(a) − 1 σ 0 cos 2 (σ 0 t) = 1 σ 0 + h x − µ 0 σ 0 sin(θ) − 1 σ 0 cos 2 (θ).
Again, we only take a particular solution:
g 11 = 1 σ 0 + r cos 2 (θ), where r is a constant. The equations b cos(θ) = g ∂ ∂θ , ∂ ∂x = g λ( ∂ ∂θ ), ∂ ∂x = σ 0 g 12 + µ 0 cos(θ) g 22 and 1 = g ∂ ∂θ , ∂ ∂θ = g λ( ∂ ∂θ ), ∂ ∂θ = σ 0 g 11 + µ 0 cos(θ) g 12
may now be solved for the remaining terms of g :
g 12 = − σ 0 µ 0 r cos(θ), g 22 = b µ 0 + σ 2 0 µ 2 0 r.
We can compute V in the same way. Using the λXP flow and initial data V (0, x) = w(x), the V -Equation (11) may be written as an ordinary differential equation:
d V dt = − sin(σ 0 t).
Thus,
V = 1 σ 0 (cos(σ 0 t) − 1) + w(a) = 1 σ 0 (cos(θ) − 1) + w x − µ 0 σ 0 sin θ .
Finally, solving for c, we get:
c(X) = K(θ, x,θ,ẋ) b cos(θ) ∂ ∂θ − ∂ ∂x
with an arbitrary function K.
The functions g, V and c produce a family of control laws via equation (3). In this example, the control laws depend on four functions, σ 0 (θ), h(x), w(x), and K(θ, x,θ,ẋ), and one constant, µ 0 . We have already chosen σ 0 and h to be constants. We will address the question of how to choose the unknown functions and parameters in order to best meet specific design criteria in a future paper. For now, we will just pick elementary functions that will insure that θ = 0, x = 0 is an asymptotically stable equilibrium. If (0, 0) is an equilibrium, grad V (0, 0) must be zero, so w ′ (0) must be zero. One standard way to insure that (0, 0) will be an asymptotically stable equilibrium is to pick positive definite g, V and g c, then H(X) ≡ 1 2 g(X, X) + V will be a Lyapunov function with time rate of change − g ( c(X), X). The Hessian of V at (0, 0) is
D 2 (0,0) = µ 2 0 σ 2 0 w ′′ (0) − 1 σ 0 − µ 0 σ 0 w ′′ (0) − µ 0 σ 0 w ′′ (0) w ′′ (0) .
So we should require:
w ′′ (0) > 0 and det D 2 (0,0) V = − 1 σ 0 w ′′ (0) > 0.
We will choose σ 0 < 0, and w(z) = 1 2 w 1 z 2 with w 1 > 0. The mass matrix should also be positive definite. Thus we should have g 11 > 0 and g 22 > 0 so that r > 0 and bµ 0 + σ 2 0 r > 0. In addition, the determinant of the mass matrix will also be positive:
b σ 0 µ 0 + br µ 0 cos 2 (θ) + σ 0 µ 2 0 r > 0.
Rearranging and using the fact that σ 0 < 0 gives:
−σ 0 bµ 0 r cos 2 (θ) > σ 2 0 r + bµ 0 .
So, µ 0 > σ 2 0 r + bµ 0 −σ 0 br cos 2 (θ) > 0, and cos 2 (θ) > σ 2 0 r + bµ 0 −σ 0 bµ 0 r .
Assuming µ 0 > 0, r > 0 and σ 0 < 0, the mass matrix will be positive definite provided condition (16) holds. When the mass matrix and potential energy of the model system are positive definite, the point (0, 0) will be Lyapunov stable for any positive semi-definite g c. In particular, when K ≡ 0 (i.e., c ≡ 0), we will get Lyapunov stability. We call controllers with c ≡ 0 conservative controllers. A direct computation shows that:
g( c(X), X) = K g b cos(θ) ∂ ∂θ − ∂ ∂x ,θ ∂ ∂θ +ẋ ∂ ∂x = K (b cos(θ) g 11 − g 12 )θ + (b cos(θ) g 12 − g 22 )ẋ = (det g) · K · (µ 0 cos θθ − σ 0ẋ ).
This will be positive semi-definite if K = Φ(θ, x)(µ 0 cos θθ − σ 0ẋ ), where Φ is any positive function. (We can do no better because λP X = σ 0 ∂ ∂θ + µ 0 cos θ ∂ ∂x will be a zero mode of any admissible c.) We will see that this is sufficient to prove that (0, 0) is an asymptotically stable equilibrium of the controlled system. Equation (3), equation (7) and our values for g, V and c combine together to give:
u = g f, ∂ ∂x = b + r det g µ 0 det g cos(θ) sin(θ) − sin(θ)θ 2 − w 1 det g σ 0 det g x − µ 0 σ 0 sin(θ) + det g Φ(θ, x) (µ 0 cos(θ)θ − σ 0ẋ ),(17)
where
det g = 1 − b 2 cos 2 (θ), det g = b σ 0 µ 0 + br µ 0 cos 2 (θ) + σ 0 r µ 2 0 .(18)
Thus, we have almost proved the following result.
Proposition 5. Let constants µ 0 , σ 0 , r, and w 1 satisfy the conditions
µ 0 > 0, σ 0 < 0, w 1 > 0, 1 > σ 2 0 r + bµ 0 −σ 0 bµ 0 r .(19)
Let Φ(θ, x) be any strictly positive function.
Then (0, 0) is an asymptotically stable equilibrium of the controlled system (14) with the control law defined by (17), (18).
Proof. Define the controlled Hamiltonian H = 1 2 g(γ,γ) + V . We have shown above that the Hessian of H is positive definite in some neighborhood of (0, 0) when conditions (19) hold. The time derivative of H is:
d dt ( H) = d dt 1 2 g(γ,γ) + V = g(∇γγ + grad V ,γ) = − g( c(γ),γ) = − det g Φ(θ, x) (µ 0 cos(θ)θ − σ 0ẋ ) 2 .
Check that there is no solution to the controlled equations satisfying µ 0 cos(θ)θ−σ 0ẋ ≡ 0.
Since Φ > 0 by assumption and det g (θ,x) > 0 for all (θ, x) sufficiently close to (0, 0), the Hamiltonian H decreases along the solutions of (17), and, therefore, may serve as a Lypunov function. The (local) asymptotic stability follows from Lyapunov's Theorem.
Remark 1. It is not possible to construct a control law so that (0, 0) is a globally asymptotically stable equilibrium.
Indeed, the solutions of the controlled system for such a control law would produce a continuous function, F : [0, ∞] × T Q → T Q so that F 0 is the identity map and F ∞ is the constant map. In other words, the flow of a vector field with a globally asymptotically stable equilibrium is a contraction. However, for the inverted pendulum cart, T Q ≃ S 1 × R 3 , which is not contractible.
Since the inverted pendulum cart cannot be globally stabilized by a control law, we can only try to maximize the size of the basin of attraction. We will compare a special case of our nonlinear control law with a linear control law that has been implemented on an inverted pendulum cart in our lab. Our cart has This is only an approximation of the actual control law that is running in the lab. In the lab we had to model the DC motor, the observer, and use a discrete control law.
Thus far, we have chosen each of the arbitrary functions in our non-linear control law to be a constant, and we took w(x) = 1 2 w 1 x 2 , since it has to have positive second derivative. As a rough guess we took µ 0 = 10, σ 0 = −.05, r = 1000, and w 1 = 1.5. We also set the function Φ = 1. These constants satisfy conditions (19). We chose the functions to be constants when possible just to simplify the exposition.
Remark 2. This would not be the best choice for engineering applications. In particular, −σ 0 controls the coefficient ofẋ in our control law. In order to stabilize the system after a large angular disturbance, we would like −σ 0 to be small so that the cart will be free to accelerate under the pendulum. On the other hand, −σ 0 needs to be much bigger in order to insure that the time constant is reasonable. We hope to describe some practical 14 approches to choosing the arbitrary functions which appear in our family of control laws so that the resulting controler will meet specific design criteria in a future paper.
For a first rough test of our control law, we ran numerical simulations of the system using both our control law and a linear control law with the arbitrary functions and parameters specified above. We used 10000 diferent sets of initial conditions. It appears that the linear controler has a shorter settling time than the nonlinear control law, when it does not blow up. However, the nonlinear control law appears to stabilize the system starting from any of the initial conditions stabilized by the linear control law, and also appears to stabilize the system starting from many initial conditions for which the linear control law produces an unbounded response. For the linear controller, x(t) increases exponentially. For the nonlinear controller, x(t) stabilizes to 0, but the system is underdamped, as we discussed previously. The graphs of the other state variables and control input are again qualitatively similar. These graphs are representative for initial conditions for which the linear control blows up, and the nonlinear control stabilizes. All of our numerical simulations were produced using MATLAB 5.1.
Our infinite dimensional family of control laws contains a one dimensional family of control laws, which was found previously. Bloch, Leonard and Marsden have developed a method for constructing control laws for mechanical systems with symmetry, [Bloch, Lenard, Marsden, 1997]. Their control law will retain the symmetry, so it cannot produce a truly Lyapunov stable equilibrium. It will only produce a Lyapunov stable equilibrium in shape space. This means that their method will not work for an inverted pendulum cart on a hill, say. If the cart is on level ground, the quantityẋ + A cos(θ)θ will be conserved. The quantityẋ + A cos(θ)θ is conserved in the closed-loop dynamics of any of the control laws developed by Bloch, Leonard and Marsden. It follows that any solution which passes through a point withθ = 0 andẋ = 0 will run off to infinity.
LINEARLY NON-CONTROLLABLE EXAMPLE
The final example that we will consider is an abstract system which is open-loop unstable with the property that the linearized system cannot be stabilized by any control law. Our method will generate a control law with a globally asymptotically stable equilibrium.
The abstract system is: g = dx 2 + dy 2 , P = dy ⊗ ∂ ∂y and V = − 3 2 x 4 + 45x 2 y 2 + 32xy 3 . In coordinates, the control problem reads:
x − 6x 3 + 90xy 2 + 32y 3 = 0 y + 90x 2 y + 96xy 2 = u.
This system is, clearly, open-loop (u = 0) unstable. For example, x = (ε −1 − √ 3t) −1 , y = 0 is a solution to the system for any positive ε. The linearized version of the system is:ẍ = 0,ÿ = u. Clearly, u can have no effect on x.
To apply our method to this system, let λP X = σ ∂ ∂x + µ ∂ ∂y , then the λ-Equation (9) reads: ∂σ ∂x = 0, ∂σ ∂y = 0.
Thus, σ is any constant and µ is any function. Pick σ = 1 and µ = 1. The g-Equation
(10) becomes: ∂ g 11 ∂x + ∂ g 11 ∂y = 0.
Pick g 11 = 2. The equations gλ = g lead to g 22 = 1 and g 12 = −1. It is easy to check that the model mass matrix is positive definite. The flow equation for V is ∂ V ∂x + ∂ V ∂y = −6x 3 + 90xy 2 + 32y 3 .
It is not hard to check that V = (x 2 − 3xy) 2 + (x 2 − 4xy − 2y 2 ) 2 is a positive definite solution to this equation. Finally, pick c = (ẏ −ẋ) ∂ ∂y as a solution to P ( c) = P (c) = 0. As before, we show that there is no non-trivial solution to the controlled equations withẏ −ẋ ≡ 0. Thus, (0, 0) is a globally asymptotically stable equilibrium.
M
= 5.02 Kg m = .454 Kg ℓ = .425 m I = .11 Kg m 2 giving b = .188. The linear control law that we obtained by placing poles at −5, −6, and a double pole at −2 is: u lin = 1021θ + 115.8x + 918.5θ + 158.2ẋ .
Figure 2
Figure 2
2shows graphs of the angular position versus time produced using the linear control law and the nonlinear control law starting from initial conditions: θ = .5,θ = −.5, x = 0, andẋ = 0. The graphs of the other state variables and control input are qualitatively similar. This output is typical of the responce obtained when both control laws appear to stabilize the system.
Figure 3
Figure 3
3shows graphs of the angular position versus time produced using the linear control law and the nonlinear control law starting from initial conditions: θ = 1.25, θ = 1.3, x = 0, andẋ = 0. For the same initial conditions,Figure 4shows graphs of the cart position versus time produced using the linear control law and the nonlinear control law.
Stabilization of Mechanical Systems Using Controlled Lagrangians. A Bloch, N Lenard, J Marsden, Proc. 1997 IEEE Conference on Decision and Control. 1997 IEEE Conference on Decision and ControlA. Bloch, N. Lenard, J. Marsden, Stabilization of Mechanical Systems Using Controlled Lagrangians, Proc. 1997 IEEE Conference on Decision and Control (1997), 2356- 2361.
N Hicks, Notes on Differential Geometry. Van NostradN. Hicks, Notes on Differential Geometry, Van Nostrad, 1965.
| {'fraction_non_alphanumeric': 0.08810806708659027, 'fraction_numerical': 0.02834484328754648, 'mean_word_length': 3.2965438539289207, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 19, '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': 'In this paper we introduce a new method to design control laws for non-linear underactuated systems. Our method will often work in situations where standard control techniques fail. Even when standard techniques apply, we believe that our approach will achieve better performance. This is because we produce an infinite dimensional family of control laws, whereas most control techniques only produce a finite dimensional family of control laws. We will describe the problem and our solution invariantly, using differential geometry and in local coordinates. We hope that this paper will be useful for both mathematicians and engineers. We include an abstract example of a system which is open-loop unstable and cannot be stabilized using any linear control law, and demonstrate that our method produces a stabilizing control law. We also apply our method to the inverted pendulum cart to compare it with some standard control techniques.An important problem in control theory is how to modify a system of ordinary differential equations so that solutions to the equations satisfy some property. In order to', 'arxivid': 'math/9901140', 'author': ['David Auckly \nCONTROL OF NONLINEAR UNDERACTUATED SYSTEMS\nDepartment of Mathematics\nDepartment of Mathematics\nKansas State University\n66506-2602ManhattanKSUSA\n', 'Lev Kapitanski \nDepartment of Mechanical and Nuclear Engineering\nKansas State University\n66506-2602ManhattanKSUSA\n', 'Warren White \nKansas State University\n66506-5106ManhattanKSUSA\n'], 'authoraffiliation': ['CONTROL OF NONLINEAR UNDERACTUATED SYSTEMS\nDepartment of Mathematics\nDepartment of Mathematics\nKansas State University\n66506-2602ManhattanKSUSA', 'Department of Mechanical and Nuclear Engineering\nKansas State University\n66506-2602ManhattanKSUSA', 'Kansas State University\n66506-5106ManhattanKSUSA'], 'corpusid': 15734511, 'doi': '10.1002/(sici)1097-0312(200003)53:3<354::aid-cpa3>3.0.co;2-u', 'github_urls': [], 'n_tokens_mistral': 9682, 'n_tokens_neox': 8446, 'n_words': 5303, 'pdfsha': '8a09cea5def19cc6d9ed5ddf1b579eaee73a2df5', 'pdfurls': ['https://arxiv.org/pdf/math/9901140v1.pdf'], 'title': [], 'venue': []} |
arxiv |
LODE: Locally Conditioned Eikonal Implicit Scene Completion from Sparse LiDAR
Pengfei Li
Ruowen Zhao
Yongliang Shi
Hao Zhao
Jirui Yuan
Guyue Zhou
Ya-Qin Zhang
LODE: Locally Conditioned Eikonal Implicit Scene Completion from Sparse LiDAR
Scene completion refers to obtaining dense scene representation from an incomplete perception of complex 3D scenes. This helps robots detect multi-scale obstacles and analyse object occlusions in scenarios such as autonomous driving. Recent advances show that implicit representation learning can be leveraged for continuous scene completion and achieved through physical constraints like Eikonal equations. However, former Eikonal completion methods only demonstrate results on watertight meshes at a scale of tens of meshes. None of them are successfully done for non-watertight LiDAR point clouds of open large scenes at a scale of thousands of scenes. In this paper, we propose a novel Eikonal formulation that conditions the implicit representation on localized shape priors which function as dense boundary value constraints, and demonstrate it works on SemanticKITTI and SemanticPOSS. It can also be extended to semantic Eikonal scene completion with only small modifications to the network architecture. With extensive quantitative and qualitative results, we demonstrate the benefits and drawbacks of existing Eikonal methods, which naturally leads to the new locally conditioned formulation. Notably, we improve IoU from 31.7% to 51.2% on SemanticKITTI and from 40.5% to 48.7% on SemanticPOSS. We extensively ablate our methods and demonstrate that the proposed formulation is robust to a wide spectrum of implementation hyper-parameters. Codes and models are publicly available at https://github. com/AIR-DISCOVER/LODE.
I. INTRODUCTION
Representing 3D data with neural implicit functions is actively explored recently due to its strong modeling capability and memory efficiency [1]- [7]. Meanwhile, it can be easily meshed and rendered to facilitate human viewing. While most methods are fully supervised [1]- [3], SIREN [6] proposes an Eikonal implicit scene completion method with weak supervision needed. It learns a signed distance function (SDF), which measures the nearest distance to the scene surface, through the process of solving an Eikonal differential equation with only points on the surface and without knowing SDF values in free space. This scheme is promising for large-scale sparse LiDAR data because (1) for non-water-tight scenes, it is hard to define signed distance in free space and (2) it requires less supervision than non-Eikonal formulations and thus is simpler, especially when considering the completion of thousands of scenes.
However, even after an exhaustive parameter search, SIREN fails to fit sparse LiDAR data ( Fig. 1-a), which limits its application in many important scenarios such as University of Chinese Academy of Sciences, China [email protected]. The implicit fitting result of SIREN [6]. Note that this is well tuned by an exhaustive parameter search. (c) The output of our method is a neural signed distance function of arbitrary resolution, i.e., implicit scene completion. (d) Our result with extended semantic parsing.
autonomous driving. This is understandable as SIREN is a pure generative model and LiDAR point clouds are extremely sparse. Specifically, the reasons are three-fold: (1) The sparsity of on-surface points amplifies the negative impact of wrongly sampled off-surface anchors. (2) The normal orientations of sparse points cannot be estimated accurately from their neighbors, which serve as a necessary boundary value constraint for SIREN fitting. (3) Without trustworthy boundary values, enforcing a hard Eikonal constraint leads to even inaccurate SDF values. As shown in Fig. 1-b, the SIREN fitting result is fragmented.
To overcome these limitations, we develop a novel Eikonal implicit formulation by introducing an intermediate embedding domain, where localized shape priors are contained. Instead of directly fitting a function to map 3D Cartesian coordinates to signed distances, we first map the Euclidean space to a corresponding high-dimensional shape embedding space, and then the signed distance space. These shape embeddings function as dense boundary values that entangle both zeroth-order (on-surface points) and first-order (normal directions) constraints, in a data-driven manner. Naturally, the issue of enforcing a hard Eikonal constraint is also alleviated. This proposed formulation is named Locally Conditioned Eikonal Formulation and abbreviated as LODE. The result of LODE is significantly better than SIREN ( Fig. 1-c). And the supplementary video demonstrates LODE performs well on in-the-wild sequential LIDAR inputs.
Specifically, to implement LODE, we propose a novel hybrid architecture combining a discriminative model and a generative model. The discriminative part of our method exploits the strong representation learning power of sparse convolution, generating latent shape embeddings from sparse point cloud input. The generative model takes as input the ground truth point cloud coordinates along with pointwise latent shape embeddings retrieved by trilinear sampling and predicts SDF values of these points. During inference, the ground truth points are replaced with the points of interest.
Furthermore, to demonstrate the flexibility of LODE, we extend our method to implicit semantic completion in two ways: (1) by adding a dense discriminative head to predict semantic labels which can be mapped to the implicit function using K-Nearest-Neighbors; (2) by adding a parallel implicit generative head to directly model the implicit semantic field. We evaluate them on SemanticKITTI and achieve results ( Fig. 1-d) comparable to state-of-the-art methods.
To summarize, our contributions are as follows:
• We develop a locally conditioned Eikonal implicit scene completion formulation that incorporates learned shape priors as dense boundary value constraints. • We apply the formulation in road scene understanding, leading to the first Eikonal implicit road scene completion method without knowing SDF values in free space. • We achieve state-of-the-art completion results on Se-manticKITTI and SemanticPOSS, outperforming the best Eikonal completion results by +19.5% and +8.2% IoU. Code, data, and models will be released.
II. RELATED WORKS Neural Implicit Representation. The general principle of neural implicit representation is to train a neural network to approximate a continuous function that is hard to parameterize otherwise. [1] proposes to learn deep signed distance functions conditioned on shape embeddings. [2] approximates occupancy functions with conditional batchnorm networks. [8] introduces data-driven shape embeddings into occupancy networks for indoor scene completion. [3] uses hyperplanes as compact implicit representations to reconstruct shapes sharply and compactly. [6] shows that using gradient supervision allows Eikonal SDF learning with only on-surface SDF value supervision and sine activations are critical to its success. [9] combines Gaussian ellipsoids and implicit residuals to represent shapes accurately. Some recent works exploit 3D implicit representations for instancelevel understanding from point cloud [10] or RGB [11] inputs. Despite these advances, there is no work yet on implicit scene completion on LiDAR point clouds where the data is extremely sparse with heterogeneous distribution. Our method bridges this gap with the proposed locally conditioned Eikonal formulation.
LiDAR-based scene understanding. While there are many advances in camera-based cognitive scene understanding [12]- [16], LiDAR point cloud provides reliably accurate 3D structural information and thus has been mainly leveraged in geometric scene understanding. Numerous LiDAR-based SLAM methods are proposed to improve the quality and realtime performance [17]- [21]. [22]- [24] further incorporate semantic understanding into LiDAR SLAM systems. These methods explicitly complete a scene with multiple frames of LiDAR data, while our goal is to achieve implicit completion with a single frame. Sparse LiDAR points are also efficiently leveraged in depth completion [25], [26], object detection [27], [28], segmentation [29]- [34]. We believe the performance of these methods will be boosted with LODE completing the original sparse representation. Moreover, by providing fine geometric details, LODE may aid in the detection of anomalous obstacles [35] and some other tasks.
III. FORMULATION A. Eikonal Implicit Completion Formulation
Eikonal completion methods aim at fitting the signed distance function (SDF) of a scene. The signed distance is the nearest distance from a point of interest to the scene surface, with the sign denoting whether the point is located outside (positive) or inside (negative) of the surface. The iso-surface where the signed distance equals zero implicitly delineates the scene. Formally, the goal of Eikonal implicit completion is to find a function Φ(x), which satisfies a set of M constraints C m , to approximate the underlying SDF. Each constraint relates the function Φ(x) or its gradient to certain input quantities a(x) on the corresponding domain Ω m :
C m (a(x),Φ(x), ∇ x Φ(x)) = 0, ∀x ∈ Ω m , m = 0, ..., M − 1.(1)
Specifically, these constraints are required:
C 0 := |∇ x Φ(x)| − 1, x ∈ Ω 0 .
(2)
C 1 := ∇ x Φ(x) − n(x), x ∈ Ω 1 .(3)C 2 := Φ(x) − SDF(x), x ∈ Ω 2 .(4)
Here, C 0 guarantees Φ(x) satisfies the Eikonal equation in the whole physical space of interest (Ω 0 ), which is a intrinsic property of SDF. C 1 forces that the gradients of Φ(x) equal the normal vectors for input on-surface points (Ω 1 ). C 2 constrains the values of Φ(x) equal the ground truth SDF for labeled anchor points (Ω 2 ). In this way, the problem can be regarded as an Eikonal boundary value problem, where the differential equation C 0 is to be solved under the firstorder constraint C 1 and the zeroth-order constraint C 2 . However, the ground truth SDF values in free space are difficult to obtain. A recent method named SIREN [6] proposes an intriguing variant where the domain of C 2 is limited to on-surface points in Ω 1 . As the ground truth SDF values of points in Ω 1 are zero, C 2 is reduced to:
C 2 := Φ(x), x ∈ Ω 1 .(5)
To remedy the lack of constraints on off-surface points, SIREN introduces another constraint:
C 3 := ψ(Φ(x)), x ∈ Ω 3 .(6)
Here, ψ pushes Φ(x) values away from 0, for randomly and uniformly sampled off-surface points (Ω 3 ⊆ Ω 0 \ Ω 1 ).
… Label Sine-MLP … Semantic Extension B 1 1 , , , Fig. 2.
Overview of our architecture. The discriminative model extracts shape priors and the generative model predicts SDF values. They are bridged by differentiable triliner sampling. Positional Encoding is used to represent more details. Two semantic extension options are outlined in small dashed boxes.
Nevertheless, these constraints fail to cope with the scenario where on-surface points in Ω 1 are sampled from sparse LiDAR point cloud data. Reasons are three-fold: (1) The sparsity of on-surface points in Ω 1 amplifies the negative impact of C 3 on the wrongly sampled off-surface anchors in Ω 3 (i.e., located on or near the surface). (2) The normal orientations of sparse points in Ω 1 cannot be estimated accurately from their neighbors, leading to an incorrect constraint C 1 .
(3) Without trustworthy boundary value constraints C 3 and C 1 , enforcing the hard Eikonal constraint C 0 leads to even inaccurate SDF values in free space.
B. Locally Conditioned Eikonal Formulation (LODE)
To overcome the aforementioned limitations, we propose a locally conditioned Eikonal formulation Φ(x, e)| e=ζ(x,Ω1) to approximate SDF. Here, we use ζ(·, ·) to first map the Euclidean space to a high-dimensional shape embedding space. It functions as a dense boundary value constraint for the differential equation. Then Φ(·, ·) maps the shape embedding space to the signed distance space. As a result, the constraints to be satisfied are formally re-written as:
C 0 := |∇ x Φ(x, e)| e=ζ(x,Ω1) | − 1, x ∈ Ω 0 .(7)C 4 := ρ(ζ(x, Ω 1 )), x ∈ Ω 0 .(8)
We use ρ(ζ(x, Ω 1 )) to represent the underlying dense constraint contained in the shape embedding space, which implicitly entangles correct C 1 , C 2 , and C 3 constraints of the modified formulations:
C 1 := ∇ x Φ(x, e)| e=ζ(x,Ω1) − n(x), x ∈ Ω 1 .(9)C 2 := Φ(x, e)| e=ζ(x,Ω1) , x ∈ Ω 1 .(10)C 3 := ψ(Φ(x, e)| e=ζ(x,Ω1) ), x ∈ Ω 3 .(11)
Here, Ω 1 contains the dense ground truth on-surface points and Ω 3 ⊆ Ω 0 \ Ω 1 . Hence the aforementioned problem of trustworthy boundary values is resolved. Naturally, the issue of enforcing a hard Eikonal constraint is also alleviated. We implement the proposed LODE formulation in a datadriven manner. The acquisition of functions ζ(·, ·) and Φ(·, ·) can be cast in a loss function that penalizes deviations from the constraints C 0 , C 1 , C 2 , and C 3 on their domain:
L LODE = λ 1 Ω0 |∇ x Φ(x, e)| e=ζ(x,Ω1) | − 1 dx + λ 2 Ω 1 (1 − ∇ x Φ(x, e)| e=ζ(x,Ω1) , n(x) )dx + λ 3 Ω 1 Φ(x, e)| e=ζ(x,Ω1) dx + λ 4 Ω 3 ψ(Φ(x, e)| e=ζ(x,Ω1) )dx,(12)
where λ 1 -λ 4 are constant weight parameters and ·, · calculates cosine similarity between two vectors.
IV. METHOD
To realize LODE, we propose a hybrid neural network architecture combining a discriminative model with a generative model, as shown in Fig. 2. The discriminative part exploits the strong representation learning power of sparse convolution, generating latent shape embeddings from sparse input Ω 1 . It together with the differentiable trilinear sampling module works as function ζ(·, ·). The generative model consists of an MLP, functioning as Φ(·, ·). It takes as input the encoded coordinates of ground truth points Ω 1 along with pointwise latent shape embeddings and predicts SDF values of these points. Using gradient descent, we can get the optimized ζ(·, ·) and Φ(·, ·) in the parameterized form.
A. Discriminative Model
Intuitively, road scenes have the characteristic of repetition. Thus convolutional neural network can be employed as the discriminative model to exploit the translation invariance.
Taking LiDAR points Ω 1 as input, we first conduct voxelization to obtain 3D occupancy volume V occ with size 1 × D occ × W occ × H occ . Then the discriminative model maps it into a shape embedding volume V se with size d se × D se × W se × H se , where d se is the dimension of the shape embedding outputs. To tackle the sparsity of V occ , we employ the sparse operations of the Minkowski Engine [36] to build the model, which is a multiscale encoderdecoder network. It extracts shape priors via a shape completion process: the encoder consisting of convolutional blocks aggregates localized features, and the decoder involving generative deconvolutional blocks generates dense results.
Yet the constant generation of new voxels will destroy the sparsity just as the submanifold dilation problem [37]. To avoid this, we use a pruning block to prune off redundant voxels. It contains a convolutional layer to determine the binary classification result of whether a voxel should be pruned, which is supervised with binary cross-entropy loss:
L com = − 1 m m i=1 1 n i ni j=1 [y i,j log(p i,j ) + (1 − y i,j )log(1 − p i,j )],(13)
where m is the count of supervised blocks, n i denotes the count of voxels in the i-th block, y i,j and p i,j are the true and predicted existence probabilities for voxel i respectively.
B. Differentiable Trilinear Sampling Module
After generating V se , pointwise shape embedding e i ∈ R dse for query point x i ∈ Ω 0 is needed. We use trilinear interpolation to sample e i for x i from its 8 nearest voxel centers to maintain the continuity of the latent shape field at the voxel borders. Formally, with the length of voxel edge normalized, the trilinear sampling for e i can be written as:
e c i = Dse m Wse n Hse k e c mnk × max(0, 1 − |x i − x m |) × max(0, 1 − |y i − y n |) × max(0, 1 − |z i − z k |),(14)
where e c i and e c mnk are shape embeddings on channel c for x i = (x i , y i , z i ) and voxel center x mnk = (x m , y n , z k ). Then the gradient with respect to e mnk for backpropagation is:
This differentiable trilinear sampling mechanism allows loss gradients to flow back to V se and further back to the discriminative model, making it possible to train discriminative model and the following generative model cooperatively.
C. Positional Encoding Module
Positional encoding has proved an effective technique in neural rendering [4] [38] for its capacity to capture highfrequency information. Thus we leverage it to represent more geometric details of the signed distance field. Specifically, the 3D Cartesian coordinate x i is encoded into high-dimensional feature y i = (γ enc (x i ), γ enc (y i ), γ enc (z i )) ∈ R denc , where γ enc (p) = (sin(2 0 πp),cos(2 0 πp), · · · , sin(2 L−1 πp), cos(2 L−1 πp)).
L is the number of frequency octaves and thus d enc = 6L.
D. Generative Model
We use an MLP as the generative model for implicit SDF representation and use sine as a periodic activation function to better model details [6]. Thus Φ can be formalized as:
Φ(x) =W n (φ n−1 • φ n−2 • · · · • φ 0 )(x) + b n , x j → φ j (x j ) = sin(W j x j + b j ),(17)
where φ j : R Mj → R Nj is the j th layer of the model. Given x j ∈ R Mj , the layer applies the affine transform with weights W j ∈ R Nj ×Mj and biases b j ∈ R Nj on it, and then pass the resulting vector to the sine nonlinearity. In our implementation, the concatenated vector [y i , e i ] is taken as input. And model weights are shared for all scenes. We use the proposed loss function (12) to optimize model weights and shape embeddings. Note that during training, x i is sampled from dense ground truth Ω 1 instead of sparse input Ω 1 . Thus, in a data-driven manner, our generative model can effectively map the shape embedding space to the signed distance space with abundant geometric information.
E. Semantic Extension
To demonstrate the flexibility of LODE, we extend our method to implicit semantic completion in two ways.
Semantic Extension A. We first semantically segment V occ with a sparse CNN. Then a dense CNN is used to predict coarse semantic completion results. Mapping it to our implicit representation using K-Nearest-Neighbor, we get the refined implicit semantic results.
Semantic Extension B. We add a parallel implicit generative head to directly model the implicit semantic label field. Its structure is similar to aforementioned generative model, except that it outputs the probabilities of label classification.
The results are supervised with a cross-entropy loss:
L seg = − 1 N seg Nseg i=1 C c=1 y i,c log(p i,c ),(18)
where y i,c and p i,c are the true and predicted probabilities. N seg points and C categories are considered.
F. Training and Inference
During training, we randomly sample N on on-surface points from Ω 1 and N off off-surface points from Ω 3 , optimizing the whole neural network with loss:
L total = L LODE + λ 5 L com + λ 6 L seg .(19)
Here, λ 5 and λ 6 are constant weight parameters. Note that λ 6 = 0 when the semantic extension is not included. During inference, we uniformly sample N 3 inf points from Ω 0 at a specified resolution. And we use a threshold v th close to zero to select the points with estimated SDF values smaller than v th as explicit surface points for evaluation.
V. EXPERIMENTS
Dataset. We evaluate the proposed LODE on Se-manticKITTI [39] and SemanticPOSS [40]. There are 22 sequences (8550 scans) and 6 sequences (2988 scans) of road scene LiDAR data in the two datasets respectively. Each scan covers a range of 51.2m ahead of the LiDAR, 25.6m to each side, and 6.4m in height. We follow the official split for training and validation. Metric. We use the interactions over union (IoU) metric for evaluation. Implementation Details. For the discriminative model, we set D occ = 256, W occ = 256, H occ = 32, m = 5. For the generative model, we use N on = N off = 16000 and N inf = 256. For training, we set λ 1 = 3000, λ 2 = 100, λ 3 = 100, λ 4 = 50, λ 5 = 100 and use the Adam optimizer with an initial learning rate of 10 −4 . When the semantic extension is included, λ 6 = 50.
A. Scene Completion Effectiveness of LODE
We compare our LODE with other recent strong Eikonal completion methods on the validation sets of SemanticKITTI and SemanticPOSS. The results of these Eikonal methods are obtained by applying them to every LiDAR sweep and taking the average. In Table.I, the first row shows that directly comparing the input sparse point cloud with completion ground truth yields 10.3% and 13.0% IoU. The existing Eikonal methods improve the IoU to 31.7% and 40.5%. Thanks to the new locally conditioned Eikonal formulation, our approach further improves IoU to 51.2% and 48.7%. These results demonstrate the effectiveness of LODE. This large improvement is better demonstrated with qualitative results in Fig. 3. Though the existing Eikonal methods are successful for clean synthetic point cloud data uniformly sampled on watertight meshes as shown in [6], fitting largescale outdoor scenes captured by LiDAR (Fig.3-a) is much more difficult. On the one hand, many regions are not sampled thus missing in the point cloud. On the other hand, caused by the mechanism of LiDAR, data sparsity increases with distance and it is extremely sparse at the far end. These result in the lack of effective boundary values for solving the Eikonal differential equation. For this reason, as pure generative models, these methods fail to fit road scenes and produce lots of artifacts (Fig.3-c,d). Our LODE, on the contrary, takes data-driven shape priors generated by a strong sparse convolutional network as the dense boundary values and successfully completes the scenes. As shown in Fig.3-e and highlighted in red boxes, both occluded and incomplete regions are better reconstructed than other Eikonal methods. In order to show that the significant margins reported in Table. I are robust to Marching Cubes thresholds, we provide an exhaustive evaluation in Fig.4. It is clear that our method outperforms other methods under all inspected thresholds.
B. Implementation Robustness of LODE
To better understand LODE and demonstrate its robustness to hyper-parameters in implementation, we provide a series of ablation studies on SemanticKITTI as follows.
Discriminative model design. We investigate two factors: (1) Where to add pruning blocks; (2) Conv layer number in the output block that generates shape embeddings. As shown in Table.II, LODE is robust to these design choices.
Does generative model capacity matter? Deeper and wider models usually achieve better results for recognition. To explore whether generative model capacity matters for LODE, we ablate the width, depth, and activations of the MLP. As shown in Table.III, different configurations produce similar results. It demonstrates the capacity of generative model is not a performance bottleneck. Interestingly, using ReLU instead of Sine activation only brings a performance drop of 1.75%. It suggests that in challenging scenarios like ours, using Sine activation is not as critical as in SIREN [6].
Which dimension of shape volume matters? To study which factor is the deciding one for the representation power of the shape volume, we evaluate different shape embedding dimensions and scale sizes. By scale size, we mean the down-sampling ratio Docc Dse . The results are summarized in Table.IV, showing that using shape embeddings of dimension 128 is already capable of representing our scenes well. But increasing scale size leads to a sharp drop of IoU, which reflects the importance of the locality of shape priors. Is trilinear sampling necessary? We justify the necessity of trilinear sampling in our method using Table. VI. A trivial nearest neighbor sampling leads to a performance drop of 2.9%. This is a clear margin that shows the benefit of smoothly interpolating shape embeddings. How to encode positional information? The goal of positional encoding is to represent fine geometric details of the scene. We investigate positional encoding levels and whether to concatenate original coordinates. As shown in Table.V, when positional encoding is not used or the encoding level is low, the completion IoU decreases dramatically. Through the qualitative results in Fig. 5-a, it is clear that leaving out positional encoding leads to the loss of details.
With these ablations and analyses, we demonstrate the role of each module and the impact of hyper-parameters in detail. Meanwhile, despite the performance fluctuation under different hyper-parameters, our quantitative results are always better than the existing Eikonal methods shown in the second to the fifth row of Table.I, which demonstrates the effectiveness and robustness of LODE. results, the semantic extension A and B achieve 20.2% and 18.0% mIoU, respectively. Although they under-perform the state-of-the-art method JS3C-Net [44], our models allow implicit completion and models the signed distance field. Qualitative results shown in Fig. 3-f demonstrate faithful semantic implicit completion. Last but not least, we map explicit semantic completion results from JS3C-Net to our implicit completion using K-Nearest-Neighbors, achieving 23.4% mIoU. These results show the flexibility of LODE as it can be easily extended to provide semantic information.
C. Flexibility of LODE
D. Other Potential Benefits of LODE
As illustrated in Fig.6, with LODE, we can get mesh reconstructions at any resolution, which is due to its continuous representation of the signed distance field. Meanwhile, as shown in Fig.1 and Fig.3, LODE enables the indiscernible LiDAR data to be converted into human-friendly visualizations, which demonstrates its capacity to enhance human understanding of robot-perceived information. Moreover, the proposed LODE has compatibility with implicit planning algorithms [45]- [49], and thus can be leveraged to facilitate downstream robotic manipulation tasks.
VI. CONCLUSION
In this study, we propose a novel locally conditioned Eikonal formulation named LODE for implicit scene completion. Learned shape embeddings are treated as dense boundary values that constrain signed distance function learning. We implement the formulation as a hybrid neural network combining discriminant and generative models. The network is trained to implicitly fit road scenes captured by sparse LiDAR point clouds, without accessing exact SDF values in free space. Large-scale evaluations on SemanticKITTI and SemanticPOSS show that our method outperforms existing Eikonal methods by a large margin. We also extend the proposed method for semantic implicit completion in two ways, achieving strong qualitative and quantitative results.
Fig. 1 .
1(a) The input is a sparsity-variant point cloud of road scenes captured by LiDAR. (b)
|x i − x m |) × max(0, 1 − |y i − y n |) × max(0, 1 − |z i − z k |).
Fig. 3 .
3Qualitative results of Eikonal implicit road scene completion on the SemanticKITTI and SemanticPOSS validation set.
Fig. 4 .
4IoU comparisons under different thresholds.
Fig. 5 .
5Qualitative results with different positional encoding strategies.
Fig. 6 .
6Scene completion results at multiple resolutions.
TABLE I SCENE
ICOMPLETION RESULTS MEASURED IN IOU (%).Method
Reference
SemanticKITTI SemanticPOSS
Input
-
10.3
13.0
SIREN [6]
NeurIPS 2020
26.3
36.0
Fourier Features [41]
NeurIPS 2020
28.6
30.9
BACON [42]
CVPR 2022
30.5
40.5
DiGS [43]
CVPR 2022
31.7
37.4
LODE (Ours)
-
51.2 (+19.5)
48.7 (+8.2)
TABLE II DISCRIMINATIVE
IIMODEL.Pruning
Blocks
Output
Block
IoU (%)
Last 1
2 convs
49.5
Last 2
2 convs
49.1
Last 3
2 convs
50.6
Last 4
2 convs
51.0
All
2 convs
51.1
All
4 convs
50.9
TABLE III
GENERATIVE MODEL.
Width Depth Activations IoU (%)
128
4
Sine
51.0
256
4
Sine
51.0
512
4
Sine
50.9
256
3
Sine
49.6
256
5
Sine
50.9
256
4
ReLU
49.3
TABLE IV SHAPE
IVEMBEDDING.Shape
Dimension
Scale
Size
IoU (%)
128
4
50.9
512
4
51.2
256
2
50.3
256
4
51.0
256
8
49.2
256
16
44.8
TABLE V
POSITIONAL ENCODING.
Positional
Encoding
Include
xyz
Encoding
Level L
IoU (%)
×
-
-
40.4
5
40.3
10
51.0
15
50.9
×
10
51.1
TABLE VI
VISAMPLING STRATEGY.
Sample Strategy
IoU (%)
Trilinear
51.0
Nearest
48.1
Table .
.VII shows semantic scene completion results on the SemanticKITTI validation set, which is evaluated on 19 categories. With little impact on original completion
TABLE VII SEMANTIC
VIISCENE COMPLETION RESULTS ON SEMANTICKITTI.Approach
ext.A ext.B JS3C LODE w/ JS3C
mIoU (%)
20.2
18.0
22.7
23.4
ACKNOWLEDGEMENTSThis work was sponsored by Tsinghua-Toyota Joint Research Fund (20223930097) and Baidu Inc. through Apollo-AIR Joint Research Center.
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| {'fraction_non_alphanumeric': 0.05792129564991922, 'fraction_numerical': 0.029519117617727426, 'mean_word_length': 4.390023650827779, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 1, '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': 'Scene completion refers to obtaining dense scene representation from an incomplete perception of complex 3D scenes. This helps robots detect multi-scale obstacles and analyse object occlusions in scenarios such as autonomous driving. Recent advances show that implicit representation learning can be leveraged for continuous scene completion and achieved through physical constraints like Eikonal equations. However, former Eikonal completion methods only demonstrate results on watertight meshes at a scale of tens of meshes. None of them are successfully done for non-watertight LiDAR point clouds of open large scenes at a scale of thousands of scenes. In this paper, we propose a novel Eikonal formulation that conditions the implicit representation on localized shape priors which function as dense boundary value constraints, and demonstrate it works on SemanticKITTI and SemanticPOSS. It can also be extended to semantic Eikonal scene completion with only small modifications to the network architecture. With extensive quantitative and qualitative results, we demonstrate the benefits and drawbacks of existing Eikonal methods, which naturally leads to the new locally conditioned formulation. Notably, we improve IoU from 31.7% to 51.2% on SemanticKITTI and from 40.5% to 48.7% on SemanticPOSS. We extensively ablate our methods and demonstrate that the proposed formulation is robust to a wide spectrum of implementation hyper-parameters. Codes and models are publicly available at https://github. com/AIR-DISCOVER/LODE.', 'arxivid': '2302.14052', 'author': ['Pengfei Li ', 'Ruowen Zhao ', 'Yongliang Shi ', 'Hao Zhao ', 'Jirui Yuan ', 'Guyue Zhou ', 'Ya-Qin Zhang '], 'authoraffiliation': [], 'corpusid': 257220114, 'doi': '10.48550/arxiv.2302.14052', 'github_urls': [], 'n_tokens_mistral': 15689, 'n_tokens_neox': 13373, 'n_words': 7744, 'pdfsha': '82f06af9baef584bd3deceddba5efb335701a911', 'pdfurls': ['https://export.arxiv.org/pdf/2302.14052v1.pdf'], 'title': ['LODE: Locally Conditioned Eikonal Implicit Scene Completion from Sparse LiDAR', 'LODE: Locally Conditioned Eikonal Implicit Scene Completion from Sparse LiDAR'], 'venue': []} |
arxiv |
DYNAMICS OF THE HEAT SEMIGROUP IN JACOBI ANALYSIS
22 Apr 2011
Francesca Astengo
Bianca Di Blasio
DYNAMICS OF THE HEAT SEMIGROUP IN JACOBI ANALYSIS
22 Apr 2011arXiv:1104.4479v1 [math.FA]
Let ∆ be the Jacobi Laplacian. We study the chaotic and hypercyclic behaviour of the strongly continuous semigroups of operators generated by perturbations of ∆ with a multiple of the identity on L p spaces.IntroductionChaos in the context of strongly continuous semigroups of bounded linear operators in Banach spaces has been introduced by W. Desch, W. Schappacher and G. Webb [5] as a generalization to continuous time of the discrete time case. In their paper [5], the authors give a sufficient condition for a strongly continuous semigroup to be chaotic in terms of the spectral properties of its infinitesimal generator. Moreover they study in detail many examples, mainly the transport equation and second order differential operators with constant coefficients (see also[4]).The purpose of this paper is to apply the criterion in [5] to study of the dynamics of the (modified) heat semigroup generated by the Jacobi operator (i.e., the generator is a perturbation of the Jacobi operator by a multiple of the identity), which is a second order differential operator with nonconstant coefficients.Jacobi analysis can be developed as a generalization of the Fourier-cosine transform and has been studied by many authors (see[12]), the main interest being the interplay between the analytic and geometric properties of the Jacobi operator. Indeed, in certain cases, the Jacobi operator is the radial part of the Laplace-Beltrami operator on Damek-Ricci spaces [2], therefore Jacobi analysis includes radial analysis on symmetric spaces of real rank one as a special case.2000 Mathematics Subject Classification. Primary: 43A32; Secondary: 43A90 47A16 .
The dynamics of the (modified) heat semigroup on non compact symmetric spaces was already studied in [9,10]. We extends the results in [9] to the context of Jacobi analysis. Therefore, as particular cases, we cover Damek-Ricci spaces and Heckman-Opdam root spaces of rank one, which were not treated in [9].
The paper is organized as follows: in Section 2 we settle notation and recall some basic facts regarding chaotic semigroups, Jacobi analyis and Lorentz spaces. In Section 3 we establish some properties of spherical functions, the weak type (1,1) boundedness of the heat maximal function and the L p,q inversion formula. In Section 4 we apply the results in the previous section to the study of the dynamics of (modified) heat semigroups.
Notation and Preliminaries
2.1. Chaotic semigroups. In this paper we follow R. L. Devaney [6], who has defined chaos in metric spaces in the following sense. A continuous map f on a metric space X is said to be chaotic if it is topologically transitive, i.e., some element has a dense orbit, and if the set of its periodic points is dense in X. These two conditions imply (see [3]) that f has sensitive dependence on initial conditions. In [5] the authors have generalized this definition to strongly continuous semigroups as follows.
Definition 1. A strongly continuous semigroup {T (t) : t ≥ 0} on a Banach space X is said to be hypercyclic if there exists f in X such that its orbit {T (t)f : t ≥ 0} is dense in X .
Definition 2.
A strongly continuous semigroup {T (t) : t ≥ 0} on a Banach space X is said to be chaotic if it is hypercyclic and the set of periodic points
{f ∈ X : ∃t > 0 such that T (t)f = f } is dense in X .
We denote by σ(B) and σ pt (B) respectively the spectrum and the point spectrum of a linear operator B on a Banach space X .
A sufficient condition for a strongly continuous semigroup to be chaotic in terms of the spectral properties of its generator was given by Desch, Schappacher, and Webb [5, which intersects the imaginary axis. For each z in Ω let φ z be a nonzero eigenvector, i.e. Many authors studied sufficient conditions for a strongly continuous semigroups to be hypercyclic (see [9] and the references therein).
Bφ z = z φ z . Suppose that for every f in the dual space X ′ of X the function F f : Ω → C,
2.2.
Lorentz spaces. Let f be a measurable function on the measure space (X, M, µ).
The nonincreasing rearrangement of f is the function f * on R + defined by
f * (t) = inf s ∈ R + : µ ({x ∈ X : |f (x)| > s}) ≤ t ∀t ∈ R + .
The function f * is nonincreasing, nonnegative, equimeasurable with f and right-continuous.
For any given measurable function f on X, we define
f L p,q = q p ∞ 0 s 1/p f * (s) q ds s 1/q 1 ≤ p < ∞, 1 ≤ q < ∞, and f L p,∞ = sup s 1/p f * (s) : s ∈ R + 1 ≤ p < ∞.
Definition 3. Let 1 ≤ p < ∞ and 1 ≤ q ≤ ∞. The Lorentz space L p,q (µ) consists of those measurable functions f on X such that f L p,q is finite.
It is easy to check that L p,p (µ) coincides with the usual Lebesgue space L p (µ), with equality of norms. Moreover, if q 1 < q 2 , then L p,q 1 (µ) is contained in L p,q 2 (µ) and, if 1 < p, q < ∞, the dual space of L p,q (µ) is L p ′ ,q ′ (µ). Here and elsewhere in this paper p ′ and q ′ are the conjugate exponents of p and q, i.e. 1 p + 1 p ′ = 1 q + 1 q ′ = 1. A good reference for Lorentz spaces is [8]. We recall the following multiplication theorem from that paper. such that for every f in L p 0 ,q 0 (µ) and m in L p 1 ,q 1 (µ)
mf L p,q ≤ C m L p 1 ,q 1 f L p 0 ,q 0 , where 1 p = 1 p 0 + 1 p 1 and 1 q = 1 q 0 + 1 q 1 .
Jacobi Analysis.
We recall some facts about Jacobi analysis which we shall need in the sequel. We follow Koornwinder [12] and the normalizations therein. Throughout this paper, α, β will be real numbers, with α ≥ β ≥ − 1 2 , α > − 1 2 . We define
A(x) = (2 sinh x) 2α+1 (2 cosh x) 2β+1 ∀x > 0.
For complex λ and ρ = α + β + 1, let
∆ = − d dx 2 − A ′ (x) A(x) d dx
and consider the differential equation
(2.1) ∆u(x) = (λ 2 + ρ 2 ) u(x) x > 0.
Using the substitution z = − sinh 2 x, we can transform equation (2.1) in the well known hypergeometric differential equation
z(1 − z) u ′′ (z) + (c − (a + b + 1)z) u ′ (z) − ab u(z) = 0 of parameters a = 1 2 (ρ − iλ), b = 1 2 (ρ + iλ), c = α + 1. Let 2 F 1 denote the Gaussian hypergeometric function. The Jacobi function ϕ λ = ϕ (α,β) λ of order (α, β) ϕ λ (x) = 2 F 1 ( 1 2 (ρ − iλ), 1 2 (ρ + iλ); α + 1, − sinh 2 x) x ∈ R
is the unique even smooth function on R which satisfies u(0) = 1 and the differential equation (2.1).
Therefore the function λ → ϕ λ (x) is analytic for all x ∈ R. Moreover {ϕ λ } is a continuous orthogonal system on R + with respect to the weight function A.
We consider on R + = (0, ∞) the measure µ which is absolutely continuous with respect to the ordinary Lebesgue measure and has density A. When 1 ≤ p ≤ ∞ we denote by L p (µ) the ordinary Lebesgue space on R + with respect to the measure µ.
Note that
(2.2) |A(x)| ≤ C x 2α+1 0 < x < 1 e 2ρx x ≥ 1.
The Jacobi transform f −→f is defined bŷ
f (λ) = R + f (x) ϕ λ (x) dµ(x),
for all functions f on R + and complex numbers λ for which the right hand side is well defined.
Let D ♯ (R) be the space of smooth even functions on R with compact support and denote by D ♯ (R + ) the space of the restrictions to R + of functions in D ♯ (R). Note that
(∆f )ˆ= (λ 2 + ρ 2 )f , ∀f ∈ D ♯ (R + ).
The following inversion formula holds for functions in D ♯ (R) [12, p. 9]
f (x) = 1 2π ∞ 0f (λ) ϕ λ (x) |c(λ)| −2 dλ, ∀x ∈ R
where c(λ) is a multiple of the meromorphic Harish-Chandra function given by the formula
c(λ) = 2 ρ−iλ Γ(α + 1)Γ(iλ) Γ( 1 2 (ρ + iλ))Γ( 1 2 (ρ + iλ) − β)
.
Moreover the Jacobi transform f −→f extends to an isometry from L 2 (µ) onto L 2 (R, 1 2π |c(λ)| −2 dλ). The following formula holds
(2.3) ϕ λ (x) ϕ λ (y) = ∞ 0 ϕ λ (u) W (x, y, u) dµ(u)
where the kernel W is explicitly known (see [12, p. 58]). Thus one can define the (generalized) translation operator
τ x f (y) = ∞ 0 f (u) W (x, y, u) dµ(u) ∀f ∈ D ♯ (R + )
and the (generalized) convolution of functions f, g in D ♯ (R + ), say, by
f ⋆ g(x) = ∞ 0 τ x f (y) g(y) dµ(y),
so that
(f ⋆ g)ˆ=fĝ.
The function W is nonnegative, supported in |x − y| ≤ u ≤ x + y and symmetric in its
three variables, so that f ⋆ g = g ⋆ f . From (2.3) with λ = iρ it follows ∞ 0 W (x, y, u) dµ(u) = 1. Moreover ∞ 0 τ x f dµ = ∞ 0 f dµ and τ 0 f = f . Finally, the Young inequality holds [12, p. 61] (2.4) f ⋆ g r ≤ C f p g q 1 p + 1 q = 1 + 1 r .
2.4.
Heat kernel. Let f be in D ♯ (R + ) and consider the initial value problem for the heat equation
(2.5) ∆u(t, x) = ∂ t u(t, x) lim t→0 + u(t, x) = f (x). t > 0, x ∈ R
One can show that the solution u to the problem (2.5) can be written as
u(t, x) = f ⋆h t (x),
where the function h t is called heat kernel and it is defined on the Jacobi transform side by h t (λ) = e −t(λ 2 +ρ 2 ) ∀t > 0, λ ∈ C.
The heat kernel h t is a nonnegative function in L p (µ) for all p ≥ 1 and h t 1 = 1 [1].
For every p in [1, ∞] denote by ∆ p the closure in L p (µ) of the operator ∆ with domain D ♯ (R + ). By the Young inequality (2.4), the family of operators {H p (t)} t>0 defined by
H p (t)f = h t ⋆ f ∀f ∈ L p (µ)
is a strongly continuous symmetric semigroup on L p (µ) whose generator is ∆ p .
The sharp estimate for the heat kernel was established in [11] (2.6)
h t (x) ≃ t −α−1 (1 + t + x) α− 1 2 (1 + x) e −ρx−ρ 2 t e −x 2 /4t ∀x > 0, t > 0. Here f (x) ≃ g(x) stands for C 1 g(x) ≤ f (x) ≤ C 2 g(x)
, for some constants C 1 and C 2 .
The L p inversion formula
Since for real values of λ, we have |ϕ λ | ≤ ϕ 0 ≤ 1 (see [12, p. 53]), the function space
L 1 0 (µ) = f : R + → C : f measurable and f ϕ 0 ∈ L 1 (µ)
contains L 1 (µ) and the Jacobi transformf is well defined as a function on R when f is in L 1 0 (µ). In the next results we show that the Lorentz spaces L p,q (µ), 1 < p < 2, 1 ≤ q ≤ ∞ are subspaces of L 1 0 (µ). When 1 ≤ p < ∞ we define
S p = λ ∈ C : |Im (λ)| ≤ 1 − 2 p ρ .
Note that S p = S p ′ , when p > 1. By S • p and ∂S p we denote respectively the interior and the boundary of S p . It is well known that from the estimate [12, p. 53]
(3.1) |ϕ λ (x)| ≤ C (1 + x) e (|Im (λ)|−ρ)x ∀x ≥ 0, λ ∈ C,
it follows that if 1 < p < 2 and λ is in S • p then ϕ λ is in L p ′ (µ). A more precise result holds (see [14] for the case of Damek-Ricci spaces).
Lemma 3.1. The following estimates hold.
(i) Let 1 < p < 2. If λ is in S • p then ϕ λ is in L p ′ ,q (µ) for any q in [1, ∞]. (ii) If 1 ≤ p < 2 and λ is in ∂S p then ϕ λ is in L p ′ ,∞ (µ). (iii) If p = 2 and λ is real, then ϕ λ /(1 + x) is in L 2,∞ (µ).
Proof. Let 1 < p < 2. It suffices to show that (i) holds with q = 1. By (3.1), when λ is in
S • p for some ε > 0, |ϕ λ (x)| ≤ C e −( 2 p ′ +2ε)ρx ∀x ∈ R + , therefore ϕ * λ (t) ≤ C ψ * (t) ∀t ∈ R + ,
where ψ(x) = e −( 2 p ′ +2ε)ρx . We now compute ψ * . By equation (2.2) we have ψ * (t) = inf s ∈ R + :
log(s −1/(2/p ′ +2ε)ρ) ) 0 A(x) dx ≤ t ≤ C t −( 1 p ′ +ε) t > 1 const t < 1.
We conclude that ψ * belongs to L p ′ ,q (µ) for any q in [1, ∞].
The proofs of (ii) and (iii) are similar. If p = 2 and λ is real we apply again (3.1).
If |Im (λ)| = (2/p − 1)ρ and 1 ≤ p < 2, we use the inequality [7] (
3.2) |ϕ λ (x)| ≤ C δ e −(|Im (λ)|+ρ)x ∀x ≥ δ > 0.
Denote by P p , 1 ≤ p < ∞, the parabolic region
P p = λ 2 + ρ 2 : λ ∈ S • p .
Corollary 3.2. Let 2 < p < ∞. For any z in the parabolic region P p there exists a nonzero φ z in L p such that ∆φ z = zφ z .
Note that Corollary 3.2 implies that the parabolic region P p is contained the point spectrum σ pt (∆ p ), for 2 < p < ∞.
Corollary 3.3. L p,q (µ) ⊂ L 1 0 (µ), 1 < p < 2, 1 ≤ q ≤ ∞.
Proof. Use the multiplication theorem with f in L p,q and m = ϕ 0 which is in L p ′ ,q ′ .
So, when f is in L p,q , we can define the Jacobi transformf as a function on R and actuallyf turns to be holomorphic in S • p .
Corollary 3.4. If f is in L p,q (µ), 1 < p < 2, 1 ≤ q ≤ ∞, then f is a bounded function in the strip S p holomorphic in S • p .
Proof. Use Lemma 3.1 and the Morera Theorem.
We include the next result, which we were unable to find in the literature (see [2] in the case of Damek-Ricci spaces).
Proposition 3.5. The heat maximal operator
H * f (x) = sup t>0 |h t ⋆ f (x)| f ∈ D ♯ (R + )
is of weak type (1, 1) and of strong type (p, p).
Proof. Since h t 1 = 1 for every t > 0 and by (2.4), the heat maximal operator is trivially bounded on L ∞ (µ). We now prove the weak L 1 -estimate. From this estimate and the Marcinkiewicz Interpolation Theorem the thesis follows.
As usual, we shall deal with the small time maximal function
H * 0 f (x) = sup 0<t≤1 |h t ⋆ f (x)| f ∈ D ♯ (R + )
and the large time maximal function
H * ∞ f (x) = sup t>1 |h t ⋆ f (x)| f ∈ D ♯ (R + ) separately. Note that H * ∞ f ≤ |f | ⋆ [sup t>1 h t ].
From the heat kernel estimate (2.6) it follows that sup
t>1 h t (x) = O((1 + x) −1/2 e −2ρx )
when x is large so that H * ∞ is of weak type (1, 1). Indeed, if k(x) = cosh(x) −2ρ , Liu [13, Lemma 3.2] proved that
τ x k(y) ≤ C cosh(x) −2ρ ∀y > 0, so that |f | ⋆ k(x) = ∞ 0 |f (y)| τ x k(y) A(y) dy ≤ cosh(x) −2ρ f 1 . Therefore µ {x : |f | ⋆ k(x) > λ} ≤ µ x : (cosh x) 2ρ < C f 1 /λ = x 0 0 A(u) du ≤ C (cosh x 0 ) 2ρ = C f 1 /λ.
where x 0 > 0 is such that (cosh x 0 ) 2ρ = C f 1 /λ (if any, otherwise x 0 = 0 and the weak type inequality is trivial).
In order to prove that H * 0 is of weak type (1, 1), we first note that the estimates of the heat kernel imply that when 0 < t ≤ 1,
0 ≤ h t (x) ≤ C e −x 2 /4 = k 1 (x) x > 1 h t (x) ≃ k t (x) 0 < x ≤ 1, where k t (x) = t −(α+1) e −x 2 /4t t, x > 0.
Let χ denote the characteristic function of the interval [−1, 1], and write
H * 0 f (x) ≤ sup 0<t≤1 |((1 − χ)h t ) ⋆ f (x)| + sup 0<t≤1 |(χh t ) ⋆ f (x)| ≤ k 1 ⋆ |f |(x) + C sup 0<t≤1 (χk t ) ⋆ |f |(x) Since the kernel k 1 is integrable, the operator f → k 1 ⋆ f is L 1 -bounded.
For the estimate of the other term we will use the weak type (1, 1) boundedness of a Hardy-Littlewood maximal function. Let X r denotes the characteristic function of [−r, r], normalized so that X r dµ = 1 and define
Mf = sup r>0 X r ⋆ |f |.
Liu [13] proved that the operator M is of weak type (1, 1).
Let ν(y) = y 0 A(x) dx and G(y) = y 0 τ x |f |(u) A(u) du. Note that G(y) ≤ ν(y) Mτ x |f |(0). Applying the size estimate (2.2), we get
(χk t ) ⋆ |f |(x) = 1 0 τ x |f |(y) k t (y) dµ(y) = − 1 0 G(y) k ′ t (y) dy + G(1)k t (1) ≤ Mτ x |f |(0) 1 0 ν(y)(−k ′ t (y)) dy + G(1)k t (1) ≤ Mτ x |f |(0) 1 0 ν ′ (y)k t (y) dy = Mf (x) 1 0 k t (y) A(y)dy ≤ Mf (x).
The thesis follows.
The standard method of approximation using the heat kernel gives the following inversion formula for functions in the Lorentz space L p,q (µ), with 1 < p < 2, q ≥ 1 (see [14] for the case of Damek-Ricci spaces).
Proposition 3.6. Let f be in L p,q (µ), with 1 < p < 2, q ≥ 1 or f in L 1 ∪ L 2 (µ). Iff is in L 1 (R, |c(λ)| −2 dλ), then for almost every x in R + ,
(3.3) f (x) = 1 2π ∞ 0f (λ) ϕ λ (x) |c(λ)| −2 dλ.
Proof. We can write f = f 1 +f 2 , where f 1 is in L 1 (µ) and f 2 is in L 2 (µ) [8]. Then for every
t > 0, we have f ⋆ h t = f 1 ⋆ h t + f 2 ⋆ h t .
Since the heat maximal operator is of weak type (1, 1) and g ⋆ h t (x) → g(x), for t → 0, whenever g is smooth and compactly supported, it follows that there exist measurable sets E 1 , E 2 of null measure such that
f 1 ⋆h t (x) → f 1 (x) for all x in c E 1 and f 2 ⋆ h t (x) → f 2 (x) for all x in c E 2 , as t → 0. This implies that if x is not in E 1 ∪ E 2 then f ⋆ h t (x) → f (x) as t → 0. Moreover µ(E) ≤ µ(E 1 ) + µ(E 2 ) = 0.
Sincef is in L 1 (R, |c(λ)| −2 dλ), for every test function ψ we have
f ⋆ h t , ψ = 1 2π ∞ 0 ∞ 0f (λ) e −t(λ 2 +ρ 2 ) ϕ λ (x) |c(λ)| −2 dλ ψ(x) dµ(x).
Using the Dominated Convergence Theorem we now get the result.
Dynamics of the heat semigroup
For every p in [1, ∞] denote by ∆ p the closure in L p (µ) of the operator ∆ with domain D ♯ (R + ). As previously observed the family
e −t∆p f = h t ⋆ f ∀f ∈ L p (µ)
is a strongly continuous semigroup on L p (µ).
In this section we study the dynamics of the shifted semigroup
e −t(∆p−θ) : L p (µ) → L p (µ).
Theorem 4.1. Let 2 < p < ∞. Then for all θ > θ p = ρ 2 − ρ 2 2 p − 1
Proof. We apply Theorem 2.1. By Corollary (3.2), the parabolic region P p is contained in the point spectrum σ pt (∆ p ) and the corresponding eigenfunctions are given by the appropriate Jacobi functions. The vertex of the parabolic region P p is at the point
θ p = ρ 2 − ρ 2 2 p − 1 2 = 4ρ 2 1 pp ′
and hence the point spectrum of (∆ p − θ) intersects the imaginary axis for all θ > θ p .
Suppose now that θ > θ p and let Ω θ denote the set (P p − θ) \ {z ∈ R : z ≤ ρ 2 − θ}, i.e.
Ω θ = z ∈ C : z = λ 2 + ρ 2 − θ |Im (λ)| < (1 − 2 p )ρ \ {z ∈ R : z ≤ ρ 2 − θ}.
Then Ω θ is an open, connected subset of the point spectrum of (∆ p − θ) that intersects the imaginary axis.
Since (Ω θ + θ − ρ 2 ) ∩ {x ∈ R : x < 0} = ∅ we choose an analytic branch z + θ − ρ 2 of the square root so that Re z + θ − ρ 2 > 0, for every z in Ω θ . Note that z −→ z + θ − ρ 2 maps Ω θ onto the open strip λ ∈ C : Re λ > 0, |Im λ| < ρ (1 − 2 p ) .
For every z in Ω θ we choose the eigenfunction φ z defined by (4.1) φ z = ϕ λ where z = λ 2 + ρ 2 − θ, so that (∆ p − θ)φ z = zφ z . As in Theorem 2.1, for every f in L p ′ (µ), we define the function
F f : Ω θ −→ C by F f (z) = f, φ z =f ( z + θ − ρ 2 ).
Then by Corollary 3.4 the function F f is holomorphic since it is the composition of two holomorphic functions. Moreover F f = 0 implies f = 0 by the inversion formula (3.3).
In the following theorem we prove that the semigroup e −t(∆p−θ) : L p (µ) → L p (µ)
is not chaotic when 1 < p ≤ 2 for every θ in R. does not have periodic elements. Moreover when 1 < p < 2 it is not hypercyclic.
Proof. Let 1 < p ≤ 2, θ in R and f a periodic point in L p (µ) for e −t(∆p−θ) . Then there exists t > 0 such that h t ⋆ f = f or equivalently e −t(λ 2 +ρ 2 −θ) − 1 f (λ) = 0 for every λ in S o p when 1 < p < 2 and for almost every real λ when p = 2. By the inversion formula (3.3) f = 0 and the first part follows.
Let 1 < p < 2 and assume that the semigroup e −t(∆p−θ) is hypercyclic. Then, the dual operator (∆ p − θ) ′ = ∆ p ′ − θ of its generator would have empty point spectrum [5,Theorem 3.3]. This a contradiction and the thesis follows.
F
f = f, φ z is analytic and does not vanish identically unless f = 0. Then the semigroup {T (t) : t ≥ 0} is chaotic.
]
Let p 0 , p 1 and q 0 , q 1 be in[1, ∞]. Then there exists a constant C
Theorem 4. 2 .
2Let 1 < p ≤ 2 and θ in R. Then the semigroupe −t(∆p−θ) : L p (µ) → L p (µ)
Theorem 3.1]: Theorem 2.1. Let B be the infinitesimal generator of a strongly continuous semigroup {T (t) : t ≥ 0} on a separable Banach space X . Let Ω be an open connected subset of σ pt (B)
the semigroupe −t(∆p−θ) : L p (µ) → L p (µ)is chaotic.
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Heat kernel and Hardy's theorem for Jacobi transform. T Kawazoe, J Liu, Chinese Ann. Math. Ser. B. 24T. Kawazoe and J. Liu, Heat kernel and Hardy's theorem for Jacobi transform, Chinese Ann. Math. Ser. B 24 (2003), 359-366.
Jacobi functions and analysis on noncompact semisimple Lie groups, Special functions: Group theoretical aspect and applications. T H Koornwinder, R. A. Askey et al.Dordrecht-Boston; ReidelT. H. Koornwinder, Jacobi functions and analysis on noncompact semisimple Lie groups, Special functions: Group theoretical aspect and applications. R. A. Askey et al. (eds.), Dordrecht-Boston: Reidel 1984, 1-85.
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. Matematica Dipartimento Di, address: [email protected] Dodecaneso. 35Dipartimento di Matematica, Via Dodecaneso 35, 16146 Genova, Italy E-mail address: [email protected]
. Dipartimento Di Matematica E Applicazioni, address: [email protected] Cozzi. 53Dipartimento di Matematica e Applicazioni, Via Cozzi 53, 20125 Milano, Italy E-mail address: [email protected]
| {'fraction_non_alphanumeric': 0.10604056437389771, 'fraction_numerical': 0.03474426807760141, 'mean_word_length': 3.0171803046404535, 'pattern_counts': {'":': 0, '<': 55, '<?xml version=': 0, '>': 32, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 4, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 24, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Let ∆ be the Jacobi Laplacian. We study the chaotic and hypercyclic behaviour of the strongly continuous semigroups of operators generated by perturbations of ∆ with a multiple of the identity on L p spaces.IntroductionChaos in the context of strongly continuous semigroups of bounded linear operators in Banach spaces has been introduced by W. Desch, W. Schappacher and G. Webb [5] as a generalization to continuous time of the discrete time case. In their paper [5], the authors give a sufficient condition for a strongly continuous semigroup to be chaotic in terms of the spectral properties of its infinitesimal generator. Moreover they study in detail many examples, mainly the transport equation and second order differential operators with constant coefficients (see also[4]).The purpose of this paper is to apply the criterion in [5] to study of the dynamics of the (modified) heat semigroup generated by the Jacobi operator (i.e., the generator is a perturbation of the Jacobi operator by a multiple of the identity), which is a second order differential operator with nonconstant coefficients.Jacobi analysis can be developed as a generalization of the Fourier-cosine transform and has been studied by many authors (see[12]), the main interest being the interplay between the analytic and geometric properties of the Jacobi operator. Indeed, in certain cases, the Jacobi operator is the radial part of the Laplace-Beltrami operator on Damek-Ricci spaces [2], therefore Jacobi analysis includes radial analysis on symmetric spaces of real rank one as a special case.2000 Mathematics Subject Classification. Primary: 43A32; Secondary: 43A90 47A16 .', 'arxivid': '1104.4479', 'author': ['Francesca Astengo ', 'Bianca Di Blasio '], 'authoraffiliation': [], 'corpusid': 119167606, 'doi': '10.1016/j.jmaa.2012.02.033', 'github_urls': [], 'n_tokens_mistral': 9167, 'n_tokens_neox': 8074, 'n_words': 4724, 'pdfsha': 'b604f70f03448f3e16bc48dbec3de61f227dcaa7', 'pdfurls': ['https://arxiv.org/pdf/1104.4479v1.pdf'], 'title': ['DYNAMICS OF THE HEAT SEMIGROUP IN JACOBI ANALYSIS', 'DYNAMICS OF THE HEAT SEMIGROUP IN JACOBI ANALYSIS'], 'venue': []} |
arxiv |
A New Comprehensive X-ray Spectral Model from the Post-shock Accretion Column in Intermediate Polars
2012. May 2014
Takayuki Hayashi
The Institute of Space and Astronautical Science/JAXA
3-1-1 Yoshinodai, Chuo-ku252-5210Sagamihara
Department of Physics
Tokyo Metropolitan University
1-1 Minami-Osawa192-0397HachiojiTokyo
Manabu Ishida
The Institute of Space and Astronautical Science/JAXA
3-1-1 Yoshinodai, Chuo-ku252-5210Sagamihara
Department of Physics
Tokyo Metropolitan University
1-1 Minami-Osawa192-0397HachiojiTokyo
A New Comprehensive X-ray Spectral Model from the Post-shock Accretion Column in Intermediate Polars
Mon. Not. R. Astron. Soc
0002012. May 2014arXiv:1307.7881v1 [astro-ph.SR] (MN L A T E X style file v2.2)accretionaccretion discs -methods: data analysis -fundamental pa- rameters -novaecataclysmic variables -white dwarfs -X-rays: stars
We model the post-shock accretion column (PSAC) of intermediate polars (IPs) with the specific accretion rate being floated in the range between 0.0001 and 100 g cm −2 s −1 and the metal abundance in the range between 0.1 and 2 times of the solar, and taking into account the gravitational potential with radial dependence, non-equipartition between ions and electrons, and ionization non-equilibrium. We fully take into account the dipole geometry for the PSAC. The specific accretion rate significantly affects the structure of the PSAC, and there is a critical rate below which the profiles of the density and temperature distributions deviate from those of the standard model. This happens when the specific accretion rate is 1 and 30 g cm −2 s −1 for the 0.7 and 1.2 M ⊙ white dwarf (WD), respectively, or the height of the PSAC becomes 1% of the white dwarf radius. Below the critical specific accretion rate, the present standard model is no longer valid. We calculate the spectra of the PSACs with the density and temperature distributions described above. Input parameters are the mass of the WD, the specific accretion rate, and the metal abundance. The spectral shape is constant and consistent with that of the standard model if the specific accretion rate is larger than the critical value, except for density-dependent emission lines. Below the critical specific accretion rate, on the other hand, the spectra soften as the specific accretion rate decreases. Associated with this, the maximum temperature of the PSAC becomes significantly lower than that of the standard model below the critical specific accretion rate. Although the ionization non-equilibrium are also considered in the spectral calculation, the effects are limited because the radiation from ionization nonequilibrium plasma is a few percent of the whole at most.
INTRODUCTION
Magnetic cataclysmic variables (mCVs) are binary systems made up of a Roche Lobe-filling late type star and a magnetized (B > 0.1 MG) white dwarf (WD). There are two subclasses in the mCV. One is the polar in which the magnetic field of the WD is so strong (B > 10 MG) that the WD rotation and the binary motion are synchronized. The other is the intermediate polar (IP), the subject of this paper, where the WD rotation is not synchronized with the binary revolution. In IPs, matter split over the Roche lobe of the secondary star initially forms an accretion disk, is funneled by the strong magnetic field within the Alfvén radius, and falls toward the WD surface nearly at the free fall velocity. Since the accreting matter becomes highly supersonic as it descends along the magnetic field line, a strong shock is ⋆ E-mail:[email protected] formed and the matter is heated up to a temperature of order 10 keV. The high temperature plasma formed below the shock front is cooled via optically thin thermal plasma emission, and finally settles onto the WD surface. The plasma flow in the downstream side of the shock is called the postshock accretion column (PSAC). X-ray spectra emitted from the IPs reflect structure of the PSAC such as radial distributions of the temperature and density. One of the most important parameter is the maximum temperature (Tmax) of the plasma. Since this quantity reflects the depth of the gravitational potential of the WD (∝ MWD/RWD), the WD mass can be estimated from the X-ray spectra with the aid of a theoretical WD mass-radius relation, for example, described by Nauenberg (1972),
RWD = 7.8×10 8 1.44 M⊙ MWD 2/3 − MWD 1.44 M⊙ 2/3 1/2 cm.
(1) Hōshi (1973) first considered a steady and spherically symmetric accretion flow onto WDs. He discussed that near the WD surface a shock is formed and a hot plasma between the shock front and the WD surface emits thermal radiation. Although he simply carried out arithmetic operations, he estimated some physical quantities, for example, an effective temperature and emission measure of the postshock plasma. Shortly after Hōshi (1973), Aizu (1973) analytically calculated the distributions of the temperature and the density along the plasma flow under the assumption that the thickness of the emission region, which corresponds to the height of the PSAC, is negligible compared with the WD radius. This assumption means that the gravitational potential can be considered as constant throughout the PSAC. Since his assumption appeared to be reasonable for high accretion rate systems and the analytically expressed temperature and density distributions are easily accommodated to the observed spectra, Aizu model had been used for evaluation of the observed spectra and WD mass estimation until a few tens of years after his publication, for example, in Fujimoto & Ishida (1997).
After that, a lot of theoretical studies were performed for the PSAC (e.g. Imamura & Durisen 1983, Woelk & Beuermann 1996, Canalle et al. 2005and Saxton et al. 2007). They included two-fluid effects, Compton cooling, effect of gravitational potential or dipolar geometry. Their calculations are, however, too complex to be applied to observed X-ray spectra. Wu, Chanmugam, & Shaviv (1994) and Cropper, Ramsay, & Wu (1998), on the other hand, beautifully simplified the PSAC model formulation, which enables us to use it for evaluating the observed spectra, and to extract information of the PSAC and the WD. They assumed one-temperature (i.e. equipartition between ions and electrons), one-dimensional and cylindrical geometry for the PSAC. Cropper et al. (1999) continued along the path of improvements to that technique by addressing the elimination of a negligible shock height assumption. In so doing, they explored and elucidated clearly for the first time the effects of including a radially varying gravitational potential along the PSAC. The model of Cropper et al. (1999) have been used for the resent WD mass measurements with X-ray spectra (Ramsay et al. 2000, Suleimanov, Revnivtsev, & Ritter 2005, Brunschweiger et al. 2009, Yuasa et al. 2010 and is said to be the present standard model of the PSAC. In fact, the standard model can reproduce observed spectra well.
While Cropper et al. (1999) considered the radially varying-gravitational potential, they hardly discussed the difference of the mass accretion rate per unit area called "specific accretion rate". Moreover, some studies using the standard model (e.g. Suleimanov, Revnivtsev, &Ritter 2005 andYuasa et al. 2010 for light WDs) noted that the PSAC structure is not influenced so much, and hence the WD masses estimated with the observed spectra are altered by the specific accretion rate only a little. As a matter of fact, Yuasa et al. (2010) investigated the influence of the specific accretion rate a on the WD mass estimation in the range a = 0.1-10 g cm −2 s −1 . As a results, they showed that the estimated WD mass is affected by less than ∼ 30% for a WD less massive than 1.2 M⊙ where most WDs are likely to belong.
However, some observations suggest that the specific accretion rate distributes in a wider range. For instance, the accretion rate of AE Aquarii and V1223 Sagittarii are estimated atṀ ∼ 10 14 g s −1 (Eracleous, Halpern, & Patterson 1991) and 8.4 ×10 16 g s −1 (Hayashi et al. 2011) with the standard model, respectively. Furthermore, Cropper, Ramsay, & Wu (1998) obtained the specific accretion rate of EX Hydrae of 0.001 g cm −2 s −1 as the best fit parameter, which is three orders of magnitude lower than that assumed in the standard model, although they did not take into account the radially varying-gravitational potential. Although these results are indirect evaluations, they suggest the specific accretion rate may be different by more than a few orders magnitudes among the IPs.
One of the remaining major issues of the standard model is that the mass of the WD and the height of PSAC in some IPs evaluated by the standard model are inconsistent with that derived from observations in low accretion rate or massive WD systems. The maximum temperature of the plasma in the peculiar IP AE Aquarii is 4.6 keV (Itoh et al. 2006). Based on the standard model, the low maximum temperature means that the WD of the AE Aquarii is no more massive than 0.2 M⊙ (see figure 2 in Yuasa et al. 2010 ), which is only one-fourth of the 0.79 M⊙ measured with the line Doppler measurement in optical band (Casares et al. 1996). In this system, owing to its fast WD spin with a period of ∼ 33 sec (Patterson 1979), the propeller effect plays an important roll and blows away most of the accreting matter (Wynn, King, & Horne 1997), which leads very low mass accretion rateṀ ∼ 10 14 g s −1 . The WD mass of EX Hydrae, 0.42±0.02 M⊙ (Yuasa et al. 2010) estimated with the standard model contradicts the optical measurement 0.79±0.26 M⊙ (Beuermann & Reinsch 2008). We believe that the latter measurement is fairly reliable because EX Hydrae is a double-lined eclipsing IP. This system is also a low accretion-system,Ṁ = 2.8×10 15 g s −1 (Suleimanov, Revnivtsev, & Ritter 2005). The WD mass of a nova IP GK Perseus is estimated at 1.15 M⊙ with "nova universal decline law" (Hachisu & Kato 2007), which is much greater than the X-ray estimations, 0.59±0.05 M⊙ by Suleimanov, Revnivtsev, & Ritter (2005), 0.90±0.12 M⊙ by Brunschweiger et al. (2009) and 0.92 +0.39 −0.13 M⊙ by Landi et al. (2009). Although this IP is very high accretion-rate system of 81.5×10 16 g s −1 , the massive WD (Suleimanov, Revnivtsev, & Ritter 2005) causes a relatively large error in the mass estimation. As for the height of the PSAC in EX Hydrae, the standard model expects about 2×10 6 cm, comparable to 0.2% of the WD radius (Yuasa et al. 2010), Allan, Hellier, & Beardmore (1998) derived the height as tall as the WD radius. These discrepancies show that the standard model should further be improved in some cases, especially for the cases of the lower specific accretion rate and of the massive WD.
In order to resolve the issues described above, we modify the standard model in the following four points; (1) the specific accretion rate, which is fixed at 1 g cm −2 s −1 in the standard model, is floated in the wide range from 0.0001 to 100 g cm −2 s −1 . The lower specific accretion rate enhances the height of the PSAC because the density becomes lower, which results in a longer cooling time. The PSAC extension further requires;
(2) the dipole should be considered as the PSAC geometry instead of the cylinder used in the standard model, which reduces the density further because of the funnel shape of the PSAC. Due to the density reduction; (3) the non-equilibrium between ions and electrons and (4) the ionization non-equilibrium should be taken into account which are not considered in the standard model. These modifications are more important for the IPs holding the massive WD because the more massive WD is smaller in size and the PSAC height becomes more significant compared with the WD radius at a higher specific accretion rate. This paper is organized as follows. The calculation scheme is described in section 2. In section 3, we investigate the PSAC structure at various specific accretion rate and metal abundance. In section 4, the spectra emitted from the PSACs are shown and their dependence on the WD mass, the specific accretion rate and the metal abundance are discussed. Finally, we summarize our results in section 5.
MODELING THE POST-SHOCK ACCRETION COLUMN
We modeled the PSAC according to the guidelines described in the previous section. Following the method of Cropper et al. (1999) and Suleimanov, Revnivtsev, & Ritter (2005), the distributions of the density and temperature were calculated. The simultaneous differential equation involving the mass continuity equation
d dz (ρvS) = 0,(2)
the momentum equation
d dz (ρv 2 + P ) = − GMWD z 2 ρ − ρv 2 S dS dz ,(3)
the energy equation
v dP dz + γP dv dz = −(γ − 1) ε − ρv 3 2S dS dz (4)
and the ideal-gas law
P = ρkT µmH (5)
describes the PSAC structure. Here, z is the spatial coordinate shown in figure 1 whose origin is the WD center, v is the bulk velocity, ρ is the mass density, T is the averaged temperature, P is the thermal pressure of the plasma, γ = 5/3 is the adiabatic index, and µ = 0.62 is the mean molecular weight. S is the cross-section of the PSAC and assumed to be proportional to a power law function of z,
S ∝ z n .(6)
The power-law index n of 0 and 3 correspond to cylinder and dipole, respectively. ε is the cooling rate via optically thin thermal radiation given by
ε = ρ µmH 2 Λ(T ),(7)
where Λ is the cooling function. We adopt the Collisional Ionization Equilibrium (CIE) cooling function calculated by SPEX package (Schure et al. 2009). One may doubt that we should use a cooling function that reflects the nonequilibrium effects correctly. However, the part of the PSAC where the equipartition between the electron and the ion is not achieved is of lower density by a few orders of magnitude than that in the equilibrium part, and hence we can use the CIE cooling function throughout the PSAC. In fact, we substituted the emissivity of thermal bremsstrahlung for the CIE cooling function in the part where the electron temperature is below 90, 95, 99 and 99.9% of the averaged temperature. As a results, all these hybrid cooling functions gave nearly identical results. The second term of the right-hand side of the equation 3 describes the conversion of thermal energy into the kinematic energy in the PSAC because of decreasing cross-section as the Venturi nozzle. The integral of equation (2) is
ρvS = aS = a(4πR 2 WD f ) =Ṁ ,(8)
where a is the mass accretion rate per unit area or "specific accretion rate" at the WD surface. f andṀ are fractional accretion area and the mass accretion rate. In the standard model, a = 1 g cm −2 s −1 and n = 0 for any IPs.
Equations (3) and (4) can be transformed with a and z ′ = z0 − z, where z0 is the shock coordinate (see figure 1),
dv dz ′ = g(z ′ ) 1 v − 1 a dP dz ′ ,(9)dP dz ′ = (γ − 1)(ε − ρv 3 2S dS dz ′ )a + g(z ′ )γP ρ γP − av ,(10)
where
g(z ′ ) = GMWD (z0 − z ′ ) 2 .(11)
Equations (9) and (10) were solved from the top of the
PSAC, z = z0(z ′ = 0) to WD surface, z = RWD(z ′ = z0 − RWD)
with the following boundary conditions at z = z0 assuming the strong shock at the top of the PSAC,
v0 = 0.25 2GMWD/z0 (12) ρ0 = a v0 ,(13)P0 = 3av0,(14)T0 = 3 µmH k v 2 0(15)
and soft landing,
vWD = 0 (at WD surface).(16)
The boundary conditions are uniquely given by specifying MWD, a, and z0. Of them, the shock position z0 matching the boundary conditions was found by iteration (shooting method). In so doing, we utilize the mass-radius relation of WDs given by equation 1. Unlike the standard model, we consider nonequipartition between ions and electrons. After ions are immediately heated up by the strong shock, electrons are heated by Coulomb scattering with the ions. Time variation of the electron temperature is written as
dTe dt = Ti − Te teq ,(17)
where teq is the time scale of equipartition calculated with (Spitzer 1962). Here, T (in Kelvin), A and Z are temperature, atomic weight and charge, and subscript e and i imply electron and ion, respectively. ni is the number density of the ion in a unit of cm −3 . For practical calculation, we followed the method of Wong & Sarazin (2009) in which eq.(17) can be rewritten as
teq = 5.87 AeAi ni cm −3 Z 2 e Z 2 i ln Λ Ti K Ai + Te K Ae 3/2 s. (18)dτ dt = 2 ln Λ 503 Z 2 i Ai n T 3/2 τ −3/2 (1 − τ ) s −1(19)
where T is the averaged temperature;
T = neTe + niTi ne + ni(20)
and τ is relative electron temperature,
τ ≡ Te T .(21)
ln Λ is the Coulomb logarithm and can be approximated as ln Λ ∼ 15.9 + ln Te
10 8 K − ln ne 10 16 cm −3 1/2 .(22)
The ionization of heavy element proceeds by impacts of electrons heated up by the interaction with the ions. Therefore, the ionization temperature trails the electron temperature. The angle bracket term is the mean value of the ratio of the square of ion charge and the atomic number, which equals 1 for our model of a pure hydrogen and helium gas. We set Ae = 1/1836.
CALCULATION RESULTS
3.1 Consistency with Previous Works Figure 2 shows the temperature and density distributions of the cylindrical PSAC in the case of a = 1 g cm −2 s −1 , M = 0.7 M⊙, and Z = Z⊙. The temperature is normalized by their maximum values. The density, on the other hand, is normalized at a point one thousandth of the PSAC height, since it diverges at the WD surface. The profiles of these distributions closely resemble those from the previous studies of Suleimanov, Revnivtsev, & Ritter (2005) and Yuasa et al. (2010). Moreover, the shock height of our calculation is 0.017 RWD, which is close to that of Suleimanov, Revnivtsev, & Ritter (2005), 0.018 RWD and Yuasa et al. (2010), 0.013 RWD. The electron temperature does not catch up the averaged temperature in the top of 20% of the PSAC. However, the density of that region is low. This suggests that the effect of non-equipartition between ions and electrons on the X-ray spectrum is limited.
[!h] 0 5×10 −3 0.01 0.015 0.01 0.1 1 T/T max , ρ/ρ max (z−R WD )/R WD M WD = 0.7 (M O . ) R WD = 0.112 (R O . ) a = 1 (g cm −2 s −1 ) h = 0.017 (R WD ) T max = 28.2 keV T/T max T e /T max ρ/ρ max
Dependency on the specific accretion rate
We investigated influence of the specific accretion rate, which was fixed at a = 1 g cm −2 s −1 in the standard model, on the PSAC structure in the range between 0.0001 and 100 g cm −2 s −1 .
3.2.1 Density distribution and critical specific accretion rate Figure 3 shows the density distributions of the cylindrical and dipolar PSAC with the specific accretion rate in the case of MWD = 0.7 M⊙ of 0.0001, 0.01, 1 and 100 g cm −2 s −1 . In this figure, the right ends of each profile corresponds to the shock front. The other ends are terminated at 0.1% of a PSAC height of each case. This figure means that the PSAC becomes taller with a lower specific accretion rate due to a longer cooling time. When the specific accretion rate is suf- ficiently high (a 1 g cm −2 s −1 for the 0.7 M⊙ WD), the density increases toward the WD surface with a power-law function of the distance from the WD surface, and its profile agrees between the dipolar and cylindrical geometries. On the other hand, when the specific accretion rate is sufficiently low (a ≪ 1 g cm −2 s −1 for the 0.7M⊙ WD) the density distribution deviates from the power law. At the same time, the density distributions of the dipolar PSACs become different from those of the cylindrical because the difference between the two geometries emerges when the PSAC extends upwards and becomes compatible with the WD radius. We hereafter refer to the specific accretion rate below which the density profile of the cylindrical PSAC starts to deviate from that of the dipolar PSAC as the critical specific accretion rate acrit. This definition of acrit implies that the standard model is no longer valid in the regime a < acrit. We systematically investigated acrit as a function of the WD mass and found that a = acrit occurs when the PSAC height ≃ 0.01RWD, irrespective of the WD mass; for instance it is 1 and 30 g cm −2 s −1 for the 0.7 and 1.2M⊙ WD, respectively. We refer to the readers to consult figure. 5 to find acrit for a WD with any given mass.
Temperature distribution
The distributions of the averaged and electron temperature are shown in figure 4. Like the density distributions, the shapes of the temperature distributions agree between the geometries if a acrit. If, on the other hand, a ≪ acrit, the averaged temperature reduces from that of the standard model over the whole PSAC and its distribution flattens. Although the averaged temperature monotonically decreases toward the WD surface in the standard model, that of our cylindrical PSAC, drawn with black color in figure 4, shows a peak in the middle of the PSAC at low enough specific accretion rate. This is because energy input by gravity overcomes cooling energy loss since the low density reduces the cooling rate, and the tall PSAC retains larger amount of gravitational energy to be released below the shock front. For the dipolar cases, on the other hand, the temperature decrease is even faster than the cylindrical. This happens whenever a cross section of a subsonic flow shrinks along the streamline, such as the Venturi nozzle. In such a case, the bulk velocity of the flow increases at the expense of the thermal energy, which reduces the temperature. The averaged temperature of the dipolar PSAC monotonically decreases as the flow descends the PSAC for the 0.7 M⊙ WD throughout the range a = 0.0001 -100 g cm −2 s −1 unlike the cylindrical case. However, that of the 1.2 M⊙ WD shows a local minimum in the middle of the PSAC, as shown in the bottom right panel of figure 4. The heat transfer from the ion to the electron becomes slower with a lower density. At a very low specific accretion rate such as 0.0001 g cm −2 s −1 the non-equipartition area extends over the 80% region of the PSAC.
PSAC height
Relations between the height of the PSAC and the specific accretion rate are shown in figure 5 for the 0.4, 0.7 and 1.2 M⊙ WDs. The PSAC constantly extends upwards with the lower specific accretion rate, but the slope of the PSAC height abruptly changes at a certain value of a. At around the high end of a, the height is proportional to a −1 , and the height of the PSAC is almost identical between the two, PSAC geometries, whereas at around the low end of a the heights are in proportion to a −0.3 and a −0.15 for the cylindrical and dipolar PSACs, respectively. It is interesting to note that the transition between these two regimes occurs at the PSAC height of about 0.2 RWD for any mass of the WDs. The transition appears in larger specific accretion rate for a more massive WD, because the PSAC height easily becomes significant relative to WD radius due to its small radius. The radial extent of the PSAC can be as large as the WD radius in the lower specific accretion rate, which may resolve the problem about the observed PSAC height referred section 1.
Maximum temperature
The maximum temperature of electrons and that averaged over ions and electrons are shown as a function of the specific accretion rate in figure 6 for the 0.4, 0.7 and 1.2 M⊙ WDs. The maximum temperature is indicative of the mass of the WD. In the region a acrit, the maximum temperatures of the two geometries are identical and constant, as is predicted by the standard model. Below the acrit, on the other hand, the maximum temperatures differ between the two geometries. In the dipolar geometry, the decrease of the maximum averaged temperature associated with the decrease of the specific accretion rate is faster than that in the cylindrical case. This is because the averaged temperature monotonically decrease for the dipolar PSAC while the hottest region emerges in the middle of the PSAC in the cylindrical geometry, as shown in the bottom four panels of figure 4. The electron maximum temperature reduces also faster for the dipolar geometry because the density is smaller than in the cylindrical geometry especially at around the top of the PSAC, which delays accomplishment of the equipartition.
Dependency on the metal abundance
We investigated influence of metal abundance on the PSAC structure as well as the specific accretion rate and the WD mass. In figure 7 we show the PSAC structures by assuming 0.1, 0.5, 1.0 and 2.0 Z⊙ with the cylindrical or dipolar geometries. The top panel of figure 7 displays examples of the cylinder PSACs on the 0.7 M⊙ WD. Note that, although cylindrical geometry is assumed, the profiles can be regarded as those of the dipolar PSACs, because the assumed specific accretion rate is sufficiently high. The middle and bottom panels show the temperature distributions for cylindrical and dipolar PSACs on the 0.7 M⊙ WD, respectively. These results mean that the influence of the metal abundance on the PSAC structure is even less significant than that of the specific accretion rate. Moreover, since the abundances of IPs are generally in the range from 0.1 to 0.6 times of solar abundance (Yuasa et al. 2010), the influence of the metal abundance on the PSAC structure is limited. However, the metal abundance significantly affects X-ray spectrum, especially line spectra (see section 4). Therefore, we remain the metal abundance as an input parameter in the following section.
Spectral calculation method
In order to calculate the spectra, the PSAC is divided into evenly spaced one hundred segments within each of which the physical quantities can be regarded as constant, and then the one hundred partial spectra are summed up. Spectra emitted from each segments are calculated with one temperature plasma emission models in SPEX package (Kaastra et al. 1996).
In calculating the partial X-ray spectra, we need to take into account the non-equilibrium between ions and electrons for part of the PSAC and hence used the following two X-ray spectrum models; one is the Cie (Collisional ionization equilibrium) model and the other is Neij (Non-Equilibrium Ionization Jump model; Kaastra & Jansen (1993)) model, both included in the SPEX package. In general, equipartition between ions and electrons and ionization state of the ions in plasma reach thermal equilibrium if the product of the electron number density and the elapsed time since the shock becomes greater than net > 10 12 cm −3 s (Masai 1984). Accordingly, for the PSAC segments at which net > 10 12 cm −3 , the Cie model is adopted where the input temperature is common among the ions, the electrons and the ionization. For the PSAC segments at which net < 10 12 cm −3 , on the other hand, the Neij model is adopted, which is characterized by the three parameters nt, Tion and T . The Tion and T are initial ionization temperature and initial common temperature of the ions and the electrons , respectively. Thus, the Neij model can only treat the case that the ions and the electrons share a common kinematical temperature, whereas our hydrodynamical model predicts that they in real have different temperatures near the top of the PSAC. However, since the ionization process is governed by electron impacts to the ions, we adopt the electron temperature for T . Tion is used as the ionization temperature that is common among all elements reaching the start points of each segment. The initial ionization temperature Tion of a segment is evaluated from the average charge of iron ion achieved by the previous PSAC segment through a relation between the average charge of iron ion and the ionization temperature, which is calculated by the SPEX (figure 8). Only for the first segment laying on the top of the PSAC, Tion is set to 0.002 keV, the limit of the Neij model. Figure 9 shows the ionization temperature calculated by the method defined above. For this figure MWD = 1.2 M⊙ and a = 0.001 g cm −2 s −1 are adopted where the nonequipartition are prominent (figure 4). The ionization temperature does not catch up that of electron in about 70% of the PSAC from its top. We note, however, that the density of the segments in the ionization non-equilibrium area is smaller than those in equilibrium by one or two orders magnitudes, which implies that the ionization non-equilibrium does not affect the resultant total X-ray spectrum significantly.
X-ray spectra
In figure 10, resultant spectra are shown for the cases MWD = 0.7 M⊙, Z = Z⊙, a = 100, 1, 0.01 and 0.0001 g cm −2 s −1 , as well as ratios of them to that of a = 1 g cm −2 s −1 . The results are shown both for the cylinder and dipole geometries. In a higher specific accretion rate domain, the flux increases in proportion to the specific accretion rate. Since the profiles of the temperature and density distributions resemble each other between the two PSAC geometries in the cases of a = 1 and 100 g cm −2 s −1 , their spectral shapes are almost identical. However, at Helike iron Kα line (∼6.7 keV), the spectral shapes of the a = 1 and 100 g cm −2 s −1 cases are considerably different in both geometries. This is because the density of PSAC reaches the critical density of iron ∼ 10 18 cm −3 between a = 1 and 100 g cm −2 s −1 and relative intensities of the He-like Kα triplet alter with the density (figure 11). On the other hand, the spectral shapes in the cases of a = 0.01 and 0.0001 g cm −2 s −1 clearly deviate from that of a = 1 g cm −2 s −1 for both PSAC geometries, because the temperature of the PSACs are significantly reduced as the specific accretion rate becomes lower ( figure 4 and 6). Change of ratios of H-and He-like iron Kα lines (figure 11) and deformation of the continuum above 10 keV energy bands are especially prominent, and observationally important because the energy bands where these two phenomena emerge are hard to be influenced by complex and heavy absorbers generally detected in the X-ray spectra of IPs. These spectral changes with the specific accretion rate is larger for the dipolar PSAC, and the spectrum of the dipolar PSAC is softer than that of the cylindrical at the same specific accretion rate. Since these deformations emerge in lower specific accretion rate systems, the specific accretion rate should be considered as an important parameter in order to extract physical parameters such as WD masses from X-ray spectra. At the same time, the spectral deformation potentially enables us to measure the specific accretion rate with X-ray spectroscopy which may give geometrical information of the height of PSAC, the accreting area on the WD surface and shape of the PSAC.
Note that influence of the ionization non-equilibrium on X-ray spectrum is not significant because of the low density of such PSAC domains for both geometries, although that effect is properly considered in this work. For example, we obtained a result that the dipolar PSAC with MWD = 0.7 M⊙, a = 0.0001 g cm −2 s −1 and Z = Z⊙ at which parameters the non-equilibrium effect manifests itself prominently, shows a spectrum more intense by about 4% around 0.7 keV and less intense by 0.01-0.02% between 10 and 100 keV at most than the ionization equilibrium spectrum. At the energy around 0.7 keV, an iron L line forest is enhanced due to the ionization non-equilibrium effect because the ionization does not proceed compared with the equilibrium case. On the other band, bremsstrahlung radiation dominating over the latter energy band (> 10 keV) is weaker for the non-equilibrium model because of low electron temperature. Even, in the energy band above 5 keV that is important for the study of the PSAC for most mCVs (Ezuka & Ishida 1999), the difference is only about 0.2%, which does not matter for the current observation quality.
The problem of the discrepancy of the WD mass measurement ( §1) may be resolved if we treat the specific accretion rate as a free model parameter. The X-ray spectrum of the PSAC is softer than that of the standard model in the domain a < acrit, which makes the resultant WD mass more massive. We will verify this possibility by applying our model to observations in the forthcoming paper.
SUMMARY
We calculated the density and temperature distributions of the PSAC in IPs by taking into account dependence on the specific accretion rate, the dipolar magnetic geometry, nonequipartition between electrons and ions, ionization nonequilibrium, and release of the gravitational potential with the form proportional to r −1 . In particular, the specific accretion rate is floated over the wide range between 0.0001 to 100 g cm −2 s −1 . This is the first comprehensive PSAC model that all these factors are fully taken into account. With the density and temperature profiles, we constructed a spectral model by dividing the PSAC radially into a hundred segments, and by integrating the spectrum from each segment calculated with the SPEX package. In our modeling, the free parameters are the WD mass, the specific accretion rate and the metal abundance. We found difference of the specific accretion rate significantly alters the profiles of the density and temperature distributions. As long as the specific accretion rate is high enough, the density and temperature distributions of our modeling are consistent with those of the present standard model. There is, however, a critical specific accretion rate below which the profiles of the density and temperature distributions significantly deviate from those of the standard model. The standard model is no longer valid, if the specific accretion rate is below the critical value acrit, which is about 1 and 30 g cm −2 s −1 for the 0.7 and 1.2M⊙ WD, respectively, or when the heigh of the PSAC reaches about 1% of the WD radius. In addition, the profiles become different between the cylindrical and the dipolar geometries. The temperature profile shows the peak in the middle of the PSAC for the cylindrical geometry whereas that for the dipole geometry declines toward the WD surface more rapidly than that of the standard model. Since the WD radius reduces for more massive WD, the critical specific accretion rate is lower for the IPs holding the less massive WD. Dipole (d) Figure 10. X-ray spectra emitted from (a) cylindrical and (b) dipolar PSACs for M WD = 0.7 M ⊙ , and a = 100, 1, 0.01 and 0.0001 g −1 cm −2 s −1 . The spectral ratios to the spectrum with a = 1 g −1 cm −2 s −1 for the (c) cylindrical and (d) dipolar. Abundance is assumed to be one solar.
The profiles of the density and temperature distributions significantly changed with the decrease of the specific accretion rate below the critical value, and mutual difference of the profiles between the two geometries is also enhanced. The non-equipartition between electrons and ions are significant in lower density domains, and hence, the nonequipartition is more significant for the dipolar PSAC. With the low specific accretion rate of 0.0001 g cm −2 s −1 the nonequipartition domain occupies 80% of the PSAC from its top for IPs holding a 0.7 M⊙ WD. We also investigated the influence of the metal abundance, and found that it hardly influences the PSAC structure between 0.1 and 2 solar abundance which covers the entire IP population so far observed.
We calculated the X-ray spectra with the density and temperature distributions and found that the X-ray spectra depend on the specific accretion rate. The X-ray spectra with the higher specific accretion rate than the critical value is almost constant except for the He-like iron Kα emission line due to the density dependence of the He-like triplet of the iron Kα line. When the specific accretion rate is smaller than the critical value, since the temperature in the PSAC reduces, the X-ray spectra soften, which is more prominent for the dipolar PSAC. Specifically, the continuum component above 10 keV and the ratio of H-like line to that of He-like reduces as the specific accretion rate decreases. The ionization non-equilibrium is not significant on X-ray spectra because of lower density than in equilibrium domain by a few orders of magnitudes.
Figure 1 .
1Geometries of the PSAC models. Dashed lines show the dipolar geometry.
Figure 2 .
2Averaged temperature (black solid line), electron temperature (black dotted line) and density (red solid line) distributions of a cylindrical PSAC for an IP of M ⊙ = 0.7 M ⊙ and a = 1 g cm −2 s −1 . The parameters are common to figure 2 of Suleimanov, Revnivtsev, & Ritter (2005) and figure 3 of Yuasa et al. (2010).
Figure 3 .
3Density distributions of the cylindrical (black) and dipolar (red) PSACs for the WD mass of 0.7 M ⊙ and a of 0.0001, 0.01, 1, 100 g cm −2 s −1 . The right ends of the distributions correspond to the tops of the PSACs. The other ends are terminated at 0.1% of the PSAC height.
Figure 4 .
4Averaged (solid) and electron (dotted) temperature distributions of the cylindrical (black) and dipolar (red) PSACs for the WD mass of 0.7 (left columns) and 1.2 (right columns) M ⊙ and a of 0.0001, 0.01, 1, 100 g cm −2 s −1 from bottom to top panels.
Figure 5 .
5The PSAC heights for the 0.4, 0.7 and 1.2 M ⊙ WDs as a function of the specific accretion rate. Black and red lines show the cylindrical and dipolar cases, respectively. The horizontal dotted line represents 20% of the WD radius, which defines the threshold specific accretion rate. Note that the heights in this figure are normalized by each WD radii.
Figure 6 .
6Averaged (solid) and electron (red) maximum temperatures relating the specific accretion rate for 0.4, 0.7 and 1.2 M ⊙ of the WDs. Black and red lines show the cylindrical and dipolar PSACs, respectively.
Figure 7 .
7Averaged (solid) and electron (dotted) temperature distributions for abundances of 0.1 (black), 0.5 (red), 1.0 (green) and 2.0 (blue) times the solar abundance. Top panel shows that the distributions for the cylindrical PSAC and a = 1 g cm −2 s −1 , which is almost identical to that of the dipolar PSACs with the same specific accretion rates. The middle and bottom panels show the distributions for the cylindrical and dipolar PSACs with a = 0.0001 g cm −2 s −1 . The WD mass of 0.7M ⊙ is adopted.
Figure 8 .
8Relation between the average charge of iron ion and the ionization temperature calculated by the SPEX.
Figure 9 .
9Averaged (black solid), electron (black dotted) and ionization (red solid) temperatures for the case M WD = 1.2 M ⊙ and a = 0.001 g cm −2 s −1 .
Figure 11 .
11Iron Kα lines (top panel) emitted from dipolar PSACs for M WD = 0.7 M ⊙ , and a = 100, 1, 0.01 and 0.0001 g −1 cm −2 s −1 . The ratio to the iron Kα lines with a = 1 g −1 cm −2 s −1 (bottom panel). Abundance is assumed to be one solar.
Photons s −1 keV −1
X-RAY SPECTRAIn this section, we calculate X-ray spectra emitted from the PSACs using the temperature and density distributions calculated in section 3.
ACKNOWLEDGEMENTThe authors would like to thank Prof. Ohashi T., Prof. Masai K. and Associate Prof. Ishisaki T. for their very useful comments.
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| {'fraction_non_alphanumeric': 0.049135727185308536, 'fraction_numerical': 0.03794826090015714, 'mean_word_length': 4.105124521072797, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 5, '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': 'We model the post-shock accretion column (PSAC) of intermediate polars (IPs) with the specific accretion rate being floated in the range between 0.0001 and 100 g cm −2 s −1 and the metal abundance in the range between 0.1 and 2 times of the solar, and taking into account the gravitational potential with radial dependence, non-equipartition between ions and electrons, and ionization non-equilibrium. We fully take into account the dipole geometry for the PSAC. The specific accretion rate significantly affects the structure of the PSAC, and there is a critical rate below which the profiles of the density and temperature distributions deviate from those of the standard model. This happens when the specific accretion rate is 1 and 30 g cm −2 s −1 for the 0.7 and 1.2 M ⊙ white dwarf (WD), respectively, or the height of the PSAC becomes 1% of the white dwarf radius. Below the critical specific accretion rate, the present standard model is no longer valid. We calculate the spectra of the PSACs with the density and temperature distributions described above. Input parameters are the mass of the WD, the specific accretion rate, and the metal abundance. The spectral shape is constant and consistent with that of the standard model if the specific accretion rate is larger than the critical value, except for density-dependent emission lines. Below the critical specific accretion rate, on the other hand, the spectra soften as the specific accretion rate decreases. Associated with this, the maximum temperature of the PSAC becomes significantly lower than that of the standard model below the critical specific accretion rate. Although the ionization non-equilibrium are also considered in the spectral calculation, the effects are limited because the radiation from ionization nonequilibrium plasma is a few percent of the whole at most.', 'arxivid': '1307.7881', 'author': ['Takayuki Hayashi \nThe Institute of Space and Astronautical Science/JAXA\n3-1-1 Yoshinodai, Chuo-ku252-5210Sagamihara\n\nDepartment of Physics\nTokyo Metropolitan University\n1-1 Minami-Osawa192-0397HachiojiTokyo\n', 'Manabu Ishida \nThe Institute of Space and Astronautical Science/JAXA\n3-1-1 Yoshinodai, Chuo-ku252-5210Sagamihara\n\nDepartment of Physics\nTokyo Metropolitan University\n1-1 Minami-Osawa192-0397HachiojiTokyo\n'], 'authoraffiliation': ['The Institute of Space and Astronautical Science/JAXA\n3-1-1 Yoshinodai, Chuo-ku252-5210Sagamihara', 'Department of Physics\nTokyo Metropolitan University\n1-1 Minami-Osawa192-0397HachiojiTokyo', 'The Institute of Space and Astronautical Science/JAXA\n3-1-1 Yoshinodai, Chuo-ku252-5210Sagamihara', 'Department of Physics\nTokyo Metropolitan University\n1-1 Minami-Osawa192-0397HachiojiTokyo'], 'corpusid': 118394517, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 13080, 'n_tokens_neox': 11049, 'n_words': 7284, 'pdfsha': '9f7c04d3840aa876f593f4baf8f967c04f492d07', 'pdfurls': ['https://arxiv.org/pdf/1307.7881v1.pdf'], 'title': ['A New Comprehensive X-ray Spectral Model from the Post-shock Accretion Column in Intermediate Polars', 'A New Comprehensive X-ray Spectral Model from the Post-shock Accretion Column in Intermediate Polars'], 'venue': ['Mon. Not. R. Astron. Soc']} |
arxiv |
Identification of the Factors Affecting the Reduction of Energy Consumption and Cost in Buildings Using Data Mining Techniques
Imtiaz Ahmed [email protected]
Department of Industrial & Management Systems Engineering
[email protected] ty
West Virginia Universi 4 Morgantown26505WV
Rahim Khanizad
School of Industrial Engineering
University of Science and Technology
TehranIran, Iran
Hadi Sahebi [email protected]
Science and Technology
sity of School of Industrial Engineering
TehranIran, Iran Univer
Hamed Khosravi [email protected]
School of Industrial Engineering
Iran University of Science and Technology
TehranIran
Identification of the Factors Affecting the Reduction of Energy Consumption and Cost in Buildings Using Data Mining Techniques
Corresponding Author: *Building Energy Consumption OptimizationConsumption Pattern IdentificationMachine Learning AlgorithmsFeature SelectionPattern Recognition
Optimizing energy consumption and coordination of utility systems have long been a concern of the building industry. Buildings are one of the largest energy consumers in the world, making their energy efficiency crucial for preventing waste and reducing costs. Additionally, buildings generate substantial amounts of raw data, which can be used to understand energy consumption patterns and assist in developing optimization strategies. Using a real-world dataset, this research aims to identify the factors that influence building cost reduction and energy consumption. To achieve this, we utilize three regression models (Lasso Regression, Decision Tree, and Random Forest) to predict primary fuel usage, electrical energy consumption, and cost savings in buildings. An analysis of the factors influencing energy consumption and cost reduction is conducted, and the decision tree algorithm is optimized using metaheuristics. By employing metaheuristic techniques, we fine-tune the decision tree algorithm's parameters and improve its accuracy. Finally, we review the most practical features of potential and nonpotential buildings that can reduce primary fuel usage, electrical energy consumption, and costs.
Introduction
Over the last few decades, there has been a marked increase in the use of energy in developing countries, a trend that is expected to continue in the near future [1]. The importance of energy conservation has recently received significant attention from a number of stakeholders, including governments, industry, academia, and other organizations. It is believed that this growing interest in energy conservation is a consequence of the growing demand for energy and the declining supply of energy resources [2]. For instance, in China, building energy consumption accounted for 28% of the total energy consumption in 2011, and it is expected to reach 35% by 2020, according to statistical data [3]. Similarly, in the United States, building energy consumption comprises approximately 39% of the total energy consumption [4]. According to the International Energy Agency (IEA), residential and commercial buildings are responsible for 32% of final energy consumption [5]. Many buildings consume approximately 20% more energy than is required as a result of incomplete construction or a failure to follow the intended design. The issue arises when facilities are not operated as originally designed, resulting in inefficient energy use [6].
In recent years, people have been spending more and more time indoors [7,8], and residential energy consumption has increased significantly. Buildings contribute significantly to global energy consumption and greenhouse gas emissions. To mitigate these negative impacts, buildings must adopt energy-efficient and sustainable practices [9]. So, predicting building energy consumption is essential for building managers to make better decisions that improve energy utilization rates [10]. However, predicting building consumption is difficult because climate, population, and seasonal variations create nonlinear patterns [11]. Currently, buildings provide data that can be used to extract useful insights, patterns, or knowledge from them [12]. It also provides an opportunity to uncover hidden data, improve our understanding of energy usage, and create strategies for reducing energy consumption. Nevertheless, extracting valuable information from the data can be challenging without advanced data analysis techniques [13][14][15].
Developing prediction models often involves the use of popular methods such as machine learning and artificial intelligence-based approaches [16][17][18]. Models based on machine learning are more effective at capturing the complex relationships between building-level characteristics and energy consumption since they have fewer restrictions regarding the statistical relationships among variables [19]. The algorithms employed for developing energy consumption prediction models possess certain advantages and disadvantages [20]. The commonly used supervised machine learning algorithms for model training include SVM, ANN, decision trees, and other statistical algorithms [21]. In addition to its flexibility, decision tree algorithm can be improved as the amount of training data increases [22]. The use of datadriven models presents a practical approach to predict energy consumption [23].
As a result, in this study, we utilize three different models (lasso regression, decision tree, and random forest) to predict energy consumption and cost in a real-world dataset. We aim to identify and rank the factors that affect energy consumption and related costs in buildings. Data visualization is used to observe and uncover valuable relationships. Three different strategies are used to select features for the study. After predicting with the selected models and features, different scenarios are evaluated and compared with several criteria. Using genetic algorithm, the performance of the decision tree algorithms for prediction are improved. Finally, an analysis of energy consumption and cost reduction based on building variables is discussed considering the most efficient models identified following the evaluation.
Literature Review
In recent times, there has been significant focus on predicting the energy consumption of buildings [24], which has led to the development and implementation of various approaches for tackling related problems [25]. To improve the operational performance of building energy systems, data mining techniques are commonly employed to extract meaningful information from large sets of building operation data [26]. Typically, these methods fall into two main categories: supervised and unsupervised [27]. The use of data mining technologies in the building industry has been extensively studied over the last ten years, with several literature reviews published on the subject.
A decision tree algorithm was utilized by Yang G et al. in 2010 to optimize building energy consumption [28]. In the same year, N. Giatani et al. analyzed energy consumption data from 1100 schools to identify patterns at the building level. They used clustering techniques (kmeans) and Matlab software to define heating energy consumption information in five clusters, which were analyzed [29]. In 2011, Wall et al. used hierarchical clustering algorithms to diagnose faults in HVAC systems, aiming to identify operational patterns [30]. The same year, R.S. Jota and colleagues used hierarchical clustering to predict building electricity consumption by identifying common consumption patterns [31]. F.W. Yu and colleagues used clustering techniques and SPSS software to evaluate the behavior and performance of the chiller system in two studies conducted in 2012 [32]. In the same year, a decision tree algorithm was applied to predict peak electrical energy demands [33]. To predict lighting energy consumption in 2013, Liu D and colleagues compared artificial neural networks with SVMs [34].
A year after, Tang et al. used the k-means algorithm to cluster the entire data before developing prediction models. They claimed that this approach reduces the prediction error and the computational burden. They employed this approach for modeling and predicting HVAC systems [35]. In 2015, hierarchical clustering was utilized to identify typical energy consumption patterns, demonstrating that the proposed method can effectively predict energy consumption and peak demand with high accuracy [36]. A method based on artificial neural networks was proposed by Deb et al. (2016) for forecasting cooling load in the building sector in the presence of data related to energy consumption. The authors used R 2 value to evaluate the results [37]. Li et al. in 2017 compared four machine learning models in order to forecast the energy consumption in a retail building. As a result of their analysis, the Extreme Learning Machine (ELM) model was found to be the most efficient model in terms of forecasting [38].
In 2018, Ma et al. evaluated building energy performance. They utilized a hierarchical clustering algorithm in their research. In addition, they used advanced techniques such as dendrograms and heat maps to understand energy consumption behaviors in the building [39]. Liu et al. conducted accuracy analyses and compared models in 2019. The accuracy analyses were based on different types of buildings. As epidemic models, they compared artificial ANNs and SVMs based on their prediction process complexity, the accuracy of the results, and the number of inputs required [40]. One of the crucial aspects of machine learning models is parameter tuning. Consequently, Seyedzadeh et al. proposed a method for optimizing machine learning models for predicting heating and cooling loads in building energy consumption. This method employed multi-objective optimization techniques with evolutionary algorithms to explore the parameter space [41]. Random Forest, a widely used and significant machine learning model, was utilized by Pham et al. in the same year for short-term energy consumption prediction. The proposed model estimated multiple buildings' hourly energy consumption [42].
In 2021, by combining ensemble learning and pattern categorization, Dong et al. were able to predict an office building's hourly energy consumption [43]. A prediction model based on machine learning in the same year was trained using a vast dataset consisting of 3-month hourly data for 5760 energy-use cases that encompass various combinations of building characteristics, outdoor weather conditions, and occupant behaviors. Four machine learning algorithms were evaluated and compared during the model development process based on their prediction accuracy and computational efficiency [44]. Based on the analysis of architectural characteristics based on data mining, Shan et al. in 2022 identified the critical attributes of various types of buildings. In their study, Principal Component Analysis (PCA) and Random Forest Analysis were used to identify significant architectural characteristics associated with various levels of energy consumption [45]. The study conducted by Li et al. was a case analysis of an educational building. They introduced a novel method for forecasting building electricity load that involves using similarity judgement and an improved TrAdaBoost algorithm (iTrAdaBoos) and found that their proposed method had a simple structure, making it easy to implement for engineering purposes, compared to other advanced models [46]. In 2023, a supervised machine-learning model was developed by kapp et al. using data from 45 manufacturing plants, which were obtained from industrial energy audits. The goal was to create a general predictor of industrial building energy consumption [47]. In another research, an energy consumption benchmark for university buildings in Brazil was established. Three machine learning techniques were evaluated for this purpose, and SVM method was found to have the lowest mean absolute error and root mean absolute error. As a result, the SVM method was chosen to develop the benchmark and efficiency scales [48]. The summary of the recent literature can be seen in Table 1.
Mehtodology
The current study employed the CRISP-DM methodology, which is an industry-agnostic process model commonly used in data mining. It consists of six iterative phases that direct the data mining process, beginning with a comprehension of the business context and concluding with the implementation of the outcomes [49][50][51]. Each phase is thoroughly described in this section, along with its utilization in the current study.
Business Understanding
One of the crucial stages in a data mining project is comprehending the business context. It dictates the data to be gathered, the analysis techniques to be employed, and the manner in which the findings should be presented [52]. The aim of this study is to identify buildings with potential for decreasing their energy consumption and costs while examining the factors that impact their energy usage.
Data Understanding
During this phase, data is gathered, explored, and described while ensuring its quality. The task of describing the data can involve the use of statistical analysis techniques to identify attributes and correlations, as specified in the user guide [52]. For this study, the data was obtained from New York State Office of Information Technology Services [53], which includes 26 columns and 57925 rows pertaining to a particular building plan. As part of the Existing Residential Home Design initiative, goldstar building performance contractors were hired to implement and construct comprehensive energy efficiency enhancements. Table 2 provides a description of the database.
Data Preparation
The process of preparing data for analysis involves employing data mining techniques. This phase typically takes up a significant amount of time during the analysis. It encompasses activities such as merging, cleansing, converting, and downsizing data [52]. The present study outlines three crucial steps within this phase: data transformation, data correction, and data reduction.
Data transformation
In order to improve the data presentation in the project management system, various changes have been implemented to specific columns. For instance, the Project Completion Date column now only shows the year of completion, omitting the month and day information. Additionally, the Customer Type column has been modified to use the numbers 1 and 0 to indicate "assisted" and "market" respectively. Similarly, the Low-Rise or Home Performance Indicator column now uses the numbers 1 and 0 to denote "Home Performance Indicator" and "Low-Rise" correspondingly. Previously, the Homeowner Received Green Jobs-Green NY Free / Reduced Cost Audit (Y / N) column had only two categories, but it now includes the numbers 1 and 0 to signify "usage" and "non-usage" respectively. To simplify the data analysis, categorical columns with more than two categories have been transformed into more intelligible columns by incorporating dummy variables into the software.
Data correction
During our analysis, we observed inconsistencies in the capitalization of the word "Gas" in the phrase "Natural Gas" in some instances. As the software is sensitive to such differences and treats them as separate lines, we standardized the capitalization by replacing instances of lowercase "gas" with uppercase "Gas". Additionally, we noticed a row with the value 1347 in the electric type column. As there was no discernible difference between the two categories, we assumed this value also belonged to the same category. Therefore, we treated it accordingly. Furthermore, as per the description provided, the Type of Program Financing column was identified to have empty values indicating that conventional financial resources were not supported. Consequently, we filled these empty values with "not financed".
Data reduction
As a first step, the Location column was removed from the dataset as it contained no useful information for the problem. The columns displaying data submission date, project ID, home location ID, and project codec were also excluded, as they were deemed irrelevant. It was decided to create only ten new columns using dummy variables to address the issue of categorical columns with many categories. This approach was adopted to avoid excessive columns, considering only the ten most frequent categories would be considered. In comparison, the remaining categories would be categorized as "zero" or "not belonging to the ten most frequent categories".
Modeling
At this stage in the Crisp method, the prepared data can be analyzed using different data mining methods to achieve the project's main objective and intended outcome. It is necessary to test different methods and compare their outputs. Sometimes it is necessary to return to the previous step and prepare some data algorithms differently to achieve the desired results [52]. In this paper, three different models with three different feature selection methods were considered. Feature selection is a popular data preprocessing approach that has shown effectiveness and efficiency in diverse machine learning and data mining applications, as indicated by various studies [54]. Its usage has been observed across multiple domains, including social media [55,56], healthcare [57,58], and biometrics [59,60]. The overall steps of the modeling are as follows:
I. Data Preprocessing
• Split the dataset into test, validation, and training sets with a 20%, 20%, and 60% ratio.
• Identify the optimal number of features to include in the algorithm by applying a threshold to the training set to select relevant features.
II. Hyperparameter Tuning
• Utilize a grid technique to optimize the algorithm's hyperparameters, considering a predefined set of values for each hyperparameter. • Consider the validation set to find the most optimal combination of hyperparameters that maximize the model's performance.
III. Model Training
• Fit the model to the training set using the optimized hyperparameters obtained from the hyperparameter tuning step. • To ensure model robustness and generalizability, we utilize cross-validation IV.
Model Evaluation
• Employ the trained model to predict the target variable for the test set. • Evaluate the model's performance by calculating relevant metrics, such as R2 score, mean squared error (MSE), root mean squared error (RMSE), and mean absolute error (MAE), for the training, test, and validation sets. • Assess the possibility of overfitting or underfitting by comparing the model's performance on different sets.
V. Model Comparison
• Calculate each model's Akaike Information Criterion (AIC) [61] to assess their relative performance and determine the best-performing model.
• Consider the AIC as a critical criterion for model selection, as it accounts for the goodness of fit and model complexity, providing a balanced approach to model comparison.
Evaluation
Evaluation involves comparing the outcomes with the predetermined business objectives; thus, interpretation is required to determine the next step. Moreover, a comprehensive review of the entire process should be conducted to identify potential improvement areas [52]. This phase is addressed and discussed in the discussion and result sections of the study.
Stability
An overview of the deployment phase is provided, which may be in the form of a final report or a software component. This phase encompasses activities such as planning the deployment, as well as monitoring and maintenance [52]. This phase is also presented in the discussion and result sections of the study.
Result
In this section, we began by utilizing data visualization and exploration techniques to gain a better understanding of the data. Next, we employed three feature selection methods, namely Forward Selection, Binary Genetic Algorithm, and Particle Swarm Optimization, to identify the relevant features. We then applied three regression models, namely Lasso, Decision Tree, and Random Forest, to predict the target features. The models' parameters were set using the grid technique, and we compared the nine resulting models using the AIC criterion. To further optimize the Decision Tree model's performance, we used the Genetic Optimization Algorithm to fine-tune its parameters. Finally, we conducted a comprehensive analysis of the decision tree models.
Data Exploration
Through the exploration of the database, valuable information and meaningful relationships can be discovered. Obtaining an overview of the data is especially beneficial during the subsequent modeling process. We conducted data exploration using different strategies, including single-feature, two-feature, and multiple-feature analysis, to uncover hidden and valuable information in the data .
Single-feature analysis
Upon analyzing the "number of building units" column, it was observed that over 95% of the projects were single-unit buildings. More than 90% of the projects were categorized as Home Performance projects. Further analysis revealed that approximately 99% of the improvements were focused on the building's body. In comparison, around 1% were related to the ventilation system, and a small percentage were related to the water heater. Notably, 111 buildings were estimated to cost zero dollars after the project. The "Measure Type" column indicated that building improvements were primarily focused on three categories: building body, ventilation system, and water heater, with only six buildings related to water heaters, 667 buildings related to ventilation systems, and the remaining projects related to building bodies.
Two-feature analysis
Upon reviewing the data in the columns labeled "Customer Type" and "Total Incentives", it becomes evident that the average amount of financial incentives provided to owners of government-subsidized buildings is approximately five times higher than the incentives provided to owners of buildings eligible for market-type incentives. This is illustrated in Figure 1. The analysis of Figure 2 and analyzing the relation between the "Pre-Retrofit Home Heating Fuel Type" factor with target features indicate that buildings that use electricity as their primary heating fuel source have remarkably higher electrical storage than other buildings. The same buildings, however, exhibit negative average values in terms of storage fuel (measured in MMBtu), indicating poor performance in this area. Additionally, buildings that use oil as their primary fuel source incurred the highest savings. According to Figure 3, after thoroughly analyzing the total project cost concerning the three stated objectives, it becomes apparent that there is a clear and direct correlation between this feature and the targets of fuel storage and cost reduction. However, the same level of correlation was not observed with the reduction of electrical energy consumption, whether in direct or indirect form. Figure 5 illustrates the scatter plot of cost saved and fuel storage along with three features indicating customer type (C_T), whether a Green Jobs Cost Audit plan was received (y/n), and the number of units. The analysis indicates that the 4-unit buildings which received the Green Jobs Cost Audit plan were able to achieve reduced fuel consumption. In general, the 3-unit buildings that received government subsidies (customer type = 1) performed the best in terms of cost savings and reducing fuel consumption.
Multiple-feature analysis
Comparison of the Models
We have compared several models for forecasting the three objectives of the problem, utilizing three separate feature selection methods. Tables 3 to 5 provide the Root Mean Square Error (RMSE), the number of selected features, and the AIC criteria for each of the nine models considered. The AIC is applied, which takes into account both the number of features and the level of RMSE. The lower the AIC value, the better the forecasting model. The analysis provides a comprehensive overview of the relative strengths and weaknesses of the different models, enabling us to identify the most effective approach for each of the three objectives. Table 3 demonstrates that in comparison with the other methods, the Forward Selection technique has selected a smaller subset of features. The Genetic Algorithm method resulted in a consistent number of selected features and the lowest RMSE was observed. When combined with Genetic Algorithm and Particle Swarm Optimization for feature selection, the Lasso algorithm produced the best RMSE results. Additionally, the models using Particle Swarm Optimization as the feature selection method showed the most consistent RMSE performance. To further facilitate the comparison of the models based on the AIC criterion, we provide Figure 6, which presents a comprehensive view of the models' relative performance across the feature selection methods. According to Figure 6, the Lasso and RFs models exhibit pretty similar performance. Considering feature forward selection, the Lasso regression model demonstrates the most favorable performance with the lowest AIC value. This model and the random forest method utilizing particle swarm optimization are the most effective predictive models. On the other hand, the decision tree algorithm does not perform as well as the other two methods. Table 4 illustrates that compared to the other techniques, the Forward Selection approach has chosen a smaller set of features and achieved the lowest RMSE. Among all three feature selection methods, Random Forest yielded the most satisfactory RMSE results compared to the other algorithms. Furthermore, models utilizing Particle Swarm Optimization as the feature selection method exhibited superior performance in terms of RMSE. Figure 7 provides a comprehensive overview of the relative performance across the methods used for selecting features based on the AIC. Figure 7, it can be concluded that the RF model, which utilizes forward selection, performs significantly better than the other models. The Lasso regression model appears to have relatively similar performance across all three feature selection methods. Finally, the decision tree model with genetic algorithm yields the least desirable results with the highest AIC value. Table 4 demonstrates that similar to the case with other targets, the Forward Selection technique has opted for a reduced number of features. Among the nine models, the RMSE values are fairly similar, with the best-performing model being the Random Forest (19.87) and the Decision Tree (25.31) performing the worst, which has the fewest features, both utilizing forward selection. Figure 8 offers a comprehensive summary of the comparative performance of the feature selection methods based on AIC. Figure 8, it can be inferred that the Random Forest (RF) model, which uses forward selection, outperforms the other models. The Lasso regression model, which also employs the same feature selection method, ranks second. Random Forest with Particle Swarm Optimization and Lasso with Genetic Algorithm yield comparable results. In contrast, the Decision Tree models demonstrate the poorest performance and have the highest AIC values.
Cost saved
Electrical energy
Primary Fuel
Optimization of Decision Tree Algorithm using Genetic Algorithm
We used a genetic algorithm to improve the performance of the decision tree algorithm for prediction, as the original algorithm performed poorly. To execute the GA, we employed the DEAP library [62] and considered a complete list of parameters related to the decision tree, along with their possible values. We began by creating chromosomes and defining separate populations. The Creator function evaluated fitness and defined a single chromosome. Since our populations were not homogeneous, we created custom individuals using the tools function, Initcycle, which determined the possible values of different genes on the chromosome. Rather than initializing the population with attributes, we filled it with individuals, creating a bag full of a specific number of individuals in no particular order. While DEAP has internal functions for mutation and evaluation, we had to define custom functions due to chromosome heterogeneity. Tournament selection involved selecting a user-defined number of chromosomes and running matches between them. The winner of each tournament was the most suitable chromosome, which was then transferred to the crossover. A custom mutation function was called, randomly selecting and mutating one of the individual chromosome genes assigned to it. The modified individual was then returned. The parameters transferred to the decision tree models were evaluated for each individual chromosome, and the resulting MSE score was used as the fitness score. Finally, we combined these functions to create GA, specifying population size, probability of crossing, probability of mutation, and the number of generations. Larger populations allowed for more exploration of the search space but required more computational time. We used the genetic algorithm for each target column to determine the best decision tree, and the findings are presented in the table below.
Decision Tree Analysis
Decision trees, which have their roots in machine learning theory, are effective tools for solving classification and regression problems. A decision tree regression approach is based on the implicit assumption that relationships between features and target objects are either linear or nonlinear [63]. For this analysis, the algorithms with the fewest features are considered.
Cost saved
The decision tree analysis revealed that the average saving cost was $624.67. Saint Lawrence was one such gas provider. Group buildings, which did not use pre-refined fuel, were funded and had Hudson as their gas supplier, which had the lowest reserves (average 316.50). Comparing these two groups, we found that the type of fuel has the greatest impact on storage costs, and the gas supplier is also influential. The details can be found in Figure 9.
Electrical energy
The decision tree analysis revealed that the average electrical reserve was 440.18 Kwh. Financial support had been provided. Group buildings, which had their gas suppliers on Long Island but did not receive electricity from them and were also using fuel before the electricity reform, had the lowest electrical reserves (average -8956.6). Comparing these two groups, we found that gas suppliers and financial support significantly impact the number of electrical reserves. The details can be found in Figure 10.
Primary fuel usage
Based on the decision tree analysis, the average annual primary fuel reserve was 29.08 MMBTU. There was no electricity reform, the buildings were located in the Jefferson area, and their gas supplier was not Long Island. Comparing the two groups, we found that financial support, the year the project was completed (before or after 2016), and the type of fuel used before the energy reform all played a significant role in storage. Figure 11 provides further details. Table 7 presents the effective variables of each model for all three target columns. This section considers the most effective of the three feature selection modes as well as the decision tree algorithms after optimization. According to Table 7, the type of fuel used, financial support, gas and electricity suppliers, total cost of the project, year of construction, and size of the home are found to be the most influential factors. Buildings that rely on electricity as their primary fuel source have not been successful in reducing fuel consumption even after making corrections, so investing in such buildings to reduce energy consumption is not recommended. Financial support plays a crucial role in the success of building energy efficiency projects, as buildings that receive funding have the potential to save more electrical energy and fuel, and therefore costs, compared to those that do not receive financial support. The choice of gas and electricity suppliers is also significant, as buildings with Long Island Power Authority as their electricity supplier do not perform well in electrical energy consumption but are the most suitable option for cost and fuel reduction compared to other suppliers. Furthermore, buildings with higher-than-average total costs have more potential for energy savings and fuel reduction than those with lower costs, which could potentially make them more attractive for loans. Older buildings built before 1980 also have the potential to reduce energy consumption, fuel consumption, and costs saved. Additionally, larger homes with over 540 square feet are generally better at reducing fuel consumption compared to smaller homes. Therefore, it is recommended to invest in larger homes to achieve greater primary fuel consumption reduction.
Discussion
Conclusion
In this study, we use three different algorithms with three different feature selection methods to identify the factors that affect energy consumption and costs. The results show that the type of fuel used, financial support, gas and electricity suppliers, total cost of the project, year of construction, and size of the home are all important factors. Specifically, the type of fuel used before modification has the greatest impact on all three target fields.
The analysis showed that different models and different feature selection methods can have a significant impact on the accuracy of the prediction. This is because different models and feature selection methods can extract different information from the data, which can lead to different predictions. This emphasizes the importance of choosing the right model and feature selection method. We also highlight the importance of using an efficient hyperparameter selection method and how it can improve the performance of an algorithm.
It should be noted that this study has certain limitations. Data used in the study lacks information regarding consumption culture, climate, level of education of residents, the age range of residents, occupancy status, and initiatives to reduce energy consumption and associated costs. Future studies may consider employing alternative data mining techniques and gathering additional data to address these limitations.
Figure 1 :
1The relationship between Customer Type and Total Incentives columns
Figure 2 :
2Investigation of the "Pre-Retrofit Home Heating Fuel Type" feature
Figure 4
4illustrates the correlation between the features. Fuel storage and cost savings features have a higher correlation (0.64) among the three target variables. Project cost features and loan amount have the strongest correlation (0.65).
Figure 3 :
3The correlation between total project cost and the objectives
Figure 4 :Figure 5 :
45The heatmap depicting correlation among features The relationship between cost saved and fuel storage, considering customer type, the status of the Green Jobs Cost Audit plan, ad the number of units
Figure 6 :
6Comparison of models based on AIC criteria for predicting the cost savings
Figure 7 :
7Comparison of models based on AIC criteria for predicting the electrical energy consumption
Figure 8 :
8Comparison of models based on AIC criteria for predicting the primary fuel usage
Figure 9 :
9Decision tree plot for the cost saved
Figure 10 :
10Decision tree plot for the electrical energy
Figure 11 :
11Decision tree plot for the primary fuel usage
Table 1 .
1A list of data mining-based methods for predicting building energy loads.Ref.
Year
Algorithm
Focus
[28]
2010
DT
Optimizing building energy consumption
[29]
2010
K-means
Identifing the patterns of energy
consumption
[30]
2011
Hierarchical Clustering
Identify the operational patterns
[31]
2011
Hierarchical Clustering
Predicting building electricity consumption
[32]
2012
Clustering techniques
Examining the chiller system
[33]
2012
DT
Predicting peak electrical energy demands
[34]
2013
SVM, ANN
Prediction of lighting energy consumption
[35]
2014
K-means
Modeling and predicting HVAC
[36]
2015
Hierarchical Clustering
Predicting energy consumption and peak
demand
[37]
2016
ANN
Forecasting cooling load
[38]
2017
Backward propagation neural
network (BPNN), support vector
regression (SVR), adaptive
network-based fuzzy inference
system (ANFIS) and ELM
Forecasting the energy consumption in a
retail building
[39]
2018
Hierarchical Clustering
Understanding energy consumption
behaviors
[40]
2019
ANN, SVM
Accuracy analyses and model comparisons
[41]
2020
Multi-objective optimization
techniques with evolutionary
algorithms
Predicting heating and cooling loads
[42]
2020
RF
Short-term energy consumption prediction
[43]
2021
Ensemble learning models
Predicting an office building's hourly energy
consumption
[44]
2021
Classification and regression trees
(CART), ensemble bagging trees
(EBT), ANN, and deep neural
networks (DNN)
Occupant-behavior-sensitive energy
consumption prediction
[45]
2022
PCA, RF
Identifing the significant architectural
characteristics
[46]
2022
iTrAdaBoos
Forecasting building electricity load
[47]
2023
A supervised machine-learning
model (SVM)
Creating a general predictor of industrial
building energy consumption
[48]
2023
Multiple linear regression (MLR),
SVM, and ANN
Establishing an energy consumption
benchmark
Table 2 :
2Description of the datasetThe type of building in the project (each has benefits in terms of receiving facilities and loans)Indicates the type of program financing (if it is empty, ie it does not support conventional programs)The main improvement of the project is on which part of the buildingVariable description
Data type
Variable name
Date reported
(
ordinal
)
numeric
Reporting Period
Unique project ID
categorical
Home Performance Project ID
Unique house location ID
categorical
Home Performance Site ID
The area where the project was done
categorical
Project County
The city where the project was done
categorical
Project City
Project zip code
categorical
Project Zip
Name of gas supplier for the project site
categorical
Gas Utility
Name of electricity supplier for the project site
categorical
Electric Utility
Project completion date
(
ordinal
)
numeric
Project Completion Date
Incentives or subsidies paid by the government
categorical
Customer Type
categorical
Low-Rise or Home Performance
Indicator
Project cost in US dollars
(
ordinal
)
numeric
Total Project Cost
Financial incentives received by the building
owner
(
integer
)
numeric
Total Incentives
categorical
Type of Program Financing
Project loan amount in dollars
(
integer
)
numeric
Amount Financed Through
Program
The type of fuel used in the heating system
before the building was remodeled.
categorical
Pre-Retrofit Home Heating Fuel
Type
Date of construction of the house
(
ordinal
)
numeric
Year Home Built
House area in square feet
(
integer
)
numeric
Size of Home
Approximate volume of home air conditioning
(
integer
)
numeric
Volume of Home
Approximate volume of home air conditioning
(
integer
)
numeric
Number of Units
categorical
Measure Type
Annual electrical storage in kilowatt hours
(
integer
)
numeric
Estimated Annual kWh Savings
Annual primary fuel storage in MMBtu
(
integer
)
numeric
Estimated Annual MMBtu
Savings
Estimate the amount of cost saved in dollars
(
integer
)
numeric
First Year Energy Savings $
Estimate
Indicates whether the landlord has used the plan
(Green Jobs-Green NY Free / Reduced Cost
Audit)
categorical
Homeowner Received Green
Jobs-Green NY Free/Reduced
Cost Audit (Y/N)
Table 3 :
3Comparison of the models considering different feature selection methodsAIC
Number of Features
RMSE
Model
Feature selection method
254306.01
7
1505.55
Lasso
Forward feature selection
257901.13
4
1669.92
Decision Tree
Forward feature selection
242964.17
10
1086.15
Random Forest
Forward feature selection
212459.47
39
450.82
Lasso
Genetic binary algorithm
220202.57
38
563.29
Decision Tree
Genetic binary algorithm
213229.61
39
460.88
Random Forest
Genetic binary algorithm
255035.37
41
1534.47
Lasso
Particle swarm optimization
256211.96
41
1587.31
Decision Tree
Particle swarm optimization
256379.46
27
1596.27
Random Forest
Particle swarm optimization
Table 4 :
4Comparison of the models for electrical energy considering different feature selection methodsAIC
Number of Features
RMSE
Model
Feature selection method
2114118.16
14
438.11
Lasso
Forward feature selection
222761.49
4
607.55
Decision Tree
Forward feature selection
211674.05
8
401.5
Random Forest
Forward feature selection
256023.16
30
1579.67
Lasso
Genetic binary algorithm
261399.24
45
1842.40
Decision Tree
Genetic binary algorithm
254176.50
33
1497.71
Random Forest
Genetic binary algorithm
21294.14
45
457.12
Lasso
Particle swarm optimization
220325.17
38
565.44
Decision Tree
Particle swarm optimization
211688.31
43
440.79
Random Forest
Particle swarm optimization
From
Table 5 :
5Comparison of the models for primary fuel usage considering different feature selection methodsAIC
Number of Features
RMSE
Model
Feature selection method
104511.89
11
20.22
Lasso
Forward feature selection
112307.11
7
25.31
Decision Tree
Forward feature selection
103903.04
8
19.87
Random Forest
Forward feature selection
104811.61
49
20.35
Lasso
Genetic binary algorithm
108832.72
32
22.86
Decision Tree
Genetic binary algorithm
105645.56
39
20.85
Random Forest
Genetic binary algorithm
105860.10
40
20.98
Lasso
Particle swarm optimization
109612.52
40
23.37
Decision Tree
Particle swarm optimization
104618.76
43
20.42
Random Forest
Particle swarm optimization
Table 6 :
6Improvement of the decision tree algorithm with the help of the genetic algorithmAIC(Genetic)
AIC(grid)
Feature
216336.48
220202.57
First-Year Energy Savings $ Estimate
254295.17
256211.96
Estimated Annual kWh Savings
107336.28
108832.72
Estimated Annual MMBtu Savings
Table 7 :
7Influential factors of each model on the target variables Low-Rise or Home Performance Indicator Total Project Cost Pre-Retrofit Home Heating Fuel TypeMMBTU
Electrical energy
Cost saved
Target
RF
DT
Lasso
RF
DT
Lasso
RF
DT
Lasso
Feature/Model
*
*
*
*
*
*
Project County
*
*
*
*
Project City
*
*
*
*
*
*
*
Gas Utility
*
*
*
*
Electric Utility
*
*
Project Completion year
*
*
Customer Type
*
*
*
*
*
*
*
*
*
*
*
*
Total Incentives
*
*
*
*
Type of Program
Financing
*
*
*
*
*
*
*
Amount Financed
Through Program
*
*
*
*
*
*
*
*
*
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| {'fraction_non_alphanumeric': 0.0537901236370006, 'fraction_numerical': 0.05152758112309691, 'mean_word_length': 4.61494959429555, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 54, 'lorem ipsum': 0, 'www.': 2, 'xml': 0}, 'pii_count': 4, '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': "Optimizing energy consumption and coordination of utility systems have long been a concern of the building industry. Buildings are one of the largest energy consumers in the world, making their energy efficiency crucial for preventing waste and reducing costs. Additionally, buildings generate substantial amounts of raw data, which can be used to understand energy consumption patterns and assist in developing optimization strategies. Using a real-world dataset, this research aims to identify the factors that influence building cost reduction and energy consumption. To achieve this, we utilize three regression models (Lasso Regression, Decision Tree, and Random Forest) to predict primary fuel usage, electrical energy consumption, and cost savings in buildings. An analysis of the factors influencing energy consumption and cost reduction is conducted, and the decision tree algorithm is optimized using metaheuristics. By employing metaheuristic techniques, we fine-tune the decision tree algorithm's parameters and improve its accuracy. Finally, we review the most practical features of potential and nonpotential buildings that can reduce primary fuel usage, electrical energy consumption, and costs.", 'arxivid': '2305.08886', 'author': ['Imtiaz Ahmed [email protected] \nDepartment of Industrial & Management Systems Engineering\[email protected] ty\nWest Virginia Universi 4 Morgantown26505WV\n', 'Rahim Khanizad \nSchool of Industrial Engineering\nUniversity of Science and Technology\nTehranIran, Iran\n', 'Hadi Sahebi [email protected] \nScience and Technology\nsity of School of Industrial Engineering\nTehranIran, Iran Univer\n', 'Hamed Khosravi [email protected] ', '\nSchool of Industrial Engineering\nIran University of Science and Technology\nTehranIran\n'], 'authoraffiliation': ['Department of Industrial & Management Systems Engineering\[email protected] ty\nWest Virginia Universi 4 Morgantown26505WV', 'School of Industrial Engineering\nUniversity of Science and Technology\nTehranIran, Iran', 'Science and Technology\nsity of School of Industrial Engineering\nTehranIran, Iran Univer', 'School of Industrial Engineering\nIran University of Science and Technology\nTehranIran'], 'corpusid': 258714623, 'doi': '10.48550/arxiv.2305.08886', 'github_urls': [], 'n_tokens_mistral': 20197, 'n_tokens_neox': 16712, 'n_words': 9297, 'pdfsha': '7a1806abbd5862f036a03fc079688da48ee9fdf3', 'pdfurls': ['https://export.arxiv.org/pdf/2305.08886v1.pdf'], 'title': ['Identification of the Factors Affecting the Reduction of Energy Consumption and Cost in Buildings Using Data Mining Techniques', 'Identification of the Factors Affecting the Reduction of Energy Consumption and Cost in Buildings Using Data Mining Techniques'], 'venue': []} |
arxiv |
MB-DECTNet: A Model-Based Unrolled Network for Accurate 3D DECT Reconstruction
Tao Ge
Department of Electrical & Systems Engineering
Washington University in St. Louis
Maria Medrano
Department of Electrical & Systems Engineering
Washington University in St. Louis
Rui Liao
Department of Electrical & Systems Engineering
Washington University in St. Louis
David G Politte
Radiation Oncology
Washington University in St. Louis
Mallinckrodt Institute of Radiology c Washington University in St. Louis
63110St. LouisMO
Jeffrey F Williamson
Bruce R Whiting
Department of Radiology
University of Pittsburgh
15213PittsburghPA
Joseph A O'sullivan
Department of Electrical & Systems Engineering
Washington University in St. Louis
MB-DECTNet: A Model-Based Unrolled Network for Accurate 3D DECT Reconstruction
dual-energy computed tomographydeep learningmodel-based learning
Numerous dual-energy CT (DECT) techniques have been developed in the past few decades. Dual-energy CT (DECT) statistical iterative reconstruction (SIR) has demonstrated its potential for reducing noise and increasing accuracy. Our lab proposed a joint statistical DECT algorithm for stopping power estimation and showed that it outperforms competing image-based material-decomposition methods. However, due to its slow convergence and the high computational cost of projections, the elapsed time of 3D DECT SIR is often not clinically acceptable. Therefore, to improve its convergence, we have embedded DECT SIR into a deep learning model-based unrolled network for 3D DECT reconstruction (MB-DECTNet) that can be trained in an end-toend fashion. This deep learning-based method is trained to learn the shortcuts between the initial conditions and the stationary points of iterative algorithms while preserving the unbiased estimation property of modelbased algorithms. MB-DECTNet is formed by stacking multiple update blocks, each of which consists of a data consistency layer (DC) and a spatial mixer layer, where the spatial mixer layer is the shrunken U-Net, and the DC layer is a one-step update of an arbitrary traditional iterative method. Although the proposed network can be combined with numerous iterative DECT algorithms, we demonstrate its performance with the dual-energy alternating minimization (DEAM). The qualitative result shows that MB-DECTNet with DEAM significantly reduces noise while increasing the resolution of the test image. The quantitative result shows that MB-DECTNet has the potential to estimate attenuation coefficients accurately as traditional statistical algorithms but with a much lower computational cost.
INTRODUCTION
Dual-energy computed tomography (DECT) has been widely investigated to generate informative and accurate images. 1, 2 Dual-energy CT (DECT) statistical iterative reconstructions (SIR) have demonstrated their potential to reduce noise and increase accuracy. [3][4][5][6] For instance, our lab proposed a joint statistical DECT algorithm, dual-energy alternating minimization (DEAM), which outperforms competing methods and achieves sub-percentage uncertainty in estimating proton stopping power. [7][8][9][10] However, 3D DECT SIRs are usually time-consuming, even with multiple acceleration techniques, including multi-GPU acceleration 11 and ordered-subsets method. 12 Compared to single-energy CT (SECT), DECT algorithms reconstruct two measured sinograms scanned at two different spectra, which at least doubles the system operations per iteration. Moreover, since approximations are usually used in DECT SIRs to evaluate the gradient of the polychromatic forward models, DECT SIRs always converge much slower than single-energy algorithms. The low convergence rate and high computational cost of projections make it difficult to get an accurate DECT result within a clinically acceptable time. To reduce the elapsed time while retaining the estimation accuracy, we incorporate DECT SIR into a deep learning model-based unrolled network for DECT reconstruction (MB-DECTNet). This work is based on deep unfolding, 13 which simulates iterative algorithms using a series of convolutional neural networks (CNN). 14, 15 We substitute the data consistency layer in deep unfolding with DECT gradients. Then, in MBDECTNet, each individual block can be considered an iteration of JSIDECT, while the step size and the image prior are determined by the CNN.
METHODS
Problem Description
In contrast to single-energy CT, DECT estimates various types of images from two sinograms measured at different spectra. According to Beer's law, the transmission data can be modeled as a Poisson random vector with the entry
d j (y) ∼ Poisson I 0,j (y, E) exp − X h(x, y)µ(x, E)dx dE ,(1)
where x ∈ X denotes spatial location or voxel index in the image domain, y ∈ Y denotes the index (fan angle, gantry angle, and bed position) of the measurement, j = L, H denotes the spectrum, E denotes the energy, I 0 is the photon counts in the absence of an object, which contains information about the bowtie filter and spectrum, etc., h(x, y) denotes the system operator, and µ(x, E) ∈ R + denotes the linear attenuation coefficient (LAC) of the object at location x and energy E. Traditional SIR algorithms solve the DECT material-decomposition problem by iteratively minimizing the objective function [3][4][5]7 arg min
c j y D{d j (y), g j (y : c)} + R(c),(2)
where D : (Y, Y ) → R and R : X → R denote the data fidelity and penalty term, respectively; c = {c i } denotes image of basis weights, and i indexes over basis materials, g j (y : c) is the sinogram predicted by the discretized forward model operating on the current basis-material-decomposition, c
g j (y : c) = E I 0,j (y, E) exp − x h(x, y) i c i (x)µ i (E) ,(3)
where µ i (E) is the LAC of the i-th basis material at energy E.
Prior work from our lab shows that DECT SIR algorithms can reconstruct images with subpercentage accuracy from uncorrected transmission data. [8][9][10] However, the elapsed time of statistical algorithms for 3D DECT is usually time-consuming because of the high computational cost of the system operator as well as the low convergence rate. To address this issue, we introduce the unrolled network for 3D DECT reconstruction in the next subsection.
Model-Based Unrolled Network
The basic idea of MB-DECTNet is using a set of stacked neural networks to simulate the iterative update process of the traditional SIR algorithms. Figure 1 shows the flowchart of the proposed pipeline. The data is firstly reconstructed by an analytical algorithm combined with material decomposition to generate the initial condition for the network. The unrolled network consists of several update blocks, and each update block mimics a single iteration of the traditional statistical algorithm.
The structure of the update block is shown in Figure 2. The data consistency (DC) layer takes the image from the previous block and the transmission data as the inputs, and outputs the reference gradient, defined as the difference between and the minimizer of the surrogate data fidelity term and the previous iterate c k−1 :
DC(c k−1 , d) = arg min c j yD {d j (y), g j (y : c)} − c k−1(4)
where k denotes the index of the current block, andD denotes the surrogate function of the data fidelity term at c = c k−1 .
Then, the DECT image and DC output are stacked along the channel dimension as the input to the truncated U-Net. We also employ residual learning 16 and group normalization 17 to make training more efficient. The output of the k-th update block Γ k can be written as
Γ θ k (c k−1 ) = UNET k (stack(DC(c k−1 , d), c k−1 )) + c k−1 .(5)
Since the training process with intermediate loss functions has been observed to converge much faster than training with only the final loss, our network is trained under the supervision of the weighted sum of a set of intermediate loss functions as arg min
θ N −1 n=0 w e,n · ||Γ θn • Γ θn−1 · · · • Γ θ0 (c init ) − c truth ||,(6)
where w e,n ∈ [0, 1] denotes the scalar that controls the backpropagation weight, N denotes the number of stacked blocks, e denotes the index of the current training epoch, θ = {θ 0 , θ 1 . . . } denotes the set of trainable parameters in all update blocks, || · || denotes the L2 norm, and • denotes the function composition.
The GPU memory footprint is a critical design issue for 3D unrolled networks since the training process stores all feature maps for backpropagation, and 3D unrolled networks generate more and larger latent maps. This work mainly uses two techniques to reduce the GPU memory footprint.
1) We partition the image volume into multiple small stacks consisting of 8 slices. The transmission data corresponding to each image stack is also truncated accordingly. Prior to passing to the DC layer, each stack is padded by 10 slices on each side to address the margin effect of helical CT reconstruction. 2) The double 3D convolutional layer with 3×3×3 kernels is substituted with a single 3D convolutional layer with 5×5×5 kernels. Compared to the original U-Net, the number of starting feature maps is also reduced from 64 to 40.
These two modifications reduce the GPU memory footprint of the network by approximately 70%.
RESULTS
Data Acquisition
In this work, model-based reconstructed images were taken as the reference. Five helical scans at 90 and 140 kVp of a head-and-neck cancer patient were sequentially acquired by a Philips Brilliance Big Bore CT scanner at a 0.75 mm×16 collimation. These helical scans were then reconstructed by a joint statistical DECT algorithm, DEAM, 8 with the motion-compensation technique 18 and a sufficient number of iterations. We selected four patient data as the training set and one as the test set. The dimension of the sinogram was 16×816×52800 with 40 rotations. The image size was 610×610×340, and the image resolution was 1×1×1.034 mm 3 . The training set had 168 samples in total. Each SIR takes 45 hours on 4× NVIDIA V100 32 GB GPUs (400 iterations with 33 ordered subsets, plus 1000 iterations without ordered subset).
Results
We used a pretraining strategy to reduce the training time. Firstly, the first block of MB-DECTNet was trained individually for 2000 iterations. In this step, the reference update from the DC layer can be precomputed to save time. Then, the weights of the pretrained block were broadcasted to other blocks, and the entire network was trained end-to-end. The MB-DECTNet consisted of four updating blocks.
In the inference mode, MB-DECTNet generates a 610×610×340 image in approximately 350 seconds, which is 462-fold times shorter than the DEAM. Figure 3 shows the image of a slice as an example of the inference result from MB-DECTNet, compared to iFDK (input) and DEAM (ground truth) images. iFDK refers to the 3D analytical DECT reconstruction that combines the FDK algorithm and the iterative filtered back-projection. 19 MB-DECTNet image is much less noisy that the initial FDK estimate while preserving contrast and image sharpness. Indeed, the vertical profile suggests that MB-DECTNet improves the spatial resolution compared to the iFDK image.
More quantitatively, figure 4 shows the relative mean absolute error (RMAE) for three different tissue ROIs relative to the LAC derived from their expected ICRU compositions, 20 defined as
RMAE(E) = 1 N x |c 1 (x)µ 1 (E) + c 2 (x)µ 2 (E) − µ(x, E)| µ(x, E) .(7)
MB-DECTNet shows its potential to estimate accurate LACs for energies at 20-150 keV, and the performance of MB-DECTNet is close to the performance of the selected DECT SIR, with a much lower computational cost.
CONCLUSIONS
To the best of our knowledge, it is the first time that a model-based unrolled network is proposed and trained end-to-end to estimate accurate basis components from dual-energy CT sinograms. Our proposed MB-DECTNet is capable of reducing noise and increasing resolution, and can be combined with numerous joint statistical DECT algorithms. The quantitative result shows that MB-DECTNet has the potential to estimate LAC accurately for energies from 20 to 150 keV as the selected traditional statistical algorithm with a much lower computational cost.
Further author information: Tao Ge: [email protected] Joseph A. O'Sullivan: [email protected]
Figure 1 .
1Flowchart of the proposed pipeline. Each blue dashed box denotes an update block, which mimics an iteration in DECT SIR. DC is the data consistency layer, and the orange boxes evaluate the loss functions for training.
Figure 2 .
2Structure of a single update block.
Figure 3 .
3Images and middle vertical profiles for a selected slice in the test dataset.
Figure 4 .
4Plots of relative mean absolute error (RMAE) versus energy for selected regions of interest.
ACKNOWLEDGMENTSThis project is supported by R01 CA212638 from the United States National Institutes of Health. We thank the Siteman Cancer Center for their help in the acquisition of clinical data.
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| {'fraction_non_alphanumeric': 0.06071445604998331, 'fraction_numerical': 0.026994801354509466, 'mean_word_length': 4.612419700214133, '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': 'Numerous dual-energy CT (DECT) techniques have been developed in the past few decades. Dual-energy CT (DECT) statistical iterative reconstruction (SIR) has demonstrated its potential for reducing noise and increasing accuracy. Our lab proposed a joint statistical DECT algorithm for stopping power estimation and showed that it outperforms competing image-based material-decomposition methods. However, due to its slow convergence and the high computational cost of projections, the elapsed time of 3D DECT SIR is often not clinically acceptable. Therefore, to improve its convergence, we have embedded DECT SIR into a deep learning model-based unrolled network for 3D DECT reconstruction (MB-DECTNet) that can be trained in an end-toend fashion. This deep learning-based method is trained to learn the shortcuts between the initial conditions and the stationary points of iterative algorithms while preserving the unbiased estimation property of modelbased algorithms. MB-DECTNet is formed by stacking multiple update blocks, each of which consists of a data consistency layer (DC) and a spatial mixer layer, where the spatial mixer layer is the shrunken U-Net, and the DC layer is a one-step update of an arbitrary traditional iterative method. Although the proposed network can be combined with numerous iterative DECT algorithms, we demonstrate its performance with the dual-energy alternating minimization (DEAM). The qualitative result shows that MB-DECTNet with DEAM significantly reduces noise while increasing the resolution of the test image. The quantitative result shows that MB-DECTNet has the potential to estimate attenuation coefficients accurately as traditional statistical algorithms but with a much lower computational cost.', 'arxivid': '2302.00577', 'author': ['Tao Ge \nDepartment of Electrical & Systems Engineering\nWashington University in St. Louis\n\n', 'Maria Medrano \nDepartment of Electrical & Systems Engineering\nWashington University in St. Louis\n\n', 'Rui Liao \nDepartment of Electrical & Systems Engineering\nWashington University in St. Louis\n\n', 'David G Politte \nRadiation Oncology\nWashington University in St. Louis\nMallinckrodt Institute of Radiology c Washington University in St. Louis\n63110St. LouisMO\n', 'Jeffrey F Williamson ', 'Bruce R Whiting \nDepartment of Radiology\nUniversity of Pittsburgh\n15213PittsburghPA\n', 'Joseph A O'sullivan \nDepartment of Electrical & Systems Engineering\nWashington University in St. Louis\n\n'], 'authoraffiliation': ['Department of Electrical & Systems Engineering\nWashington University in St. Louis\n', 'Department of Electrical & Systems Engineering\nWashington University in St. Louis\n', 'Department of Electrical & Systems Engineering\nWashington University in St. Louis\n', 'Radiation Oncology\nWashington University in St. Louis\nMallinckrodt Institute of Radiology c Washington University in St. Louis\n63110St. LouisMO', 'Department of Radiology\nUniversity of Pittsburgh\n15213PittsburghPA', 'Department of Electrical & Systems Engineering\nWashington University in St. Louis\n'], 'corpusid': 256459277, 'doi': '10.1117/12.2654252', 'github_urls': [], 'n_tokens_mistral': 6081, 'n_tokens_neox': 5201, 'n_words': 3179, 'pdfsha': '17bcf14af316dd842ee75f6de8f9b7d84352a678', 'pdfurls': ['https://export.arxiv.org/pdf/2302.00577v1.pdf'], 'title': ['MB-DECTNet: A Model-Based Unrolled Network for Accurate 3D DECT Reconstruction', 'MB-DECTNet: A Model-Based Unrolled Network for Accurate 3D DECT Reconstruction'], 'venue': []} |
arxiv |
User Fairness Non-orthogonal Multiple Access (NOMA) for 5G Millimeter-Wave Communications with Analog Beamforming
7 Nov 2018
Senior Member, IEEEZhenyu Xiao
Lipeng Zhu
Member, IEEEZhen Gao
Fellow, IEEEDapeng Oliver Wu
Fellow, IEEEXiang-Gen Xia
User Fairness Non-orthogonal Multiple Access (NOMA) for 5G Millimeter-Wave Communications with Analog Beamforming
7 Nov 20181
The integration of non-orthogonal multiple access in millimeter-Wave communications (mmWave-NOMA) can significantly improve the spectrum efficiency and increase the number of users in the fifth-generation (5G) mobile communication. In this paper we consider a downlink mmWave-NOMA cellular system, where the base station is mounted with an analog beamforming phased array, and multiple users are served in the same time-frequency resource block. To guarantee user fairness, we formulate a joint beamforming and power allocation problem to maximize the minimal achievable rate among the users, i.e., we adopt the max-min fairness. As the problem is difficult to solve due to the non-convex formulation and high dimension of the optimization variables, we propose a sub-optimal solution, which makes use of the spatial sparsity in the angle domain of the mmWave channel. In the solution, the closed-form optimal power allocation is obtained first, which reduces the joint optimization problem into an equivalent beamforming problem. Then an appropriate beamforming vector is designed. Simulation results show that the proposed solution can achieve a near-upper-bound performance in terms of achievable rate, which is significantly better than that of the conventional mmWave orthogonal multiple access (mmWave-OMA) system.IndexTerms-millimeter-wave communications, Nonorthogonal multiple access, mmWave-NOMA, user fairness, analog beamforming, power allocation.Z. Xiao and L. Zhu are with the
I. INTRODUCTION
W ITH the rapid growth of mobile data traffic, higher data rate is an insistent requirement in the fifth generation (5G) of mobile communication [1]. Millimeter-Wave (mmWave) communications, with frequency ranging from 30-300 GHz, provides abundant spectrum resources and is perceived as a candidate key technology for 5G [1]- [3]. In addition to the large amount of bandwidth, the mmWave-band signal has a shorter wavelength compared with the traditional microwave-band signal, which makes it possible to equip a large antenna array in a small area. Considerable beam gain can be obtained to overcome the high propagation loss in the mmWave-band [3].
Although more spectrum resources are available in the mmWave band, multiple access is still an important issue to in-crease the spectrum efficiency and the number of users/devices to support 5G Internet of Things (IoT). Non-orthogonal multiple access (NOMA), considered as another candidate technology for 5G, has drawn widespread attention in both academia and industry [4]- [11]. Different from the conventional orthogonal multiple access (OMA) schemes, NOMA serves multiple users in a single resource block (time/frenquency/code) and distinguishes them in power domain. Successive interference cancellation (SIC) is required at the receivers. In general, the users are sorted by an increasing order of channel gains. The one with lower channel gain is prior, i.e., its signal is decoded and removed first with the signals of the other users treated as noise [5]- [10]. In this way, NOMA can increase the spectrum efficiency and break the limit that the maximal number of users is no larger than the number of radio-frequency (RF) chains in OMA networks [7]- [9], [12]- [15].
To make full use of the spectrum resource, we investigate NOMA in mmWave communications (mmWave-NOMA) in this paper. The combination of the two candidate technologies for 5G has been preliminarily explored in several literatures. In [12], the coexistence of NOMA and mmWave communications was considered, where random beamforming was used in order to reduce the system overhead. The results demonstrated that the combination of NOMA and mmWave communications yields significant gains in terms of sum rates and outage probabilities, compared with the conventional mmWave-OMA systems. In [13], the new concept of beamspace multipleinput multiple-output NOMA (MIMO-NOMA) with a lensarray hybrid beamforming structure was proposed to use multi-beam to serve multiple NOMA users with arbitrary locations. With this method, the number of supported users can be larger than the number of RF chains in the same time-frequency resource block. Beamforming, user selection and power allocation were considered for mmWave-NOMA networks in [16], where random beamforming was adopted first. Then a power allocation algorithm that leverages the branch and bound (BB) technique and a low complexity user selection algorithm based on matching theory were proposed. A NOMA based hybrid beamforming design was proposed in [17], where a user pairing algorithm was proposed first and then the hybrid beamforming and power allocation algorithm was proposed to maximize the sum achievable rate. In [18], the NOMA-mmWave-massive-MIMO system model and a simplified mmWave channel model were proposed. Whereafter, theoretical analysis on the achievable rate was considered in both the noise-dominated low-SNR regime and the interference-dominated high-SNR regime. To further improve the data rate, power allocation and beamforming were jointly explored in [19] and [20] for a 2-user downlink and uplink mmWave-NOMA scenario, respectively, where the key technique is the multi-directional beamforming design with a constant-modulus (CM) phased array.
Different from these works [12], [13], [16]- [19], we consider user fairness for downlink mmWave-NOMA networks in this paper. To improve the overall data rate, we maximize the minimal achievable rate among multiple users, i.e., we adopt the max-min fairness 1 . Due to the requirement of low hardware cost and power consumption, an analog beamforming structure with a single RF chain is utilized, where both implementations of single phase shifter (SPS) and double phase shifter (DPS) are considered [22], [23]. In the formulated problem, power allocation and beamforming are jointly optimized. As the problem is non-convex and the dimension of the optimization variables is large, it is difficult to solve this problem with the existing optimization tools. To this end, we solve this problem with two stages and obtain a sub-optimal solution. In the first stage, we obtain closed-form optimal power allocation with an arbitrary fixed beamforming vector, which reduces the joint optimization problem into an equivalent beamforming problem. Then, in the second stage, we propose an appropriate beamforming algorithm utilizing the spatial sparsity in the angle domain of the mmWave channel. Finally, we verify the performance of the proposed joint beamforming and power allocation method for user fairness mmWave-NOMA by simulations. The results show that the proposed solution can achieve a near-upper-bound performance in terms of achievable rate, which is significantly better than that of the conventional mmWave-OMA system.
The rest of the paper is organized as follows. In Section II, we present the system model and formulate the problem. In Section III, we propose the solution. In Section IV, simulation results are given to demonstrate the performance of the proposed solution, and the paper is concluded finally in Section V.
Symbol Notation: a and a denote a scalar variable and a vector, respectively. (·) T and (·) H denote transpose and conjugate transpose, respectively. | · | and · denote the absolute value and Euclidean norm, respectively. E(·) denotes the expectation operation.
[a] i denotes the i-th entry of a. C N denotes an N -dimension linear space in complex domain.
II. SYSTEM MODEL AND PROBLEM FORMULATION
A. System model
In this paper, we consider a downlink mmWave communications system. As shown in Fig. 1, the base station (BS) is equipped with a single RF chain and an N -antenna phased array. K users with a single antenna are served in the same resource block. Each antenna is driven by the power amplifier (PA) and phase shifter (PS). 1 We adopt the max-min fairness because it is a typical and extensively used fairness rule in NOMA [10], [21]. Besides the max-min fairness, there are also other fairness rules in NOMA, like proportional fairness, etc. [10]. The BS transmits a signal s k to User k (k = 1, 2, · · · , K) with transmission power p k , where E(|s k | 2 ) = 1. The total transmission power of the BS is P . The received signal for User k is
y k = h H k w K k=1 √ p k s k + n k ,(1)
where h k is the channel response vector between the BS and User k. w is the antenna weight vector (AWV), i.e., analog beamforming vector, and n k denotes the Gaussian white noise at User k with power σ 2 . Two PS structures, named SPS implementation and DPS implementation, are considered. For the SPS implementation, each antenna branch has a single PS as shown in Fig. 1(a). The elements of the AWV are complex numbers, whose modulus and phase are controlled by the PA and PS respectively. To reduce hardware complexity, all the PAs have the same scaling factor in general. Thus, the AWV has CM elements, which is denoted by
|[w] i | = 1 √ N , i = 1, 2, ..., N.(2)
The above constraint is non-convex, which results in a major challenge of AWV design, i.e., we can only adjust the phase but not the amplitude of the signal. To reduce the design difficulty, a new implementation named DPS was proposed in [22], [23], which is shown in Fig. 1(b). For the DPS implementation, each antenna is driven by the summation of the two independent PSs. Although the modulus of each PS is constant, the phases of two PSs can be adjusted to achieve different modulus in each antenna branch. Thus, the modulus constraint is relaxed to
|[w] i | ≤ 2 √ N , i = 1, 2, ..., N.(3)
By doubling the number of PSs, the new constraint becomes convex and therefore make it more tractable to develop lowcomplexity design approaches. The channel between the BS and User k is a mmWave channel. 2 Subject to the limited scattering in mmWave-band, multipath is mainly caused by reflection. As the number of the multipath components (MPCs) is small in general, the mmWave channel has directionality and appears spatial sparsity in the angle domain [24], [26]- [30]. Different MPCs have different angles of departure (AoDs). Without loss of generality, we adopt the directional mmWave channel model assuming a uniform linear array (ULA) with a half-wavelength antenna space. Then a mmWave channel can be expressed as [24], [26]- [30]
h k = L k ℓ=1 λ k,ℓ a(N, Ω k,ℓ ).(4)
where λ k,ℓ , Ω k,ℓ are the complex coefficient and cos(AoD) of the ℓ-th MPC of the channel vector for User k, respectively.
We have
L k l=1 E(|λ k,ℓ | 2 ) ∝ 1 d 2 k
, where d k is the distance between the BS and User k. L k is the total number of MPCs for User k, a(·) is a steering vector function defined as
a(N, Ω) = [e jπ0Ω , e jπ1Ω , e jπ2Ω , · · ·, e jπ(N −1)Ω ] T ,(5)
which depends on the array geometry. Let θ k,ℓ denote the real AoD of the ℓ-th MPC for User k, then we have Ω k,ℓ = cos(θ k,ℓ ). Therefore, Ω k,ℓ is within the range [−1, 1].
In general, the optimal decoding order of NOMA is the increasing order of the effective channel gains, i.e., h H k w 2 .
However, we cannot determine the order of the effective channel gains before beamforming design. For simplicity, we utilize the increasing order of uses' channel gains as the decoding order. We will illustrate the rational of selecting the increasingchannel-gain decoding order in Section III-C, and verify that it can achieve near optimal performance by simulations. Without loss of generality, we assume h 1 ≥ h 2 ≥ · · · ≥ h K . Therefore, User k can decode s n (k + 1 ≤ n ≤ K) and then remove them from the received signal in a successive manner.
The signals for User m (1 ≤ m ≤ k − 1) are treated as noise. Thus, the achievable rate of User k is given by
R k = log 2 (1 + h H k w 2 p k h H k w 2 k−1 m=1 p m + σ 2 ).(6)
B. Problem Formulation As aforementioned, both beamforming and power allocation have an important effect on the performance of the mmWave-NOMA system. To improve the overall data rate and guarantee user fairness, we formulate a problem to maximize the minimal achievable rate (the max-min fairness) among the K users in this paper, where beamforming and power allocation are jointly optimized. The problem is formulated as
Max {p k },w min k {R k } s.t. C 1 : p k ≥ 0, k = 1, 2, · · · , K C 2 : K k=1 p k ≤ P, C 3 : w ≤ 1, C 4 : |[w] i | = 1 √ N or |[w] i | ≤ 2 √ N , i = 1, 2, ..., N(7)
where R k denotes the achievable rate of User k as defined in (6) and min k {R k } is the minimal achievable rate among the K served users. The constraint C 1 indicates that the power allocation to each user should be positive. C 2 is the transmission power constraint, where P is the total transmission power. C 3 is the norm constraint on the AWV, and C 4 is the additional modulus constraint on the AWV for SPS or DPS implementation. The above problem is challenging, not only due to the non-convex formulation, but also due to that the variables to be optimized are entangled with each other. It is computationally prohibitive to directly search the optimal solution, because the dimension of the optimization variables is N + K, which is large in general. Next, we will propose a sub-optimal solution with promising performance but low computational complexity.
III. SOLUTION OF THE PROBLEM
As the modulus constraints for SPS and DPS implementations are different, we first solve the problem without considering the constraint C 4 . As thus, Problem (7) is simplified as Max
{p k },w min k {R k } s.t. C 1 : p k ≥ 0, k = 1, 2, · · · , K C 2 : K k=1 p k ≤ P, C 3 : w ≤ 1.(8)
We will solve Problem (8) first, and then particularly consider the modulus constraints in Section III-D. Problem (8) is still difficult due to the non-convex formulation, so we propose a sub-optimal solution with two stages. In the first stage, we obtain the closed-form optimal power allocation with an arbitrary fixed AWV. Then, in the second stage, we propose an appropriate beamforming algorithm utilizing the angle-domain spatial sparsity of the mmWave channel.
A. Optimal Power Allocation with an Arbitrary Fixed AWV
First, we introduce a variable to simplify Problem (8). Denote the minimal achievable rate among the K users as r. Then Problem (8) can be re-written as
Max {p k },w,r r s.t. C 0 : R k ≥ r, k = 1, 2, · · · , K C 1 : p k ≥ 0, k = 1, 2, · · · , K C 2 : K k=1 p k ≤ P, C 3 : w ≤ 1,(9)
where the constraints C 0 : R k ≥ r, (k = 1, 2, · · · , K) are necessary and sufficient conditions of the fact that r is the minimal achievable rate among the served users. On one hand, as r is the minimal rate, the achievable rate of each user should be no less than r. On the other hand, there is at least one user, whose achievable rate R km is equal to r; otherwise we can always improve r to minish the gap between R km and r.
We give the following Theorem to obtain the optimal solution of power allocation of Problem (9) with an arbitrary fixed AWV.
Theorem 1. Given an arbitrary fixed w 0 , the optimal power allocation of Problem (9) is
p 1 = η σ 2 h H 1 w 0 2 , p 2 = η(p 1 + σ 2 h H 2 w 0 2 ),
. . .
p K = η( K−1 m=1 p m + σ 2 h H K w 0 2 ),(10)
where η = 2 r − 1, and with the optimal power allocation, R k = r (k = 1, 2, · · · , K).
Before proving Theorem 1, we give Lemma 1 for the summation of the optimal power allocation in (10), which is a function of η. (10) is
Lemma 1. The summation of power allocation in
g(η) K k=1 p k = K k=1 η(1 + η) K−k σ 2 h H K w 0 2 .(11)
Proof. We prove Lemma 1 with mathematical induction.
When m = 1, (11) is easy to verify
p 1 = η σ 2 h H 1 w 0 2 .(12)
When m = n (n ≥ 1), assume that
n k=1 p k = n k=1 η(1 + η) n−k σ 2 h H k w 0 2 .(13)
When m = n + 1, based on (13), we have
n+1 k=1 p k = n k=1 p k + η( n k=1 p k + σ 2 |h H n w 0 | 2 ) =(1 + η) n k=1 p k + η σ 2 |h H n w 0 | 2 =(1 + η) n k=1 η(1 + η) n−k σ 2 h H k w 0 2 + η σ 2 |h H n w 0 | 2 = n+1 k=1 η(1 + η) n+1−k σ 2 h H k w 0 2 .(14)
Finally, we can conclude that (11) is true.
Based on Lemma 1, the proof of Theorem 1 is presented in Appendix A. According to Theorem 1 and Lemma 1, Problem (9) can be equivalently written as
Max w,η η s.t. K k=1 p k = K k=1 η(1 + η) K−k σ 2 h H k w 2 ≤ P, w ≤ 1,(15)
where η = 2 r − 1.
Hereto, the first stage to solve Problem (8) is finished, where the optimal power allocation is obtained, and thus the original problem with entangled power allocation and beamforming is reduced to a pure beamforming problem as shown in (15), which will be solved in the next subsection.
B. Beamforming Design with Optimal Power Allocation
The remaining task is to to solve Problem (15) and obtain w; then the closed-form expression of {p k (k = 1, 2, · · · , K)} can be obtained by (10). The main challenge is that the first constraint is non-convex, where w and η are entangled. As the dimension of w, i.e., N , is large in general, it is computationally prohibitive to directly search the optimal solution. However, the introduced variable η is only 1-dimensional. We can search the maximal value of η in the range of [0, Γ] with the bisection method, where Γ is the search upper bound. According to the definition of η = 2 r − 1, η in fact represents the minimal signal to interference plus noise power ratio (SINR) among the K users. If we allocate all the beam gain and power to the user with the best channel condition, i.e., User 1, then User 1 can achieve the highest (N σ 2 ). Thus, we select Γ as the search upper bound. Given a fixed η, we judge whether an appropriate w can be found in the feasible region of Problem (15). Thus, we need to solve the following problem
SINR Γ = ( N n=1 |[h 1 ] n |) 2 P/Min w f (w) K k=1 η(1 + η) K−k σ 2 h H k w 2 s.t. w ≤ 1.(16)
Given η, if the minimal value of the objective function in Problem (16) is no larger than P , which means that a feasible solution can be found with the given η, we enlarge η and solve Problem (16) again. If the minimal value of the objective function in Problem (16) is larger than P , i.e., a feasible solution cannot be found with the given η, we lessen η and solve Problem (16) again. The stopping criterion of the bisection search is that η meets an accuracy requirement.
To solve Problem (16), some approximate manipulations are required to simplify the beamforming problem. Retrospecting the characteristic of the mmWave channel, the channel response vectors of different users are approximatively orthogonal due to the spatial sparsity in the angle domain, which is
h H m h H m h n h n ≈ 1, If m = n; 0, If m = n.(17)
With this approximation, { h k h k , k = 1, 2, · · · , K} can be considered as an orthonormal basis of a subspace in C N . We say the subspace expanded by { h k h k , k = 1, 2, · · · , K} is a channel space. In Problem (16), most beam gains are inclined to focus on the users' directions. Thus, the AWV should be located in the channel space, which can be written as
w = K k=1 α k h k h k ,(18)
where {α k , k = 1, 2, · · · , K} are the coordinates of w in the channel space. Substituting (18) into Problem (16), we have
Min {α k } K k=1 η(1 + η) K−k σ 2 α 2 k h k 2 s.t. K k=1 α 2 k = 1.(19)
Note that the norm constraint for w ≤ 1 is replaced by w = 1 here, because the norm of optimal w is surely 1.
Assuming that w ⋆ is optimal and w ⋆ < 1, we can always normalize the AWV to get a better solution of w ⋆ w ⋆ .
To solve Problem (19), we define the Lagrange function as
L(α, λ) = K k=1 η(1 + η) K−k σ 2 α 2 k h k 2 + λ( K k=1 α 2 k − 1).(20)
The Karush-Kuhn-Tucker (KKT) conditions can be obtained by the following equation [31],
∂L ∂α k = 0, k = 1, 2, · · · , K ∂L ∂λ = 0.(21)
From the KKT conditions, we can obtain the solution of Problem (19), which is given by
∂L ∂α k = 0 ⇒ −2η(1 + η) K−k σ 2 α 3 k h k 2 + 2λα k = 0 ⇒α k = 4 η(1 + η) K−k σ 2 λ h k 2 ⇒α k ∝ 4 η(1 + η) K−k h k 2 .(22)
Thus, the designed AWV in Problem (16) is given by
w = K k=1 4 η(1 + η) K−k h k 2 h k h k , w =w w .(23)
In summary, we give Algorithm 1 to solve Problem (15).
Algorithm 1: AWV design
Input: Channel response vectors: h k , k = 1, 2, · · · , K; Total transmission power: P ; Noise power: σ 2 ;
The search accuracy ǫ. Output:
η and w. 1: η min = 0, η max = Γ. 2: while η max − η min > ǫ do 3: η = (η max + η min )/2;
4:
Calculate w according to (23) and the objective function in Problem (16): f (w).
5:
if f (w) > P then 6: η max = η. Hereto, we have solved Problem (8) and obtain the solution {p ⋆ k , w}, where the AWV is obtained in Algorithm 1 and the power allocation is given in (10). The AWV is approximately optimal while the power allocation is optimal for the designed AWV. A leftover problem is to verify the rational of the decoding order. We will consider this problem next.
C. Decoding order
When formulating Problem (7), we assumed that the decoding order of signals is the increasing order of the channel gains. Next, we will verify that the order of the effective channel gains after beamforming design is the same with the channel-gain order. The effective channel gain for User k is
|h H k w| 2 ∝ |h H kw | 2 = K m=1 4 η(1 + η) K−m h m 2 h H k h m h m 2 (a) = 4 η(1 + η) K−k h k 2 h H k h k h k 2 = η(1 + η) K−k h k ,(24)
where (a) is according to the orthogonal assumption of the channel response vectors. As η = 2 r − 1 > 0, η(1 + η) K−k is decreasing for k. We have assumed that the order of the users' channel gains is h 1 ≥ h 2 ≥ · · · ≥ h K . Thus, under the orthogonal assumption of the channel response vectors, the order of users' effective channel gains is
|h H 1 w| 2 ≥ |h H 2 w| 2 ≥ · · · |h H K w| 2 .(25)
As shown in (25), the order of the effective channel gains is the same with that of channel gains. However, this property may not hold if we utilize other decoding orders, which indicates that the increasing-channel-gain decoding order is more reasonable. In the simulations, we will compare the performance of different decoding orders and find that the performance of increasing-channel-gain decoding order is very close to the performance of the optimal decoding order.
D. Consideration of Modulus Constraints
When solving Problem (8), the additional modulus constraints on the AWV were not considered. Next, we will consider the modulus constraints and solve the original problem, i.e., Problem (7). As we have shown in the system model, the modulus constraints on the elements of the AWV are (2) and (3) for SPS and DPS implementations, respectively. Some additional normalized operations on the designed AWV are required to satisfy the constraints. For the SPS implementation, the constant modulus normalization is given by
[w S ] i = [w] i √ N [w] i , i = 1, 2, · · · , N.(26)
where w S denotes the AWV for SPS implementation. For the DPS implementation, the modulus normalization is given by
[w D ] i = [w] i , If [w] i ≤ 2 √ N ; 2 √ N , If [w] i > 2 √ N .(27)
where w D denotes the AWV for DPS implementation. Each element of w D is the sum weight of the corresponding antenna branch, and it needs to be decomposed into two components, which can be expressed as
[w D ] i a i e jθi = 1 √ N e j(θi+ϕi) + 1 √ N e j(θi−ϕi) ,(28)
where a i ∈ [0, 2
[w D ] 2i−1 = 1 √ N e j(θi+ϕi) , [w D ] 2i = 1 √ N e j(θi−ϕi) .(29)
E. Computational Complexity
As we obtained the closed-form optimal power allocation with an arbitrary fixed AWV, the computational complexity is mainly caused by the beamforming algorithm in the second stage. In Algorithm 1, the total search time for η is T = log 2 ( Γ ǫ ), where Γ is the search upper bound and ǫ is the search accuracy. Thus, the computational complexity of the proposed method is O(T ), which does not increase as N and K. However, if we directly search the solution of Problem (7) and obtain the globally optimal solution, the total complexity is O(( 1 ǫ ) N +K ), which exponentially increases as N and K.
IV. PERFORMANCE SIMULATIONS
In this section, we provide simulation results to verify the performance of the proposed joint beamforming and power allocation method in the mmWave-NOMA system. We adopt the channel model in (4) in the simulations, where the users are uniformly distributed from 10m to 500m away from the BS, and the channel gain of the user 100m away from the BS has an average power of 0dB. The number of MPCs for all the users are L = 4. Both LOS and NLOS channel models are considered. For the LOS channel, the average power of the NLOS paths is 15 dB weaker than that of the LOS path. For the NLOS channel, the coefficient of each path has an average power of 1/ √ L. The search accuracy in Algorithm 1 is ǫ = 10 −6 .
We first show the power allocation and the effective channel gains in Figs. 2 and 3, respectively, where the LOS channel model is adopted 3 . Each point is an average result from 10 4 channel realizations. From Fig. 2 we can find that most power is allocated to User 4, the user with the lowest channel gain. Less power is allocated to the users with higher channel gains, so as to reduce interference. Despite all this, it can be observed from Fig. 3 that the effective channel gain of User 4 is still the lowest. The user with a better channel gain have a higher effective channel gain with the proposed solution, which verifies the conclusion in Section III-C about the decoding order. It is noteworthy that the effective channel gains of User 1 and User 4 go increasing and decreasing, respectively, when P/σ 2 becomes higher, which is the result of joint power allocation and beamforming. It indicates that when the total power is high, power and beam gain should be jointly allocated to enlarge the difference of the effective channel gains to achieve a larger minimal user rate.
Next, we compare the performance between the considered mmWave-NOMA system and a mmWave-OMA system. We give the following method to calculate the minimal achievable rates in a K-user mmWave-OMA system, where time division multiple access (TDMA) is used without of generality.
If all the time slots are allocated to User k, the achievable rate for User k isR
k = log 2 (1 + h H k w 2 P σ 2 ).(30)
Assume that the time division is ideal, which means that the time slot can be allocated to the users with any proportion. To maximize the minimal achievable rate of the K users, more time should be allocated to the users with lower channel gains, such that the achievable rates of the K users are equal. Thus, the time allocation for User k is
β k = 1/R k K m=1 1/R m .(31)
Then the achievable rate of User k in the mmWave-OMA system is
R OMA k = β kRk = 1 K m=1 1/R m ,(32)
where all the users have the same achievable rate. Fig. 4 shows the comparison result of the minimal achievable rates between the mmWave-NOMA and mmWave-OMA systems with varying total power to noise ratio. The minimal achievable rates of Ideal NOMA/OMA, SPS-NOMA/SPS-OMA and DPS-NOMA/DPS-OMA are based on the beamforming given in (23), (26) and (27), which are corresponding to the beamforming without CM constraint, with SPS implementation and with DPS implementation, respectively. Each point in the figure is the average performance of 10 4 LOS channel realizations. We can find that the minimal achievable rates of SPS-NOMA are lower than that of DPS-NOMA, which is very close to the minimal achievable rates of Ideal NOMA, this is because the strict modulus normalization on the AWV for SPS results in significant performance loss, while the modulus normalization on the AWV for DPS is more relaxed and has little impact on the rate performance. In addition, the minimal achievable rates of the mmWave-NOMA system is distinctly better than those of the mmWave-OMA system for all the cases, and superiority is more significant when the total power to noise ratio is higher. Fig. 5 compares the minimal achievable rates between mmWave-NOMA and mmWave-OMA systems with varying number of users. For fairness, the total transmission power is proportional to the number of users, and the average transmission power to noise for each user is 20 dB. Each point in Fig. 5 is the average performance of 10 4 LOS channel realizations. It can be observed again that the minimal achievable rate of mmWave-NOMA is better than that of mmWave-OMA for both SPS and DPS implementations, and the minimal achievable rates of DPS-NOMA is very close to that of Ideal NOMA. On the other hand, the minimal achievable rates of both mmWave-NOMA and mmWave-OMA decreases as the number of users increases. This is mainly due to that the orthogonality of the channel vectors of the users become weakened, which deteriorates the beamforming performance and in turn the minimal achievable rate performance. realizations. It can be seen that the moduli of the AWV's elements are mainly distributed around 1/ √ N , and almost all of them have a modulus less than 2/ √ N . The results in Fig. 6 demonstrate that the modulus normalization for the DPS implementation has a limited impact on the performance.
In the second stage of the proposed solution, we have assumed that the channel response vectors are orthogonal and then found an appropriate AWV in (16). To evaluate the impact of this approximation, we compare the performance of the proposed solution with the upper-bound performance. We solve Problem (16) using particle swarm optimization, where the density of particles is sufficiently high, and thus the obtained minimal achievable rate can be treated as the upper bound. Limited by the computational complexity, we provide the simulation results with a relatively small-scale antenna array, i.e., N = 8, 16. The comparison result is shown in Fig. 7, where each point is averaged from 10 3 LOS channel realizations. The minimal achievable rate of Ideal NOMA is based on the beamforming given in (23), which is corresponding to the beamforming without the CM constraint and the orthogonality assumption of the channel vectors between the NOMA users. As we can see, when N = 8, the performance gap between the proposed solution and the upper bound is no more than 0.25 bps/Hz. When N = 16, the performance gap is even smaller, i.e., no more than 0.2 bps/Hz. The reason is that the orthogonality of the channel vectors becomes stronger when N is larger. Thus, the approximation of the beamforming design in Problem (16) has limited impact on the system performance, and the proposed sub-optimal solution can achieve an nearupper-bound performance, especially when N is large. Fig. 8 compares the minimal achievable rates of mmWave-NOMA under the LOS and NLOS channel models with varying total power to noise ratio. The number of antennas is N = 16, 64, 256, respectively. The number of users is K = 4. Each point in Fig. 8 is the average performance of 10 4 channel realizations. It can be seen that the performance of DPS-NOMA with the LOS channel model is slightly better than that with the NLOS channel model, because the channel power is more centralized for the LOS channel. However, the performance gap between them is quite small, especially when N is large. The reason is that according to (24), the effective channel gain is linear to h k , the norm of the channel vector, rather than that of the power of the strongest path. Thus, the performance gap of DPS-NOMA with the LOS and NLOS channel models is small. The simulations above are all based on the increasingchannel-gain decoding order. Next, we will show the impact of the decoding order on the mmWave-NOMA system. Fig. 9 shows the performance comparison between different decoding orders with varying total power to noise ratio, where N = 32 and K = 4. There are 24 decoding orders in total for the 4 users. Each point in Fig. 9 is the average performance of 10 4 LOS channel realizations. The minimal achievable rates of the 24 decoding orders are all calculated. The order with the highest minimal achievable rate is chosen as the optimal order and the order with the lowest minimal achievable rate is chosen as the worst order. The increasingchannel-gain order is the one adopted in our solution, while the decreasing-channel-gain order is one for comparison. From the figure we can find that there is a significant performance gap between the optimal order and the worst order, which means that the decoding order has an important impact on the performance of mmWave-NOMA. Moreover, the performance with the increasing-channel-gain order is almost the same as the optimal one, while the performance with the decreasingchannel-gain order is almost the same as the worst one. This result shows the rational of adopting the increasing-channelgain order in our solution.
V. CONCLUSION
In this paper, we have investigated downlink max-min fairness mmWave-NOMA with analog beamforming. A joint beamforming and power allocation problem was formulated and solved in two stages. In the first stage, the closed-form optimal power allocation was obtained with an arbitrary fixed AWV, reducing the joint beamforming and power allocation problem into an equivalent beamforming problem. Then, an appropriate beamforming vector was obtained by utilizing the spatial sparsity in the angle domain of the mmWave channel. Both implementations of SPS and DPS were considered with different modulus normalizations. The simulation results demonstrate that the modulus normalization has limited impact on the achievable rate performance, especially for the DPS implementation. Moreover, by using the proposed solution, the considered mmWave-NOMA system can achieve a near-upperbound performance of the minimal achievable rate, which is significantly better than that of the conventional mmWave-OMA system.
APPENDIX A PROOF OF THEOREM 1
Without loss of generality, we denote {p ⋆ k , r ⋆ } one optimal solution of Problem (9) with fixed w 0 , where the achievable rate of User k is R ⋆ k , and let η ⋆ = 2 r ⋆ − 1. With η ⋆ we can obtain another solution
{p • k , r ⋆ } , where p • 1 = η ⋆ σ 2 h H 1 w 0 2 , p • 2 = η ⋆ (p • 1 + σ 2 h H 2 w 0 2 ),
. . .
p • K = η ⋆ ( K−1 m=1 p • m + σ 2 h H K w 0 2 ).(33)
The following lemma shows that this solution is also an optimal one.
Lemma 2. The solution {p • k , r ⋆ } is also an optimal solution of Problem (9), and the achievable rates under this parameter setting always satisfy R • k = r ⋆ (1 ≤ k ≤ K). Proof. First, we need to verify that the constraints C 0 , C 1 and C 2 are all satisfied.
According to the expression of (33), it is obvious that {p • k ≥ 0}, which means that the constraint C 1 is satisfied.
In addition, according to the assumption that {p ⋆ k , r ⋆ } is an optimal solution, we have
-Fig. 1 .
1-Illustration of a mmWave mobile cell, where one BS with N antennas serves multiple users with one single antenna.
10: end while 11: return η and w.
√NFig. 2 .
2] and θ i ∈ [0, 2π) are the modulus and the phase of [w D ] i , respectively, and ϕ i = arccos( Power allocation with varying total power to noise ratio, where N = 32 and K = 4.the weights of the two PSs corresponding to [w D ] i are
Fig. 3 .
3Effective channel gains with varying total power to noise ratio, where N = 32 and K = 4.
Fig. 4 .
4Comparison of the minimal achievable rates between NOMA and OMA system with varying total power to noise ratio, where N = 32 and K = 4.
Fig. 5 .Fig. 6 .
56Comparison of the minimal achievable rates between the NOMA and OMA systems with varying number of users, where N = 32 and the average transmission power to noise for each user is 20 dB. Moduli of the elements of the AWVs, where N = 32, K = 4 and P/σ 2 = 25 dB.
Fig. 6 Fig. 7 .
67shows the modulus of the elements of AWVs , where N = 32, K = 4 and P/σ 2 = 25 dB. We show the 1st, 8th, 16th and 32th element of 200 AWVs with different channel Comparison of the minimal achievable rates between the proposed solution and the upper bound with varying total power to noise ratio, where K = 4.
Fig. 8 .
8Performance comparison between LOS and NLOS channel models with varying total power to noise ratio, where K = 4.
NOMA, Optimal order DPS-NOMA, Worst order DPS-NOMA, Increasing-channel-gain order DPS-NOMA, Decreasing-channel-gain order
Fig. 9 .
9Comparison of the minimal achievable rates under different decoding orders with varying total power to noise ratio, where N = 32 and K = 4.
In this paper, we assume the channel state information (CSI) is known by the BS. The mmWave channel estimation with low complexity can be referred in[24] and[25].
Similar results can be observed when the NLOS channel model is adopted; thus the results are not presented here for conciseness.
Next, we use mathematical induction to prove that p • k ≤ p ⋆ k (k = 1, 2, · · · , K). When k = 1, according to (34) we haveThus, we can conclude that p • k ≤ p ⋆ k (k = 1, 2, · · · , K) and we havewhich means that the constraint C 2 is satisfied.With the considered solution (p • k , r ⋆ ), we havewhere (a) is based on (33). The above equation means that the constraint C 0 is satisfied., it is also an optimal solution of Problem(9).As both {p • k , r ⋆ } and {p ⋆ k , r ⋆ } are optimal solutions of Problem (9), we will prove that they are in fact the same as each other. For this sake, we need to prove that R ⋆ k = r ⋆ (1 ≤ k ≤ K). We assume that there exists one user whose achievable is strictly larger than r ⋆ , i.e., R ⋆ k0 > r ⋆ , and we will prove that this assumption does not hold as follows.As we have assumed that R ⋆ k0 > r ⋆ , we have R ⋆ k0 > R • k0 = r ⋆ . In addition, we have proven that p • k ≤ p ⋆ k (see the proof in(35)and(36)). According to the expression of R k in(6), it is straightforward to derive p ⋆ k0 > p • k0 .We define another solution {p △ k , r △ }, where r △ = r ⋆ + δ, andwhere η △ = 2 r △ − 1 and δ > 0. Thus, we have η △ > η ⋆ . Next, we prove that {p △ k , r △ } is within the feasible region of Problem(9). Similar to the proof in Lemma 2, we can prove that {p △ k ≥ 0} and R △ k = r △ > r ⋆ (1 ≤ k ≤ K), which means that the constraints C 0 and C 1 are satisfied. According to Lemma 1, the summation of power allocation in (33) and (39) are g(η ⋆ ) and g(η △ ), respectively. As we have provenwhich is contradictory to Constraint C 2 in Problem(9). As g(η) is an increasing function for η, we can always find a small positive δ, which satisfies g(η ⋆ + δ) < P , i.e., g(η △ ) < P . Thus, the constraint C 2 is satisfied with sufficiently small δ.In brief, {p △ k , r △ } is within the feasible region of Problem (9) provided that δ is small enough. However, we have R △ k = r △ > r ⋆ (1 ≤ k ≤ K), which means that the solution {p △ k , r △ } is better than {p ⋆ k , r ⋆ }, which is contradictory to the fact that {p ⋆ k , r ⋆ } is an optimal solution. Thus, the assumption that there exists one user whose achievable is strictly larger than r ⋆ does not hold. Equivalently, the achievable rates of users under the optimal power allocation satisfy R ⋆ k = r ⋆ = R • k (1 ≤ k ≤ K). Solve the equations set above and we can obtain that {p ⋆ k , r ⋆ } is the same as {p • k , r ⋆ }, and the optimal power allocation of Problem (9) is given by(10).
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| {'fraction_non_alphanumeric': 0.06580854641386907, 'fraction_numerical': 0.024390715293480993, 'mean_word_length': 3.851570557899672, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 12, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 58, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The integration of non-orthogonal multiple access in millimeter-Wave communications (mmWave-NOMA) can significantly improve the spectrum efficiency and increase the number of users in the fifth-generation (5G) mobile communication. In this paper we consider a downlink mmWave-NOMA cellular system, where the base station is mounted with an analog beamforming phased array, and multiple users are served in the same time-frequency resource block. To guarantee user fairness, we formulate a joint beamforming and power allocation problem to maximize the minimal achievable rate among the users, i.e., we adopt the max-min fairness. As the problem is difficult to solve due to the non-convex formulation and high dimension of the optimization variables, we propose a sub-optimal solution, which makes use of the spatial sparsity in the angle domain of the mmWave channel. In the solution, the closed-form optimal power allocation is obtained first, which reduces the joint optimization problem into an equivalent beamforming problem. Then an appropriate beamforming vector is designed. Simulation results show that the proposed solution can achieve a near-upper-bound performance in terms of achievable rate, which is significantly better than that of the conventional mmWave orthogonal multiple access (mmWave-OMA) system.IndexTerms-millimeter-wave communications, Nonorthogonal multiple access, mmWave-NOMA, user fairness, analog beamforming, power allocation.Z. Xiao and L. Zhu are with the', 'arxivid': '1811.02908', 'author': ['Senior Member, IEEEZhenyu Xiao ', 'Lipeng Zhu ', 'Member, IEEEZhen Gao ', 'Fellow, IEEEDapeng Oliver Wu ', 'Fellow, IEEEXiang-Gen Xia '], 'authoraffiliation': [], 'corpusid': 53229703, 'doi': '10.1109/twc.2019.2913844', 'github_urls': [], 'n_tokens_mistral': 16126, 'n_tokens_neox': 14017, 'n_words': 9078, 'pdfsha': '92fdd470c825f85f16c6c31fdbdd1ee7dc3d8e8c', 'pdfurls': ['https://arxiv.org/pdf/1811.02908v1.pdf'], 'title': ['User Fairness Non-orthogonal Multiple Access (NOMA) for 5G Millimeter-Wave Communications with Analog Beamforming', 'User Fairness Non-orthogonal Multiple Access (NOMA) for 5G Millimeter-Wave Communications with Analog Beamforming'], 'venue': []} |
arxiv |
A Hermite pseudospectral solver for two-dimensional incompressible flows on infinite domains
29 Oct 2013
Zhaohua Yin
Institute of Mechanics
National Microgravity Laboratory
Chinese Academy of Sciences
No.15 Beisihuanxilu100190BeijingP.R. China
A Hermite pseudospectral solver for two-dimensional incompressible flows on infinite domains
29 Oct 2013arXiv:1311.0189v1 [physics.flu-dyn]Hermite functionsSpectral methodsNavier-Stokes equation
The Hermite pseudospectral method is applied to solve the Navier-Stokes equations on a two-dimensional infinite domain. The feature of Hermite function allows us to adopt larger time steps than other spectral methods, but also leads to some extra computation when the stream-function is calculated from the vorticity field. The scaling factor is employed to increase the resolution within the region of our main interest, and the aliasing error is fully removed by the 2/3-rule. Several traditional numerical experiments are performed with high accuracy, and some related future work on physical applications of this program is also discussed.
Introduction
In practice, the boundary effect of flows is inevitable and requires careful treatment. On the other hand, when people concentrate on physical mechanism of fluid, they hope that the boundary effect can be fully removed. For example, the Oseen vortex in two-dimensional(2D) fluid adopts the assumption of the infinite domain (e.g., see [1] and references therein). It has also been shown in many previous investigations on how the boundary can influence the physical mechanism of flows. Even when the region of main interest are fairly far away from the solid boundary, the drop speeds show about 15% difference in a thermocapillary migration study [2]. In the research of the axisymmetrization of an isolated 2D vortex [3], a double-sized computing domain is adopted to alleviate the vortex overrotation caused by the periodical boundary in a Fourier spectral simulation [4]. Hence, it is essential to adopt the infinite domain in such studies. When the infinite computing domain (−∞, +∞) is considered, one possible way is still using the finite-domain solver but with a certain mapping between (x min , x max ) and (−∞, +∞). Another way is something like the sponge-layer suggested in [5], which absorbs the incoming vortex filaments. The other way is Email address: [email protected]; [email protected] (Zhaohua Yin). more natural: Laguerre or Hermite functions are adopted to construct global approximation to functions defined on unbounded intervals.
The normalized Hermite function of degree n is defined as:
H n (x) = 1 √ 2 n n! e − x 2 2 H n (x),(1)
where H n (x) are the usual (unnormalized) Hermite polynomials (for instance, see [6,7]).Ĥ n (x) are orthogonal in L 2 (−∞, ∞):
+∞ −∞Ĥ k (x)Ĥ m (x) = √ πδ km , k, m ≥ 0.
In the last few decades, many investigations concerning the theory and application of Hermite functions have been carried out, and an early review can be seen in [8]. Although the computing domain is infinite for Hermite spectral methods, the region of our main interest is still finite. This fact led to the appearance of the most important concept in the practical sense: the scaling factor [9,10].
There have been many Hermite spectral (or, pseudospectral) studies which investigated partial differential equations from different fields [11,12,13,14,15,16,17,18,19,20]. Most researches in this field have so far dealt with one-dimensional problems, or multi-dimensional problems with only one Hermite direction. One most recent paper [21] deals with two-dimensional partial differential equations with two Hermite directions, and takes an elliptic equation with a harmonic potential and a class of nonlinear wave equations into consideration. However, so far as we know, there has been no effort in solving the multi-dimensional Navier-Stokes (NS) equations with pure Hermite spectral (or Hermite pseudospectral) methods, and this is the main object of this paper.
The paper is arranged as follows: the numerical details are described in Sec. 2, and the performance of some validating tests is introduced in Sec. 3, and the discussions and some related future work are presented in Sec. 4.
Numerical scheme for the Hermite pseudospectral NS solver
The 2D incompressible NS equations on the infinite domain
x = (x, y) ∈ [−∞, +∞] × [−∞, +∞]
in terms of vorticity and stream function are written as
∂ω ∂t + u · ∇ω = ν∆ω,(2)∆ψ = −ω,(3)
where u = (u, υ) is the velocity, ν the kinematic viscosity, ψ stream function, and vorticity ω = (0, 0, ω) = ∇×u. The stream function is related to velocity by u = ∂ψ/∂y and υ = −∂ψ/∂x. We use pseudospectral methods to solve Eq. (2) by expanding u, υ, ψ, and ω in a truncated Hermite series.
Expansions
In pseudo-spectral methods, the optimal pseudo-scpectral points are the roots ofĤ N +1 (x), which are denoted by {x} N j=0 with the order With the three-term recurrence:
x 0 > x 1 > . . . > x N . Since x 0 = −x N ≈ √ 2N,H 0 (x) = e −x 2 /2 ,Ĥ 1 (x) = √ 2xe −x 2 /2 , Ĥ n+1 (x) = x 2 n + 1Ĥ n (x) − n n + 1Ĥ n−1 (x), n ≥ 1,(4)
the values ofĤ n (x j ), 0 ≤ n, j ≤ N can be calculated. The 2D vorticity field is then transformed to the Hermite spectral space with the following equation:
ω(x, y) = N kx=0 N ky=0ω kx,kyĤkx (x)Ĥ ky (y),(5)
whereω kx,ky is the Hermite coefficients and k = (k x , k y ) wave numbers for x and y directions. The Hermite coefficients are determined bŷ
ω kx,ky = π (N + 1) 2 N j=0Ĥ ky (y j ) H 2 N (y j ) N i=0Ĥ kx (x i ) H 2 N (x i ) ω(x i , y j ) , 0 ≤ k x , k y ≤ N.(6)
The In this paper, the determination of L is 1) as small as possible; 2) to make sure that the vorticity outside the box of grid points is zero.
Differentials
The Hermite coefficients of first derivatives can be obtained through those of the primitive function with O(N 2 ) operations, for example:
for all 0 ≤ k x , k y ≤ N, (ω x ) kx,ky = k x + 1 2ω kx+1,ky − k x 2ω kx−1,ky , (ω y ) kx,ky = k y + 1 2ω kx,ky+1 − k y 2ω kx,ky−1 ,
with the understanding thatω −1,j =ω j,−1 =ω N +1,j =ω j,N +1 = 0, for 0 ≤ j ≤ N. And the coefficients of the higher-order derivatives can be obtained similarly from those of lower-order derivatives.
Solving the Navier-Stokes Equations: Eq. (3)
The reverse of ψ from ω by Eq. (3) is quite expensive since
−( ω) kx,ky =(ψ xx ) kx,ky +(ψ yy ) kx,ky = (7) (k x + 1)(k x + 2) 2ψ kx+2,ky − 2k x + 1 2ψ kx,ky + k x (k x − 1) 2ψ kx−2,ky + (k y + 1)(k y + 2) 2ψ kx,ky+2 − 2k y + 1 2ψ kx,ky + k y (k y − 1) 2ψ kx,ky−2 .(8)
A sparse (N + 1) 2 × (N + 1) 2 matrix with 5N(N + 8) non-zero elements has to be dealt with. Our numerical experiments show that the resultant equation system is not very well conditioned, and some pivoting is necessary for fairly large N to solve the above equations (e.g., see [24]).
Solving the Navier-Stokes Equations: Eq. (2)
The actual time integration of Eq. (2) is carried out by evaluating the Hermite expansion coefficients in course of time, and we only go back to physical space when we compute the nonlinear term J = u · ∇ω of the vorticity equation. We discretize the advection term J with a 2 nd order Adams-Bashforth scheme. The application of Eq. (2) in spectral space giveŝ
ω t+1 k −ω t k = −∆t( 3 2Ĵ t k − 1 2Ĵ t−1 k ) − ν∆tk 2ωt k ,(9)
whereω t k andĴ t k are the Hermite coefficients of ω and J at time t, respectively. The value ofĴ t k is obtained by the combination of Hermite transforms and the so-called de-aliasing technique by padding or truncation [25].
As a comparison, the time integration for Fourier or Chebyshev spectral solver normally adopts a semi-implicit scheme. For example, the so-called ABCN scheme is adopted in a Fourier code [26]:
ω t+1 k −ω t k = −∆t( 3 2Ĵ t k − 1 2Ĵ t−1 k ) − ν∆t 2 k 2 (ω t+1 k +ω t k ) .(10)
There are two reasons why the explicit time scheme is adopted in this solver:
• It has been shown that the spectral radii for the first and second Hermite differentiation matrices are O( √ N) and O(N), respectively [27]. This places rather weak stability restrictions on the Hermite method, and for the standard heat equation [6], a maximum step size in the time On the other hand, the semi-implicit scheme of the Hermite solver leads to an equation system similar to that of Eq. (7), which means extra nontrivial computation and computer memory.
Our numerical experiment shows that the adoption of the semi-implicit scheme in Hermite simulations does not lead to a larger time step.
Validation and comparison
The main purpose of this section is to validate the Hermite pseudospectral solver, and to show some improvement of it when compared with other numerical schemes. Three numerical experiments are performed:
• a steady axisymmetric solution for 2D NS equation (subsection 3.1);
• an unsteady axisymmetric solution with exact formula to describe its evolution (subsection 3.2);
• an unsteady non-axisymmetric solution without any exact formula to describe its evolutions, but its comparison is also easy since it has been widely adopted for validating new codes in many previous studies (subsection 3.3).
Burgers vortex: a steady solution
The Burgers vortex is a steady viscous vortex maintained by a secondary flow, and provides an excellent example of a balance between convection, intensification and diffusion of vorticity. In the 2D case, the corresponding governing equations are
∂ω ∂t + (−αx + u) ∂ω ∂x + (−βy + υ) ∂ω ∂y = (α + β)ω + ν∆ω,(11)∆ψ = −ω, u = ∂ψ ∂y , υ = − ∂ψ ∂x .(12)
When α = β > 0, ω(x, y) = Ωe −α(x 2 +y 2 )/2ν is an exact steady solution to the above equations [28], where the constance Ω is arbitrary and also a measure of the magnitude of the vorticity.
In the following simulation, the grid points are confined in a box of [−3.5, 3.5] × [−3.5, 3.5]. For other parameters, we choose Ω = 10.0, α = 0.012 and ν = 0.0025.
To justify the result of our simulation, the L ∞ error is introduced as L ∞ = max abs ω simulation (x, y, t) − ω exact (x, y, t) Ω , and the above definition will also be adopted in the next subsection.
Three different resolutions are adopted in this subsection. The errors between the t = 4.0 results from our simulation and the original initial input data are shown in Tab. 1. It seems that, for all simulations, the L ∞ errors are very small, and decrease with higher resolutions.
As discussed before in subsection 2.4, the time steps for different grids in Tab. 1 show that the maximum value of time step is of order O(1/N), even with the explicit time integration methods as described in Eq. (9). The same conclusion can be drawn for all simulations described in this section. The only difference in this subsection is that all time steps here are much smaller that those in Tab. 2 because of the existence of the secondary flow (see the α and β related terms in Eq. (11)).
Lamb-Oseen vortex: a self-similar unsteady solution
Eqs. ω(x, y, t) = 2π 1 + 4νt exp(− x 2 + y 2 1 + 4νt ).
This vortex is named after Horace Lamb and Carl Wilhelm Oseen (the Lamb-Oseen vortex) [29], which models a line vortex that decays due to viscosity. Note that this solution is still axisymmetric and self-similar, but can provide more confirmation about our explicit time integration methods when compared with the steady solution in the previous subsection.
In the following simulations, the grid points are confined in a box of [−2π, 2π] × [−2π, 2π], and viscosity is set to be 0.00037. Three different resolutions are adopted. The errors between the t = 4.0 results from our simulations and Eq. (13) are shown in Tab. 2. Again, for all simulations, the L ∞ errors are very small, and decrease with higher resolutions.
Axisymmetrization of an isolated vortex
In this subsection, we will conduct a study of the axisymmetrization of an isolated vortex based on that used in the 1987 paper by Melander, McWilliams, and Zabusky [3]. The governing equations are ∂ω ∂t
+ u ∂ω ∂x + υ ∂ω ∂y = −ν h ∆ 2 ω,(14)
Here, ν h is hyperviscosity.
The initial conditions are an elliptical vortex with a smooth transition between rotational and irrotational fluid:
ω(x, y, 0) = 20 − 20 exp − κ r exp 1 r−1 r < 1 0 r ≥ 0,(16)
where r ≡ x 2 2 + 2y 2 and κ = 1 2 e 2 ln(2). The similar question has been investigated by quite a few groups, and two numerical schemes are selected and compared here. One is the Fourier pseudospectral method which has high spectral accuracy but double periodical condition [3], and another is the vortex method which is also implemented on the infinite domain [4,30].
The evolution of vortex is central symmetric, but not axisymmetric any more. Three simulations are performed: Fig. 2 also shows that the gird points almost equally distribute near the center, so Run1 and Run2 can roughly be treated as simulations with the same resolution. Of course, the effective resolution of Run3 is the highest among the three simulations.
An easy self-checking technique is to study the time evolutions of maximum vorticity, which is conserved for an inviscid fluid. The relative errors of the maximum vorticity of the three runs are shown in Fig. 3. It is obvious that Run3 has the highest precision, and the error is around 10 −4 all the time. For the two compared runs, the maximum vorticity of Run2 is better conserved than Run1 throughout the whole simulations. Despite the similar effective resolutions, the spectral accuracy of our solver brings Run2 a higher precision than that of Run1. On the other hand, the corresponding error in the vortex methods is about ten times larger than our results (see, e.g. [30]).
It has been shown that for Fourier spectral methods, some artificial boundary conditions such as the sponge-layer are required to absorb the incoming vortex filaments [5]. Without any special treatment, the Fourier spectral method will cause about 17-degree overrotation at t = has the merits of both numerical schemes having been compared: the high spectral accuracy as that of Fourier spectral methods, and the infinite domain computing as that of vortex methods.
Two compared simulations with the same ν h and initial conditions as those of Run3 are performed by the Fourier Pseudo-spectral methods [26]: Except the phenomenon of overrotation, there is no big difference in the contour plots of Fourier and Hermite simulations until t = 4. The overrotation in Run4 causes some vortex filaments to drift away from the view box at t = 8, and some peripheral filaments break at t = 10 (Figs. 6). The large domain size in Run5 not only alleviates the overrotation, but also keeps the vortex filaments unbroken and well confined in the view box (Figs. 7).
Run4
The phenomenon of axisymmetrization can be quantified by considering the effective aspect ratio of vortical structures defined as
λ = G + R G − R ,(17)
where R 2 = D 2 + 4G 2 11 , D = G 20 − G 02 , G = G 20 + G 02 , and G mn = ω(x, y)x m y n dxdy. Despite different ν h s and resolutions, the time evolutions of effective aspect ratios show good agreement for all three simulations in this subsection. Run3, which has a different ν h , begins to have some difference from the other two runs after t = 5. There is almost no difference for the two compared runs until t = 13. Again, the time evolutions of aspect ratio here are in good agreement with those of previous studies (e.g., [30]).
Finally, two numerical schemes for the infinite domain can be compared.
• Vortex methods are developed as a grid-free methodology that is not limited by the fundamental smoothing effects associated with grid-based methods. The small scale and large scale are accurately simulated at the same time. They are especially well-suited to simulating filamentary motion, such as wisps of smoke, and in real-time simulations such as video games, because of the fine detail achieved by using minimal computation. • The Hermite spectral method is a grid-based method. If all non-zero vortices throughout simulations are confined in a small region, the spectral method has a higher accuracy than vortex methods. On the other hand, if some vortex filaments are drifted away from the view box, the Hermite spectral method requires a smaller scale factor to capture them. This means that the grid points are spread on a larger region, and more grid points are needed to guarantee the effective resolution within the region of our main interest. Moreover, a lot of computing time is wasted since vorticity is zero on most grid points.
Conclusions
To sum up, the Hermite spectral scheme presented in this paper can be used in solving 2D NS equations on the infinite domain. Of course, because of the definition of the Hermite function (Eq.
(1)), the resultant solutions of this solver should decay exponentially as (x, y) → ∞. The primary variable in this paper is ω, and the vorticity outside the box of grid points is almost zero, and thus the Hermite spectral methods apply. In the future, the same solver will be adopted to explore the Oseen vortex as a maximum entropy state of a two dimensional flow [1]. The Hermite spectral methods may also be used in exploring the phenomena of thermocapillary drop migration, where velocity far away from the drop is close to zero [31].
Fig. 1 .
1Distribution of grid points for the resolution of 20 × 20. Here, x 0 = 5.55 is the largest root of H 21 (x). Note that the density of points is slightly higher in the center, which is much clearer inFig. 2.
Fig. 2 .
2Distances between grid points and their neighboring points for different resolutions. All points are confined within the [−1, 1] box. The distance is scaled by the shortest distance of the corresponding resolution, which always locates in the center.
the values of vorticity in our solver are defined on the grid points in the box of [-x 0 ,x 0 ]×[-x 0 ,x 0 ] (e.g., see Fig. 1 or Figs. 6 in [7]).
above direct evaluation takes O(N 3 ) operations. With the Fast Hermite Transform [22,23], only O(N 2 log 2 N) operations are needed. In real simulations, it is impossible to focus on the infinite domain, so we always concentrate on some small regions, e.g. [−L, L] × [−L, L]. Hence, the scaling factor a = γ 0 /L is used in Eq. kyĤkx (x)Ĥ ky (y).
(11)&(12) are not real since they are maintained by a secondary flow. Without any external force or second flow, there will be no steady vortex for Eqs. (2)&(3). However, there is an exact time-dependent solution for the initial condition ω(x, y, 0) = 2π exp(−(x 2 + y 2 )):
Fig. 3 .
3The relative errors in the conservation of maximum vorticity for the three Hermite simulations in subsection 3.3.
Fig. 4 .
4Contours of vorticity for Run1 (the first row) and Run2 (the second row). All plots are rotated by 124.6 degree to compare with the results of Figs. 5.2 in[4] with the vortex methods.
Fig. 5 .
5Contours of vorticity for Run3, and all plots are rotated by 90 degree to compare with the vortex method results of Figs. 10 in[30].
Fig. 6 .
6Contours of vorticity for Run4.
Fig. 7 .
7Contours of vorticity for Run5. ∆ψ = −ω, u = ∂ψ ∂y , υ = − ∂ψ ∂x .
Run1 uses 225 × 225 grids within [−π, π] × [−π, π], ν h = 10 −7 , and the time step is 5.0 × 10 −4 ; Run2 uses 450×450 grids within [−2π, 2π]×[−2π, 2π], ν h = 10 −7 , and the time step is 2.5×10 −4 ; Run3 uses 400 × 400 grids within [−π, π] × [−π, π], ν h = 0.3125 × 10 −7 , and the time step is 1.5625 × 10 −4 . Throughout the simulations, the region of our main interest, which covers all non-zero vorticity, is confined in the box of [−2.2, 2.2] × [−2.2, 2.2] (see Figs. 5). For Run1 and Run2, the shortest distances between grid points are almost identical (≈ 0.02).
Table 1
1The error in the Burgers vortex simulations at t = 4.0. direction of order O(1/N) is required, whereas for Fourier and Chebyshev methods it is of order O(1/N 2 ), O(1/N 4 ), respectively. In the actual calculations this means that we need not even consider implicit time integration methods with the Hermite method. • For the Fourier spectral methods, the solution of Poisson equation can be obtained with trivial efforts of O(N 2 ) operations, so the semi-implicit scheme only causes few extra computations.Time step Resolution
L ∞ error
0.0005
120x120
1.03 × 10 −3
0.00025
200x200
5.62 × 10 −4
0.000125
400x400
2.70 × 10 −4
Table 2
2The error in the Lamb-Oseen vortex simulations at t = 4.0.Time step Resolution
L ∞ error
0.0025
120x120
1.38 × 10 −3
0.00125
200x200
6.54 × 10 −4
0.000625
400x400
2.32 × 10 −4
Fig. 8. Evolution of the effective aspect ration of the vorticity field from the three simulations in subsection 3.3.1.65 in
the smaller domain simulation ([−π, π] × [−π, π]), and about 3-degree overrotation for the larger
domain [−2π, 2π] × [−2π, 2π] [4]. However, no overrotation appears for Run1 and Run2 despite
the different sizes of grid boxes because there is no boundary in our simulations. Also, a good
agreement is reached when the vorticity evolutions of our simulations are compared with those
of vortex method (Figs. 4 & 5). At the late stages of simulations, more and more filaments are
generated, and it becomes a challenge for any numerical scheme to capture all these fine vortex
structures. The related vorticity contours of Run3 are presented in Figs. 5 and those very fine
filaments are captured even after t > 8. So, for the current problem, the Hermite spectral method
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t
Effective Aspect Ratio
v h =0.3125X10 −7 , 400X400
v h =10 −7 ,
450X450
v h =10 −7 ,
225X225
uses 512 × 512 grids on the domain of [−π, π] × [−π, π]; Run5 uses 1024 × 1024 grids on the domain of [−2π, 2π] × [−2π, 2π].
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Stablity and structure of stretched vortices. A Robinson, P Saffman, Studies in applied mathematics. 70A. Robinson and P. Saffman, Stablity and structure of stretched vortices, Studies in applied mathematics, 70 (1984) 163-181.
P Saffman, M Ablowitz, E Hinch, J Ockendon, P Olver, Vortex dynamics. CambridgeCambridge University PressP. Saffman, M. Ablowitz, E. Hinch, J. Ockendon, and P. Olver, Vortex dynamics, Cambridge University Press, Cambridge, (1992).
Inviscid axiymmetrization of an elliptical vortex. P Koumoutsakos, Journal of Computational Physics. 138P. Koumoutsakos, Inviscid axiymmetrization of an elliptical vortex, Journal of Computational Physics, 138 (1997) 821-857.
Z Yin, P Gao, W Hu, L Chang, Thermocapillary migrations of nondeformable drops. 2082101Z. Yin, P. Gao, W. Hu, and L. Chang, Thermocapillary migrations of nondeformable drops, Physics of Fluids, 20 (2008) 082101.
| {'fraction_non_alphanumeric': 0.06182977490097575, 'fraction_numerical': 0.037935143142369496, 'mean_word_length': 4.201675041876047, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 5, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, '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': 'The Hermite pseudospectral method is applied to solve the Navier-Stokes equations on a two-dimensional infinite domain. The feature of Hermite function allows us to adopt larger time steps than other spectral methods, but also leads to some extra computation when the stream-function is calculated from the vorticity field. The scaling factor is employed to increase the resolution within the region of our main interest, and the aliasing error is fully removed by the 2/3-rule. Several traditional numerical experiments are performed with high accuracy, and some related future work on physical applications of this program is also discussed.', 'arxivid': '1311.0189', 'author': ['Zhaohua Yin \nInstitute of Mechanics\nNational Microgravity Laboratory\nChinese Academy of Sciences\nNo.15 Beisihuanxilu100190BeijingP.R. China\n'], 'authoraffiliation': ['Institute of Mechanics\nNational Microgravity Laboratory\nChinese Academy of Sciences\nNo.15 Beisihuanxilu100190BeijingP.R. China'], 'corpusid': 15875433, 'doi': '10.1016/j.jcp.2013.10.039', 'github_urls': [], 'n_tokens_mistral': 9949, 'n_tokens_neox': 8510, 'n_words': 5038, 'pdfsha': '37a577d4e05a1fad80bb5a113bb49f5972202d3b', 'pdfurls': ['https://arxiv.org/pdf/1311.0189v1.pdf'], 'title': ['A Hermite pseudospectral solver for two-dimensional incompressible flows on infinite domains', 'A Hermite pseudospectral solver for two-dimensional incompressible flows on infinite domains'], 'venue': []} |
arxiv |
The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion
Student Member, IEEEYifeng Xiong
Senior Member, IEEESoon Xin Ng
Fellow, IEEELajos Hanzo
The Accuracy vs. Sampling Overhead Trade-off in Quantum Error Mitigation Using Monte Carlo-Based Channel Inversion
1
Quantum error mitigation (QEM) is a class of promising techniques for reducing the computational error of variational quantum algorithms. In general, the computational error reduction comes at the cost of a sampling overhead due to the variance-boosting effect caused by the channel inversion operation, which ultimately limits the applicability of QEM. Existing sampling overhead analysis of QEM typically assumes exact channel inversion, which is unrealistic in practical scenarios. In this treatise, we consider a practical channel inversion strategy based on Monte Carlo sampling, which introduces additional computational error that in turn may be eliminated at the cost of an extra sampling overhead. In particular, we show that when the computational error is small compared to the dynamic range of the error-free results, it scales with the square root of the number of gates. By contrast, the error exhibits a linear scaling with the number of gates in the absence of QEM under the same assumptions. Hence, the error scaling of QEM remains to be preferable even without the extra sampling overhead. Our analytical results are accompanied by numerical examples.
submatrix obtained by extracting the i 1 -th to i 2 -th rows and the j 1 -th to j 2 -th columns from A is denoted as [A] i1:i2,j1:j2 . The notation [A] :,i represents the i-th column of A, and [A] i,: denotes the i-th row, respectively. • The trace of matrix A is denoted as Tr{A}, and the complex conjugate of A is denoted as A † . Similarly, the complex adjoint of an operator X is also denoted as X † . • The notation A ⊗ B represents the Kronecker product between matrices A and B. The notation A B denotes the Hadamard product between matrices A and B. • The notations E{·}, Var{·}, and Cov{·} represent the expectation, the variance, and the covariance matrix of their arguments, respectively.
I. INTRODUCTION
N OISY intermediate-scale quantum computers [1] has been one of the most impressive recent advances in the area of quantum computing. In particular, a quantum computer consisting of 53 quantum bits (qubits) has been built in 2019, and has been shown being capable of performing computational tasks that are challenging to be carried out by state-of-the-art classical supercomputers [2].
Due to their limited number of qubits, noisy intermediatescale quantum computers may not be capable of supporting fully fault-tolerant quantum operations relying on quantum error correction codes [3]- [6], which are widely believed to be necessary for implementing complex algorithms that require relatively long coherence time [7]- [9], such as Shor's factorization algorithm [10] and Grover's search algorithm [11]. Alternatively, a class of algorithms tailored for these computers, namely that of the variational quantum algorithms (VQAs) [12]- [15], is receiving much attention. Briefly, VQAs aim for sharing their computational tasks between relatively simple quantum circuits and classical computers. A little more specifically, quantum circuits are employed in VQA for computing a cost function or its gradient [16], which is then fed into an optimization algorithm run on classical computers. The objective of this design paradigm is to assist near-term quantum devices in outperforming classical computers in the context of practical problems, such as solving combinatorial optimization problems using the quantum approximate optimization algorithm [14], [17], [18] and quantum chemistry problems using the variational quantum eigensolver [12].
Although the performance of VQAs has been characterized using some illustrative examples [12], [24]- [26], it is not known whether these examples could be scaled up to problems of larger size. In fact, recent analytical results in [19], [27], [28] support the opposite statement. More explictly, [19] arXiv:2201.07923v1 [quant-ph] 20 Jan 2022 [19] × No QEM Analytical and numerical S. Wang et al. [20] No QEM Analytical and numerical S. Endo et al. [21] Exact channel inversion Sampling overhead vs. accuracy trade-off
Only numerical Y. Xiong et al. [22] Exact channel inversion Analytical and numerical R. Takagi [23] Exact channel inversion Analytical and numerical
Our contributions Monte Carlo-based channel inversion Analytical and numerical proves that the magnitude of the cost function (or its gradient) computed by VQAs vanishes exponentially as the number of qubits n increases. Fortunately, the follow-up investigations [29], [30] found that this so-called "barren plateau" phenomenon may be mitigated to a certain extent by techniques borrowed from the literature of classical machine learning, such as pre-training and layer-by-layer training. However, the authors of [20] show that when decoherence is taken into account, the dynamic range of the computational results also vanishes exponentially upon increasing the circuit depth N L , even if these techniques are applied. To summarize, these results imply that when the quantum circuit is long in depth or large in the number of qubits, the computational error become excessive in practical applications.
To improve the error scaling of VQAs with respect to the depth of the circuits, a body of literature has been devoted to searching for methods that efficiently mitigate the effect of decoherence-induced impairment, without using quantum error correction codes [31]- [34]. Among these research efforts, one of the most promising methods is quantum error mitigation (QEM) [31], which aims for applying an "inverse channel" right after each quantum channel modelling the impact of decoherence. Both the numerical and experimental results of [21], [35] show that QEM is indeed capable of reducing the computational error in VQAs in the context of quantum chemistry problems.
The error reduction capability of QEM comes at the price of a computational overhead. To elaborate, QEM is implemented by sampling from a "quasi-probability representation" of the inverse channel, which would increase the variance of the computational results, hence some computational overhead (termed as "sampling overhead" [21], [22]) is required for ensuring that a satisfactory accuracy can be achieved. By appropriately choosing the total number of samples, one may strike a beneficial computational accuracy vs. overhead trade-off. Therefore, it is important to quantify the sampling overhead, before we can conclude whether QEM can play a significant role in making VQAs practical.
The literature of QEM sampling overhead analysis typically assumes that the channel inversion procedure is implemented exactly [21]- [23]. 1 Under this assumption, the sampling over- 1 We will refer to QEM based on exact channel inversion as "ideal QEM" in the rest of this treatise. head can be characterized by the sampling overhead factor [22], which is determined by the quality of the channel as well as the by basis operations implementing the channel inversion. However, exact channel inversion may be unrealistic in practical scenarios, since it requires a pre-processing stage that is computationally excessive. Moreover, the computational cost of this pre-processing stage increases rapidly with the number of gates, which may negate the benefit of QEM.
Against this background, in this treatise, we consider a practical channel inversion method based on Monte Carlo sampling, which only increases the pre-processing complexity linearly with the number of gates. The drawback of this method is that it cannot invert the channel exactly, hence there would be some residual error that accumulates during computation. Compared to the ideal QEM, this method has a less beneficial accuracy vs. overhead trade-off, because additional samples would be necessary to compensate for the residual error. To characterize this trade-off, we investigate the relationship between the residual error and the number of gates N G , the number of samples N s , and the gate error probability . We boldly and explicitly contrast our contributions to the related recent research on VQAs and QEM in Table I, which are further detailed as follows.
• We analyse the error scaling in the absence of QEM, by providing both upper and lower bounds of the magnitude of the computation error. We show that the error magnitude scales linearly with the number of gates N G , as well as with the gate error probability , when we have application of carrying out multiuser detection in wireless communication systems using the quantum approximate optimization algorithm and show that our analytical results do apply. The rest of this treatise is organized as follows. In Section II, we present the formulation of VQAs and the channels modelling the decoherence. Then, in Section III, we discuss a pair of QEM implementation strategies, namely the Monte Carlo-based QEM and the exact channel inversion. Based on this discussion, in Section IV, we analyse the error scaling behaviours of these two QEM implementations, respectively, under the assumption that they use the same number of circuit executions. We provide further intuitions concerning these analytical results in Section V, with an emphasis on the accuracy vs. sampling overhead trade-off, complemented by numerical examples in Section VI. Finally, we conclude in Section VII.
N G 1. • We propose
II. FORMULATION OF VARIATIONAL QUANTUM ALGORITHMS
A typical VQA iterates between classical and quantum devices, as portrayed in Fig. 1. The parametric state-preparation circuit (also known as the "ansatz" [17]) transforms a fixed input state to an output state, according to the parameters chosen by a classical optimizer. The output state is then measured and fed into a quantum observable, which maps the measurement outcomes to the desired computational results. The results correspond to the value of a cost function or its gradient, which in turn serve as the input of the associated classical optimization algorithm. The iterations continue until certain stopping criterion is met, for example, the computed gradient becomes almost zero.
In this treatise, we focus on the error induced by the sampling procedure in QEM, hence we consider the computational result of a single iteration, meaning that the parameters used for state preparation are fixed. We model the decoherenceinduced impairment in the parametric state-preparation circuit as quantum channels acting upon the associated quantum states at the output of perfect quantum gates, as exemplified by the simple circuit shown in Fig. 2. In this figure, C k , k = 1 . . . 4 represents the channel modelling the decoherence in the k-th quantum gate, while G k represents the k-th ideal decoherencefree quantum gate.
In the subsequent subsections, we present the mathematical formulations of the system models shown in Fig. 1 and 2.
|ψ 1 G 1 C 1 G 2 C 2 |ψ 2 G 4 C 4 |ψ 3 G 3 C 3 1 Fig. 2.
Simple example of the noisy parametric state-preparation circuit seen in Fig. 1.
A. Operator-sum Representation
Without loss of generality, we assume that the input state of the circuit is the all-zero state |0 ⊗n , where n is the number of qubits. In general, when the circuit is decoherence-free, the computational result of a variational quantum circuit may be expressed as
r = 0| ⊗n NG k=1 G † i M ob NG k=1 G NG−k+1 |0 ⊗n ,(1)
where N G is the number of gates in the circuit, G k denotes the k-th quantum gate, and the operator M ob represents the quantum observable, which describes the computational task as a linear function of the final state.
If we consider a more practical scenario, where the quantum state evolves owing to quantum decoherence as the circuit operates, the state can no longer be fully characterized using the state vector formalism. Instead, we may use the density matrix formalism. In particular, the input state may be described as
ρ 0 = (|0 0|) ⊗n .(2)
Correspondingly, the output state of the k-th imperfect quantum gate may be represented in an operator-sum form [36,Sec. 8.2.4], relying on following recursive relationship
ρ k = C k G k ρ k−1 G † k ,(3)
where the operator C k is characterized by
C k (ρ) = n k i=1 E k,i ρE † k,i ,(4)
representing the channel modelling the imperfection of the kth gate. The matrices E k,i represent the operation elements [36, Sec. 8.2.4] of the channel C k satisfying the completeness condition of
n k i=1 E † k,i E k,i = I.
Finally, when all gates completed their tasks and the measurement results have been obtained, the computational result may be expressed as
r = Tr {M ob ρ NG } .(5)
B. Pauli Transfer Matrix Representation
In the standard operator-sum form [36,Sec. 8.2.4], the quantum states are represented by matrices. However, in many applications, such as the error analysis considered in this treatise, it would be more convenient to treat them as vectors. Correspondingly, the quantum channels and gates would then be represented by matrices. To this end, the Pauli transfer matrix (PTM) representation of quantum operators was proposed in [37], which allows a quantum operator O to be expressed as
[O] i,j = 1 2 n Tr {S i O(S j )} ,(6)
where S i denotes the i-th Pauli operator in the n-qubit Pauli group. Similarly, a quantum state ρ can be expressed as
[ρ] i = 1 √ 2 n Tr {S i ρ} .(7)
Under the PTM representation, the computational result may be rewritten as
r = v T ob NG k=1 (C NG−k+1 G NG−k+1 ) v 0 ,(8)
where G k represents the k-th perfect gate, and C k represents the channel modelling the imperfection of the k-th gate. The vector v 0 denotes the initial state, whereas v ob is the vector representation of the quantum observable M ob .
To simplify the notation, we define
R k := k i=1 (C k−i+1 G k−i+1 ), R k := k i=1 G k−i+1 .(9)
Especially, for k = 0, we define R 0 = R 0 = I. The output state of the k-th quantum gate can then be expressed as
v k : = R k v 0 = C k G k v k−1 .(10)
Hence we have
r = v T ob v NG = v T ob R NG v 0 .(11)
C. Channel Model
In this treatise, we consider Pauli channels [38], for which the Pauli transfer matrices take the following form
C k = diag {c k } ,(12)
where
c k = Hp k ,(13)
with H denoting the Hadamard transform, whereas p k represents a probability distribution satisfying 1 T p k = 1, p k ≥ 0.
III. QEM AND ITS IMPLEMENTATION STRATEGIES
Ideally, for a channel C k , QEM would apply its inverse based on a linear combination of predefined quantum operations, taking the following form
C −1 k = L l=1 α (k) l O l ,(14)
where O l is the l-th quantum operation, while α k := [α 1 . . . α L ] T is the quasi-probability representation vector satisfying 1 T α k = 1. This linear combination may be rewritten as a probabilistic mixture of the quantum operations as follows:
C −1 k = α k 1 L l=1 s (k) l p (k) l O l ,(15)
where s (k) l and p (k) l are the l-th entries of s k and p k , respectively, given by
p (k) i = |α (k) i | α k 1 , s k = sgn{α k }.(16)
Note that the vector p k describes a probability distribution. Typically, the probabilistic mixture in (15) is implemented by generating a set of candidate circuits and performing postprocessing on the output of these circuits. In the following subsections, we will discuss different candidate selection strategies and their characteristics.
A. Exact Implementation and Sampling Overhead
The inverse channel C −1 k in (15) is assumed to be implemented exactly in the seminal paper [31] that proposed QEM for the first time, as well as in many other existing contributions [21]- [23]. Exact implementation implies that, each quantum operation O l should appear in exactly N p (k) l candidate circuits in every N samples of the computational result.
The assumption of exact implementation significantly simplifies the performance analysis of QEM. In particular, it leads to a clear and concise formula of sampling overhead, which describes the computational overhead imposed by the variance-boosting effect of QEM. To elaborate, assume that the variance of the computational result is σ 2 based on N 0 samples. According to (15), if we implement the inverse channel C −1 k , the variance would become α k 2 1 σ 2 . Therefore, in order to achieve the same accuracy as the case without QEM, we should acquire N 0 ( α k 2 1 − 1) additional samples. If we further assume that all gates are protected by QEM, we have the following formula for the total sampling overhead
N exact = N 0 NG k=1 α k 2 1 − 1 .(17)
The simplicity of (17) is largely due to the assumption of exact implementation. Despite its theoretical convenience, the practicality of exact implementation is doubtful. Specifically, the number of the lth candidate circuit, N p (k) l has to be an integer, which might be unrealistic for an arbitrary p (k) l . Furthermore, the number of the probability parameters p (k) l would increase exponentially as the number of QEM-protected gates increases, which may render the candidate circuit selection procedure computationally prohibitive when N G is large. Motivated by these drawbacks, we propose to use a Monte Carlo implementation of QEM, detailed in the next subsection.
B. Monte Carlo Implementation
In the Monte Carlo implementation, we first sample from the probability distribution p k for each gate, and obtain N samples constituting a set L = {l 1 , . . . , l N }, where for all k we have l k = 1, 2, . . . , L. Thus we may approximate the inverse channel as
Γ k = α k 1 N N i=1 s (k) li O li = α k 1 L l=1 s (k) l p (k) l O l ,(18)
where
p (k) m = 1 N N i=1 I{l i = m}.
The advantage of the Monte Carlo approach is that it may result in a much lower complexity for candidate circuit generation, compared to the exact implementation. To elaborate further, as a "toy" example, for a circuit consisting of two gates we have
Γ 2 G 2 Γ 1 G 1 = α 1 1 α 2 1 N N i=1 s (1) li,1 s (2) li,2 O li,2 G 2 O li,1 G 1 ,(19)
where G k = C k G k , and l i,k denotes the i-th sample drawn from the distribution p k . This implies that in order to obtain a sample for the entire circuit, we may simply generate one sample for each gate, and concatenate them as shown in the right hand side of (19). Compared to the exact implementation, the Monte Carlo implementation can generate an arbitrary number of circuit samples N , at a relatively low computational cost of O(N N G ).
The reduced complexity of the Monte Carlo implementation comes with a cost of inaccurate channel inversion, since G k is only an approximation of C −1 k . Hence there would be a residual channel for each gate, which is given by
C k = Γ k C k .(20)
A natural question that arises is, whether the additional computational error caused by these residual channels would erode the error reduction capability of Monte Carlo-based QEM. In the rest of this treatise, we will discuss the impact of these residual channels on the accuracy vs. sampling overhead tradeoff.
IV. ERROR SCALING ANALYSIS OF MONTE CARLO-BASED QEM
In this section, we discuss the error scaling behaviour of quantum circuits protected by Monte Carlo-based QEM, and contrast the results to that of circuits without QEM protection. In order to make a fair comparison, we consider the following assumptions.
A. Assumptions
Assumption 1 (Bounded gate error rate): The error probability of each quantum gate is upper bounded by u .
Since we consider Pauli channels in this treatise, the gate error probability corresponding to a quantum channel C k (under its PTM representation) may be computed as
(C k ) = 1 − 1 4 n Tr {C k } .(21)
Assumption 2 (Bounded observable): The eigenvalues of the quantum observable M ob are bounded in the interval [−1, 1].
Assumption 2 ensures the boundedness of the computation result r. In this treatise we assume that the upper and lower bounds are 1 and −1, respectively, but they may be replaced with any other constant real numbers without affecting our analytical results. The assumption may also be rewritten as
max v∈S n v T ob vv T v ob ≤ 1,(22)
where S n denotes the space of all density matrices over n qubits. Furthermore, the assumption also implies that
v ob 2 ≤ √ 2 n .(23)
This follows from the fact that v ob 2 = M ob F , and that
M ob F = 2 n i=1 λ i (M ob ) ≤ √ 2 n ,
where λ i (·) denotes the i-th largest eigenvalue of its argument. Assumption 3 (Zero bias term): We assume that
Tr {M ob } = √ 2 n [v ob ] 1 = 0.(24)
Note that [v ob ] 1 is the coefficient of the identity operator, which serves as a bias term in the computation result being constant with respect to the quantum state. Thus this assumption does not restrict the generality of our results.
B. Benchmark: Error Scaling in the Absence of QEM
In this subsections, we characterize the error scaling of quantum circuits that are not protected by QEM. The results will serve as important benchmarks in the following discussions. Let us start with a bound of the dynamic range of computational results, which will lead to a lower bound of the computational error.
Proposition 1: Assume that each qubit would be processed by at least N L gates, and that for each of these gates, the probability of each type of Pauli error (i.e., X error, Y error or Z error) on each qubit is lower bounded by l . The computational result r exhibits the following convergence behaviour:
|r| ≤ exp (−4 l N L ) .(25)
Proof: Please refer to Appendix I.
Proposition 1 implies that decoherence would force the computation result to be almost independent of the quantum observable v ob in an asymptotic sense. Indeed, as indicated by (25), when N L is large, r is only determined by the first entry of v ob . Moreover, consider the case where | r| ≥ 1 − c holds for all N G , the computational error is lower bounded as
|r − r| ≥ 1 − c − exp (−4 l N L ) .(26)
From the Taylor expansion
exp (−4 l N L ) = 1 − 4 l N L + (4 l N L ) 2 2 − · · · ,
we see that when L N L 1, the lower bound is approxi-
mately |r − r| 4 l N L − c,(27)
which increases linearly with respect to l N L . We may also provide an upper bound for the computational error as follows.
Proposition 2: The computational error can be upper bounded as
|r − r| ≤ 2 u N G .(28)
Proof: Please refer to Appendix II. Combining Propositions 1 and 2, we see that the computational error grows linearly with N G , when the number of gates in each "layer" is constant (hence N L is a constant multiple of N G . This is typically true for VQAs.
C. The Statistics of the Residual Channels
Before diving into details about the error scaling, in this subsection, we first investigate the characteristics of the residual channels of gates protected by Monte Carlo-based QEM.
According to the sampling overhead analysis in [22] based on the assumption of exact channel inversion, if we wish to execute the decoherence-free circuit N s times, we should sample from the probabilistic mixture of candidate circuits for as many as
N = N s α k 2 1(29)
times, in order to keep the variance of the computational result unchanged by the channel inversion procedure. Here we consider the Monte Carlo-based channel inversion using the same number of samples, hence we have
p (k) m = 1 N s α k 2 1 Ns α k 2 1 i=1 I{l i = m}.(30)
Of course, the Monte Carlo-based channel inversion has lower accuracy compared to the exact channel inversion, when they use the same number of samples. The accuracy could be improved by using additional samples, which will be discussed in more detail in Section V-B. After the sampling procedure, α k is approximated by α k taking the following form
α k = α k 1 · s k p k = α k 1 · s k (p k + n) = α k + α k 1 · s k n,(31)
where n denotes the sampling error. In general, the approximated inverse channel may be expressed in terms of α k as
Γ k = L i=1 [ α k ] i O i ,(32)
where {O i } L i=1 is a set of operators forming a basis of the space where the imperfect gate C k G k resides in. Interested readers may refer to Table 1 in [22] for an example of such operator sets. For the Pauli channels considered in this treatise, Γ has a simpler form. Specifically, using (12), the quasiprobability representation vector may be expressed as
α k = H −1 (1/c k ),
where H is the Hadamard transform over n qubits, and H −1 is the corresponding inverse transform given by H −1 = 1 4 n H. The approximated inverse channel can now be expressed as
Γ k = diag −1 { H α k },(33)
and thus the residual channel takes the following form
C k = diag −1 { H α k c k }.(34)
To simplify our further analysis, we introduce c k := H α k c k , where c k may be further expressed as
c k = 1 + α k 1 · H(s k n) c k = 1 + α k 1 · c k n,(35)
and n := H(s k n). Note that the vector p k is a multinomial distributed random vector, satisfying
E{ p k } = p k , Cov{ p k } = 1 N s α k 2 1 P k − p k p T k ,(36)
where P k = diag {p k }. Therefore, the vector n satisfies
E{ n} = 0, Cov{ n} = HCov{ p k } H,(37)
since the sign vector s k does not have an effect on the covariance matrix. Thus we have the following results for c k :
E{ c k } = 1, Cov{ c k } = α k 2 1 · Cov{c k n} = 1 N s H P k − p k p T k H c k c T k .(38)
For simplicity of further derivation, we use the notation of Ξ k := Cov{ c k }.
D. Error Scaling in the Presence of Monte Carlo-based QEM
In this subsection, we investigate the scaling law of computational error when the quantum circuit is protected by Monte Carlo-based QEM, based on the above discussions concerning the residual channels in the previous subsection.
We note that for QEM-protected circuits, the computational result is a random variable due to the randomness in the sampling procedure, given by
r = v T ob v NG ,(39)
where
v k = R k v 0 = C k G k v k−1 .(40)
After defining these quantities, we may obtain the following bound on the RMSE of the computational result r.
Proposition 3 (Square-root Increase of QEM Inaccuracy): For a quantum circuit consisting of N G gates which is protected by QEM, the RMSE of the computational result is upper bounded by
E{(r − r) 2 } ≤ 2 n/2 exp(2N G N −1 s ) − 1.(41)
This result is not restricted to Pauli channels. In fact, it applies to all completely positive trace-preserving channels, when the basis operators {O i } L i=1 are all completely positive tracenonincreasing operators.
Proof: Please refer to Appendix III for the proof of this proposition, as well as additional discussions on Pauli channels as a special case.
Note that by applying the Taylor expansion to exp(2N G N s ), we have
exp(2N G N s ) − 1 = 2 N s N G + 1 2 2 N s N G 2 + · · · ,
which is approximately 2N G N −1 s , when N G N s . This means that when the RMSE is far less than 1, its scaling law is given by
O( √ N G / √ N s )
. This is particularly useful, since in typical applications (e.g., variational quantum algorithms), having an RMSE close to 1 would be excessive.
In Proposition 3, the dependence of the RMSE on the error probability of quantum gates is not demonstrated. According to (91) of the Appendix, for Pauli channels, this dependence mainly relies on the term Ξ k max . Next we expound a little further on this issue based on Assumption 1.
Proposition 4 (Improved Bound for Pauli Channels): Under Assumption 1, we have the following refined upper-bound for the RMSE of the computational result under Pauli channels:
E{(r − r) 2 } ≤ 2 n/2 exp ˜ N G N −1 s − 1 ≈ 2 n/2 exp 10 u N G N −1 s − 1(42)
where˜ is given by˜
:= 5 2 σ u + 1 4 σ 2 u ,(43)
and
σ u := 4 u · 1 − u (1 − 2 u ) 2 .(44)
The approximation is valid when u 1. Proof: Please refer to Appendix IV. Verification of the approximation in (42) is straightforward: one may simply substitute (43) and (44) into (42). This proposition implies that, when u N G N s , the RMSE is on the order of O( √ u N G / √ N s ). Engendered by our specific proof technique, the factor 2 n/2 in (41) and (42) seems to be an artifact. According to the numerical results which will be presented in Section VI, we conjecture this factor is essentially unnecessary, implying that
E{(r − r) 2 } ≤ exp N G N −1 s − 1.(45)
Regretfully, it seems to be technically challenging to remove this factor from the bounds. Further investigations into this issue will be left for our future research.
V. DISCUSSIONS
A. Intuitions about the Error Scaling with the Circuit Size
As indicated by the results in Section IV, with respect to N G , we observe an O( √ N G ) scaling of the computational error of circuits protected by Monte Carlo-based QEM, when the number of samples is the same as that of QEM based on exact channel inversion. By contrast, when QEM is not applied, the computational error scales as O(N G ), as discussed in Section IV. Thus we may conclude that, although there are residual channels due to the inexact channel inversion, Monte Carlo-based QEM can still slow down the accumulation of computational error.
Revisiting the low-complexity example of a single-qubit circuit, we may understand these error scaling behaviours more intuitively. Specifically, the entire space of all legitimate single-qubit quantum states can be described by the celebrated Bloch sphere [36, Sec. 1.2]. As demonstrated in Fig. 3, the Bloch sphere would shrink as N G increases when no QEM is applied, since the completely positive trace-preserving quantum channels are contractive transformations. This is in stark contrast with the case where Monte Carlo-based QEM is applied, when the Bloch sphere becomes "blurred" as N G increases, since it is not determined whether the sphere will expand or shrink after each gate. Consequently, the sphere may expand after one gate and then shrink after another, hence the corresponding computational errors would cancel each other to a certain extent.
In light of the aforementioned intuition, we may interpret the error scaling of Monte Carlo-based QEM in following informal way. Assume that every gate k would transform the Bloch sphere in a way that its radius becomes (1 + λ k ) times that of its original value, where λ k is a zero-mean random variable with variance σ 2 k . If additionally all λ k values are mutually independent, we can see that
1 N G NG k=1 ln(1 + λ k ) ∼ N − 1 2 σ 2 , σ 2
holds asymptotically as N G → ∞ by applying the central limit theorem, where we have:
σ 2 = 1 N G NG k=1 σ 2 k .
Hence the radius of the Bloch sphere after k gates, denoted by a k , tends to be a log-normally distributed random variable characterized by
E{a k } = 1, Var{a k } = exp N G σ 2 − 1.
Therefore, the standard deviation of the Bloch sphere's radius tends to be exp (N G σ 2 ) − 1, which is on the order of O √ N G σ 2 when N G σ 2 1. This agrees with our formal analytical results.
The linear error scaling experienced in the case where no QEM is applied may be interpreted by considering the graphical illustration in Fig. 4. Since the computational result r converges exponentially fast to zero as indicated by Proposition 1, it deviates from r linearly when N G is relatively small, which may be viewed as a lower bound of the computational error. Additionally, the actual evolution of r is also bounded by the tangent line of it at N G = 0, which gives rise to the error upper-bound in Proposition 2.
B. The Accuracy vs. Sampling Overhead Trade-off
If we denote by the average error probability of each gate, from the discussion in Section IV we see that the computational error roughly scales as Θ( ) when QEM is not applied, whereas it scales as O( √ ) when Monte Carlobased QEM is applied, according to Section IV-D. This may be understood by considering the variance of the samples, which is proportional to . Hence the RMSE is proportional to √ . Since is typically far less than 1, it seems that Monte Carlo-based QEM has a less preferable performance. Nevertheless, it is noteworthy that the error scaling in the QEMprotected case is actually O( N −1 s ), where the number N s of effective circuit executions is a configurable parameter. The O( N −1 s ) dependency on N s originates from the fact that the sampling variance scales as O(N −1 s ). Therefore, our results should not be viewed as indicating the superiority of non-QEM-based solutions. Rather, they should be viewed as a suggestion on the specific selection of N s , in the sense that it should be on the order of to ensure the error scaling is as beneficial as that of the family of non-QEM-based solutions.
Similarly, by increasing N s as a function of N G , one could also improve the error scaling of Monte Carlo-based QEM with the circuit size. Indeed, since the error of Monte Carlobased QEM scales as O( N G N −1 s ), we can choose an N s that is proportional to N G in order to attain a constant error with respect to N G . Note that the exact channel inversion also has a constant error with respect to N G in the asymptotic limit of N G → ∞. Therefore, using Monte Carlo-based QEM, we could use N G times the number of samples to attain the same error scaling as that of QEM based on exact channel inversion. In practical scenarios, however, this may be an excessive sampling overhead. Fortunately, even if we use the same number of samples as that of the exact channel inversion, Monte Carlo-based QEM still exhibits a quadratic error scaling improvement compared to the no-QEM-based case.
C. The Intrinsic Uncertainty of the Computational Results
In the previous discussions, we followed the definition of computational results in (5). But even if the gates are decoherence-free, the intrinsic uncertainty of quantum states may bring some randomness to the computational result. To be specific, for a quantum state ρ, the variance of a quantum observable O may be computed as follows [13]:
Var ρ {O} = Tr O 2 ρ − (Tr {Oρ}) 2 ,(46)
which quantifies the intrinsic uncertainty of the state ρ under the observable O. If the quantum circuit is executed N s times, the variance is then given by N −1 s Var ρ {O}, and hence the mean-squared error (MSE) may be expressed as
MSE = (r − r) 2 + 1 N s · Var ρ {O}.(47)
We first consider the case where QEM is not applied. Since in VQAs, the observable M ob is typically implemented using a Pauli operator decomposition, its variance may also be decomposed as
Var ρ N G {M ob } = 4 n i=1 1 2 n [v ob ] 2 i Var ρ N G {S i }.(48)
For each Pauli operator, we have
Var ρ N G {S i } = Tr S 2 i ρ NG − (Tr {S i ρ NG }) 2 = 1 − (Tr {S i ρ NG }) 2 .(49)
Hence
Var ρ N G {M ob } = 4 n i=1 1 2 n [v ob ] 2 i (1 − (Tr {S i ρ NG }) 2 ) = v T ob 1 2 n I − V 2 NG v ob ,(50)
where V NG = diag {v NG }. Note that from (7) we have
[v NG ] 2 i ≤ 2 −n for all i, hence it follows that 0 ≤ Var ρ N G {M ob } ≤ 1.(51)
Thus the MSE of the computational result is bounded by
(r − r) 2 ≤ MSE ≤ (r − r) 2 + 1 N s .(52)
When circuits are protected by QEM, it has been shown that [31] if the number of effective executions is N s , the variance equals to that in the case where QEM is not applied. Thus the total error scales on the order of
O N G N s + O 1 N s .
This implies that, the effect of QEM may not be very significant when N G 1. But note that when ρ NG corresponds to one of the eigenstates of all Pauli operators i having non-zero
coefficient [v ob ] i , we have Var ρ N G {S i } = 1 − (Tr {S i ρ NG }) 2 = 0,
which follows from that fact that Pauli operators only have eigenvalues of ±1. Therefore, QEM would be more effective when the final state ρ NG is close to one of these eigenstates.
VI. NUMERICAL RESULTS
In this section, we evaluate the analytical results presented in the previous sections via numerical examples. If not otherwise stated, the following parameters and assumptions will be used throughout the section.
• The number of effective circuit executions is N s = 5000; • For Monte Carlo-based QEM, we use the same number of samples (i.e., actual circuit executions) as that of QEM based on exact channel inversion; • The quantum channels modelling the gate imperfections are single-qubit depolarising channels having gate error probability 10 −3 .
A. Rotations Around the Bloch Sphere
We first consider the simplest scenario, where the quantum circuits are constituted of single-qubit gates, because these simple circuits allow us to clearly observe the error scaling described in the previous sections. In particular, we consider the circuits shown in Fig. 5. The quantum observable M ob in this example is the Pauli Z operator Z on the qubit, which satisfies
Z |0 = |0 , Z |1 = − |1 .
The corresponding PTM representation is given by
v ob = [0 0 0 √ 2] T . |0 X X · · · X 1 (a) Repeated Pauli X gates. Repeat N G /3 times |0 H R z (θ) H (b)
Repeated θ-rotations around the X-axis. For the circuit consisting of repeated Pauli X gates shown in Fig. 5a, the RMSE of the computational results both with and without QEM protection is demonstrated in Fig. 6a, as a function of N G . As it can be observed from the figure, when N G is relatively small, the RMSE of circuits operating without QEM protection grows linearly with N G , while the RMSE of circuits protected by Monte Carlo-based QEM scales as O( √ N G ). The RMSE of QEM based on exact channel inversion scales as O( √ N G ) for small N G , but converges to a constant (≈ N −1 s ) when N G is large. Furthermore, when N G is large, the RMSE of circuits operating without QEM protection converges to a constant. This agrees with Proposition 1, which indicates that their computational results converge to zero regardless of the quantum observable.
The RMSE scalings with respect to the gate error probability are shown in Fig. 6b, where we choose N G = 10, while the number of effective circuit executions, namely N s = 5000, does not vary as the gate error probability increases. It is noteworthy that when is small, the RMSE of circuits operating without QEM protection is lower than that of their counterparts protected by QEM. This phenomenon may be understood from our discussion in Section V-B, where we have indicated that the error scaling of QEM-protected circuits is O( N −1 s ). Compared to the O( ) scaling of non-QEM-protected circuits, the RMSE may be higher when is much smaller than N s . Interestingly, as seen from the figure, the square root scaling with respect to becomes preferable to the linear scaling when is relatively large. For the circuit comprising repeated rotations around the Xaxis, as illustrated in Fig. 5, we set θ = π/256, and the results are plotted in Fig. 7. Observe that the envelope of the RMSE curves exhibit similar scaling behaviours as those in Fig. 6b, but there are some oscillations. To understand the RMSE oscillations of circuits protected by Monte Carlo-based QEM, from (90) we may express the covariance matrix of v k as follows
Σ k = (11 T + Ξ k ) G k Σ k−1 G † k + Ξ k µ k µ T k .
(53) Note that the term Ξ k ⊗ µ k µ T k varies with k under the observable M ob = Z, and hence the RMSE is oscillatory.
The RMSE oscillation of non-QEM-protected circuits may be better understood by investigating the evolution of the computational result r as N G increases, which is portrayed in Fig. 8. It can be seen that the mean values of the non-QEM-protected circuits fit nicely within the bounds given by Proposition 1. Furthermore, the RMSE of QEM-protected cir- cuits is mainly contributed by the variance of the computation results, while the RMSE of circuits not protected by QEM is mainly determined by the mean value, since in the latter case the bias is far larger than the standard deviation. As the dynamic range of the mean values is reduced, by coincidence, there are multiple intersections of the ground truth and the mean values, and thus the computational error of non-QEMprotected circuits oscillates as N G increases. Finally, we demonstrate that some non-Pauli channels may also exhibit the O( √ N G ) error scaling. In particular, we consider amplitude damping channels [39] having the following PTM representation
C damp = 1 0 0 0 0 √ 1 − γ 0 0 0 0 √ 1 − γ 0 γ 0 0 1 − γ ,(54)
where γ is the amplitude damping probability. Here, we set the amplitude damping probability to γ = 1 × 10 −3 . The RMSE scalings with respect to the number of gates N G are shown in Fig. 9a for the circuit comprising repeated Pauli X gates, and in Fig. 9b for the circuit consisting of repeated (π/256) rotations around the X-axis. We observe that the curves corresponding to QEM based on exact channel inversion and those corresponding to Monte Carlo-based QEM exhibit the O( √ N G ) scaling behavior, while the non-QEMprotected curves scale as O(N G ), which is similar to the error scaling behaviour under Pauli channels as portrayed in Fig. 6 and Fig. 7.
B. The Quantum Approximate Optimization Algorithm Aided Multi-User Detection
In this subsection, we apply our analytical results to a practical variational quantum algorithm, the quantum approximate optimization algorithm [14], which aims for solving combinatorial optimization problems assuming the following form
max z∈{−1,+1} n K k=1 w k n k i=1 z l k,i ,(55)
where z = [z 1 . . . z n ] T , and l k,i ∈ {1, 2, . . . , n}. In the formulation of the quantum approximate optimization algorithm, the problem (55) is transformed into the maximization of ψ| H |ψ , where the quantum observable H is given by
H = K k=1 w k n k i=1 Z l k,i .(56)
The trial state |ψ is prepared using a parametric circuit having an alternating structure, so that
|ψ = e −iβ P B e −iγ P H · · · e −iβ1B e −iγ1H |+ ⊗n ,
where P is the number of stages in the alternating circuit, and B is the "mixing Hamiltonian" [17] given by B = n i=1 X i . The parameters β = [β 1 . . . β P ] T and γ = [γ 1 . . . γ P ] T are typically obtained using via an optimization procedure implemented on classical computers [12]. For the purpose of this treatise, here we do not optimize the parameters, but use the following (suboptimal) adiabatic configuration [40] instead
γ k = kP −1 , β k = 1 − kP −1 .
We consider the multiuser detection problem of wireless communications [41]. In particular, assuming that the modulation scheme is BPSK, in a spatial division multiple access system, the signal received at a base station equipped with m antennas from n single-antenna uplink transmitters may be expressed as
y = Hx + ω,
where H ∈ R m×n denotes the channel, x ∈ {−1, +1} n represents the transmitted signal, and ω ∈ R m is the noise. We assume here that the noise is i.i.d. Gaussian. Hence the maximum likelihood estimate of x is given bŷ
x ML = arg max x∈{−1,+1} n 2(Hy) T x − x T H T Hx.
This may be further reformulated as the maximization of the quadratic form ψ| H |ψ , where
H = 1 Z n i=1 [H T y] i Z i − n−1 i=1 n j=i+1 [H T H] i,j Z i Z j ,(57)
and Z is a normalizing coefficient ensuring that the quantum observable H satisfies our Assumption 2.
In this illustrative example, we consider the case where m = n = 4, and [ω] i ∼ N (0, 0.0631), ∀i, such that the signal-tonoise ratio is 12dB. We assume furthermore that the channels between each pair of antennas are uncorrelated non-dispersive Rayleigh channels, hence the entries of the channel H are i.i.d. Gaussian variables with zero mean and variance m −1 [42]. For the quantum circuits, we choose gate error probability = 3 × 10 −4 . Under these assumptions, the RMSE scalings with respect to P of non-QEM-protected circuits and that of circuits protected by Monte Carlo-based QEM are portrayed in Fig. 10. It can be observed that the non-QEM-protected circuits exhibit an O(P ) scaling, while the QEM-protected circuits exhibit an O( √ P ) scaling, as indicated by Propositions 2 and 4, respectively.
To illustrate the evolution of the computational results during the execution of circuits, we plot the objective function values (i.e., ψ| H |ψ ) computed at each stage k of the circuits in Fig. 11, for the case where P = 225. Note that the results computed by the QEM-protected circuits converge monotonically towards the optimum, for which the main source of error is the variance. By contrast, for the non-QEM-protected circuits, the results were on the right track for k < 100, but soon they deviate from their QEM-protected counterparts, and start to converge to zero. In this example, the bound (25) is not as tight as it was in Section VI-A, but it still indicates that the dynamic range of the results computed by non-QEM-protected circuits decays exponentially as k increases.
VII. CONCLUSIONS
The trade-off between the computational overhead and the error scaling behaviour of both quantum circuits protected Proof: First observe that the matrix representation of a perfect gate G i as well as that of a channel C i take the following block-diagonal form
G i = 1 0 T 0 U i , C i = 1 0 T 0 D i ,(58)
where U i is a unitary matrix, whereas D i is a diagonal matrix having diagonal entries taking values in the interval [0, 1]. Since the matrix R NG is the product of several G i and C i , it becomes clear that its largest singular values satisfies σ 1 (R NG ) = 1, and its second largest singular value satisfies
σ 2 (R NG ) ≤ NG i=1 D i 2 .(59)
Furthermore, we have
r − 1 2 n Tr {M ob } ≤ σ 2 (R NG )(60)
due to the "bounded observable" Assumption 2. Note that the quantity N L defined in this proposition is related to the depth of the circuit. To elaborate, if we say that "a layer of gates" is executed if each qubit has been processed by at least one gate, then the entire circuit consists of at least N L layers. For each single-qubit channel C in these layers, due to the assumption that each single-qubit Pauli error occurs at probability of at least l , the following bound holds:
C = I − 2diag {[p X + p Z p Y + p Z p X + p Y ]} (1 − 4 l ),(61)
where p X , p Y and p Z are error probabilities corresponding to the Pauli-X, Y and Z errors, respectively. Thus we obtain
σ 2 (R NG ) ≤ (1 − 4 l ) NL = exp{N L ln(1 − 4 l )} ≤ exp(−4 l N L ).(62)
Hence the proof is completed.
APPENDIX II PROOF OF PROPOSITION 2
Proof: In this proof, we will work under the operator-sum representation of quantum channels. Since we consider Pauli channels, the recursion (3) may be rewritten as
ρ k = 4 n i=1 [p k ] i S i G k ρ k−1 G † k S i = [p k ] 1 G k ρ k−1 G † k + 4 n i=2 [p k ] i S i G k ρ k−1 G † k S i .(63)| r − Tr {M ob ρ NG }| ≤ (| r| + M ob 2 ) 1 − NG k=1 [p k ] 1 ≤ 2 1 − NG k=1 [p k ] 1 .
(64) According to Assumption 1, for any k, the vector p k satisfies
[p k ] 1 ≥ 1 − u , 4 n i=2 [p k ] i ≤ u .(65)
Therefore, we have
| r − Tr {M ob ρ NG }| ≤ 2(1 − (1 − u ) NG ) ≤ 2 u N G .(66)
Hence the proof is completed.
APPENDIX III PROOF OF PROPOSITION 3
Proof: We first expand the expression of MSE as follows
E{(r − r) 2 } = E v T ob v NG − r 2 = v T ob E v NG v T NG v ob + r 2 − 2 rv T ob E{v NG }.
(67) Hence the RMSE is given by
E{(r − r) 2 } = v T ob A k v ob + r 2 − 2 rv T ob µ k ,(68)
where A k := E{v k v T k } and µ k := E{v k }. Using (32) and (40)
, we have v k = L i=1 [ α k ] i O i C k G k v k−1 .(69)
This implies the following recursive relationships:
A k = L i=1 L j=1 e (k) ij O i C k G k A k−1 G T k C T k O T j ,(70a)µ k = G k µ k−1 ,(70b)
where e (k)
ij = [E k ] ij := E{[ α k ] i [ α k ] j }.
The matrix E k may be expressed as
E k = E{ α k α T k } = α k α T k + α k 2 1 Cov{ p k } = α k α T k + 1 N s P k − p k p T k .(71)
For the simplicity of further derivation, we denote E k := 1 Ns P k − p k p T k .
Let us now consider the case of k = 1. In this case, v 0 of (8) is a deterministic vector, thus we have
A 0 = v 0 v T 0 , µ 0 = v 0 .(72)
Using the recursive relationship of µ k = G k µ k−1 , we now see that v T ob µ NG = r. Hence we may simplify (67) as
E{(r − r) 2 } = v T ob A NG v ob + r 2 − 2 rv T ob µ NG = v T ob A NG v ob − (v T ob µ NG ) 2 = v T ob A NG − µ NG µ T NG v ob .(73)
Observe that the term A NG − µ NG µ T NG is in fact the covariance matrix of v NG , upon defining
Σ k := A k − µ k µ T k ,(74)
and substituting into (73) we arrive at
E{(r − r) 2 } = v T ob Σ NG v ob .(75)
The covariance matrix can be further formulated as
Σ k = A k − µ k µ T k = A k − R k v 0 v T 0 R T k .(76)
It now suffices to compute A k . Taking trace from both sides of (70a), we have
Tr {A k } = L i=1 L j=1 e (k) ij Tr O i C k G k A k−1 G T k C T k O T j .
(77) Next we consider the decomposition e (k)
ij = [α k ] i [α k ] j + E k ij .(78)
Observe that the term
[α k ] i [α k ] j satisfies L i=1 L j=1 [α k ] i [α k ] j Tr O i C k G k A k−1 G T k C T k O T j = Tr {A k−1 } ,(79)
since
C −1 k = L i=1 [α k ] i O i . Thus we have Tr {A k } − Tr {A k } = L i=1 L j=1 E k ij Tr O i C k G k A k−1 G T k C T k O T j ≤ L i=1 L j=1 E k ij Tr C k G k A k−1 G T k C T k ,(80)
where the third line follows from the fact that all the basis operators O i , i = 1 . . . L are trace-nonincreasing operators, hence represent contractive transformations. Since unitary transformations preserve the trace, we further obtain 14 Note that
Tr {A k } ≤ Tr {A k−1 } 1 + λ max {C T k C k } L i=1 L j=1 E k ij .(81)L i=1 L j=1 E k ij ≤ 1 N s vec{P k } 1 + vec{p k p T k } 1 = 2 N s ,(82)
and that λ max {C T k C k } ≤ 1 since C k is a completely positive trace-preserving channel, hence is contractive. In light of this, the upper bound of Tr {A k } can now be simplified as follows:
Tr {A k } ≤ Tr {A k−1 } 1 + 2 N s .(83)
From (72) we have Tr {A 0 } = 1 since v 0 is a unit vector, hence we obtain
Tr {A NG } ≤ NG k=1 1 + 2 N s ≤ exp 2N G N −1 s .(84)
Using (76), we have
Tr {Σ NG } = Tr {A NG } − Tr v 0 v T 0 ≤ exp 2N G N −1 s − 1.(85)
Note that Σ NG is a positive semidefinite matrix, hence we have
Tr {Σ NG } ≥ λ max (Σ NG ),(86)
where λ max (·) denotes the maximum eigenvalue of a matrix. This implies that
E{(r − r) 2 } = v T ob Σ NG v ob ≤ Tr {Σ NG } · v ob ≤ exp 2N G N −1 s − 1 · v ob .(87)
Hence the proof is completed by applying (23).
Especially, for Pauli channels, we have the following simplified recursions:
A k = E{ c k c T k } G k A k−1 G T k , µ k = G k µ k−1 .(88)
In fact, we have
E{ c k c T k } = 11 T + Ξ k ,(89)
which follows from (38). Substituting (89) into (88), we obtain
A k = G k A k−1 G T k + Ξ k G k A k−1 G T k .(90)
Following the same line of reasoning as we used in the general case, we have where q k := [p
(2) k . . . p (4 q ) k ] T ∈ R 4 q −1 . Note that vec{q k q T k } 1 = 1 T q k q T k 1 ≤ ( √ 1 + σ u − 1) 2 ,(101)
implying that
vec{P k − p k p T k } 1 ≤ 5 2 σ u + 1 4 σ 2 u ,(102)
which follows from that fact that
√ 1 + x − 1 ≤ x 2
holds for all x ≥ 0. Hence we have
Ξ k max ≤ 1 N s 5 2 σ u + 1 4 σ 2 u ,(103)
which proves (42). Thus the proof is completed.
Fig. 3 .
3Schematic illustration of the Bloch sphere undergoing a sequence of N G imperfect single-qubit gates. The Bloch sphere shrinks when QEM is not applied, whereas it becomes "blurred" when the Monto Carlo-based QEM is applied.
Fig. 4 .
4Demonstration of the evolution of computational result r in the absence of QEM, as a function of N G . The upper and lower bound of computational error correspond to the results in Proposition 2 and Proposition 1, respectively.
Fig. 5 .
5Circuits implementing rotations around the X-axis of the Bloch sphere.
versus N G .
versus gate error probability (N G = 10).
Fig. 6 .
6The RMSE of the results computed by quantum circuits consisting of repeated Pauli X gates (as demonstrated inFig. 5a).
Fig. 7 .
7The RMSE of the results computed by quantum circuits carrying out repeated rotations around the X-axis of the Bloch sphere (as shown inFig. 5b), as functions of N G , when θ = π/256.
Fig. 8 .
8The computational results of QEM-protected and non-QEM-protected circuits configured for carrying out repeated rotations (θ = π/256) around the X-axis of the Bloch sphere (shown inFig. 5b), as functions of N G .
Repeated (π/256)-rotations around the X-axis.
Fig. 9 .
9The RMSE of the results computed by quantum circuits demonstrated inFig. 5, which are contaminated by amplitude damping channels.
Fig. 10 .
10The RMSE of the results computed by QEM-protected and non-QEM-protected circuits implementing the quantum approximate optimization algorithm based on (57), as functions of the number of stages P .
Fig. 11 .
11The objective function values computed at the k-th stage of the quantum approximate optimization algorithm (for which P = 225) implemented based on (57).
TABLE I COMPARISON
IBETWEEN THE CONTRIBUTIONS OF THIS TREATISE AND EXISTING LITERATURE EVALUATING THE PERFORMANCE OF VQAS AND QEM. Only numerical J. R. McClean et al.Circuit condition
Subject of Analysis
Method of performance
evaluation
Noisy?
QEM implementation
J. R. McClean et al. [13]
×
No QEM
Only accuracy
an upper bound on the root-mean-square error (RMSE) of the computational error in the presence of Monte Carlo-based QEM. Specifically, we show that the RMSE is upper bounded by the square root of N G as well as , when N G 1. This implies that when we use the same number of samples as the ideal QEM, Monte Carlo-based QEM can still provide a quadratic error reduction versus N G , compared to the case of no QEM.• We provide an intuitive interpretation of the proposed error scaling laws, by visualizing the decoherence-induced impairments on the Bloch sphere as the quantum circuit executes. • We demonstrate the analytical results using various numerical examples. Specifically, we consider a practicalFig. 1. The structure of a typical implementation for variational quantum algorithms.Parametric
State-
Preparation
Circuit
(Ansatz)
Quantum
Observable
Fixed Input State
Output
State
Quantum Device
Optimization
Algorithm
Classical Device
Adjust parameters
Cost Function
Value/Gradient
by Monte Carlo-based QEM and their non-QEM-protected counterparts was investigated. As for the non-QEM-protected circuits, we have shown that the dynamic range of the noisy computational results shrinks exponentially as the number of gates N G increases, implying a linear error scaling with N G . By contrast, the error scales as the square root of N G in the presence of Monte Carlo-based QEM, at the same computational cost as that of QEM based on exact channel inversion. Moreover, the error scaling of Monte Carlo-based QEM can be further improved with increased computational cost.We have also demonstrated the analytical results both for low-complexity examples and for a more practical example of the quantum approximate optimization algorithm employed for multi-user detection in wireless communications. It may be an interesting future research direction to apply the results to other practical examples, or verify them using experimental approaches. ACKNOWLEDGMENTS Y. Xiong would like to acknowledge Daryus Chandra for helpful conversations regarding quantum channel estimation and quantum error correction codes. APPENDIX I PROOF OF PROPOSITION 1
Assumption 2 implies that M ob 2 ≤ 1, meaning that Tr {M ob ρ} ≤ 1 holds for any legitimate density matrix ρ. Note that terms such as S i G k ρ k−1 G † k S i in (63) are indeed legitimate density matrices. Thus the computational result satisfies
the "max norm" · max is defined asFrom(38)we obtainwhere the third line follows from the fact that c k represents a contractive transformation, so that c k 1. Note that every entry in H has an absolute value of 1, and henceandHence we arrive at exactly the same bound as given in(87).APPENDIX IV PROOF OF PROPOSITION 4Proof: We start the proof from revisiting the inequality in (92), and arrive at:Next we construct an upper bound for the term vec{P k − p k p T k } 1 . According to the sampling overhead of QEM in[22], Assumption 1 implies thatSince αThis further implies thatTherefore, upon taking the entry-wise absolute value, we obtain(99) Here, the symbol "≤" stands for entry-wise "not larger than". Observe that summing up the first row, the first column and the main diagonal, by applying (98), we see that
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He is currently pursuing the PhD degree with Next-Generation Wireless within the School of Electronics and Computer Science, University of Southampton. His research interests include quantum computation, quantum information theory, graph signal processing. Yifeng Xiong received his B.S. degree in information engineering, and the M.S. degree in information and communication engineering from Beijing Institute of Technology (BIT). Beijing, Chinaand statistical inference over networksYifeng Xiong received his B.S. degree in informa- tion engineering, and the M.S. degree in information and communication engineering from Beijing Insti- tute of Technology (BIT), Beijing, China, in 2015 and 2018, respectively. He is currently pursuing the PhD degree with Next-Generation Wireless within the School of Electronics and Computer Science, University of Southampton. His research interests include quantum computation, quantum information theory, graph signal processing, and statistical infer- ence over networks.
He is currently a Professor of Next Generation Communications at the University of Southampton. His research interests include adaptive coded modulation, coded modulation, channel coding, space-time coding, joint source and channel coding, iterative detection, OFDM, MIMO, cooperative communications, distributed coding, quantum communications, quantum error correction codes, joint wireless-and-optical-fibre communications, game theory, artificial intelligence and machine learning. U K Southampton, Soon Xin Ng (S'99-M'03-SM'08) received the B.Eng. degree (First class) in electronic engineering and the Ph.D. degree in telecommunications from the University of Southampton. John Wiley/IEEE Pressthe IEEE Access and the KSII Transactions on Internet and Information Systems. He is one of the Founders and Officers of the IEEE Quantum Communications & Information Technology Emerging Technical Subcommittee (QCIT-ETCSoon Xin Ng (S'99-M'03-SM'08) received the B.Eng. degree (First class) in electronic engineering and the Ph.D. degree in telecommunications from the University of Southampton, Southampton, U.K., in 1999 and 2002, respectively. From 2003 to 2006, he was a postdoctoral research fellow working on collaborative European research projects known as SCOUT, NEWCOM and PHOENIX. Since August 2006, he has been a member of academic staff in the School of Electronics and Computer Science, University of Southampton. He was involved in the OPTIMIX and CONCERTO European projects as well as the IU-ATC and UC4G projects. He was the principal investigator of an EPSRC project on "Cooperative Classical and Quantum Communications Systems". He is currently a Professor of Next Generation Communications at the University of Southampton. His research interests include adaptive coded modulation, coded modula- tion, channel coding, space-time coding, joint source and channel coding, iterative detection, OFDM, MIMO, cooperative communications, distributed coding, quantum communications, quantum error correction codes, joint wireless-and-optical-fibre communications, game theory, artificial intelligence and machine learning. He has published over 260 papers and co-authored two John Wiley/IEEE Press books in this field. He is a Senior Member of the IEEE, a Fellow of the Higher Education Academy in the UK, a Chartered Engineer and a Fellow of the IET. He acted as TPC/track/workshop chairs for various conferences. He serves as an editor of Quantum Engineering. He was a guest editor for the special issues in IEEE Journal on Selected Areas in Communication as well as editors in the IEEE Access and the KSII Transactions on Internet and Information Systems. He is one of the Founders and Officers of the IEEE Quantum Communications & Information Technology Emerging Technical Subcommittee (QCIT-ETC).
He is a Foreign Member of the Hungarian Academy of Sciences and a former Editor-in-Chief of the IEEE Press. He has served several terms as Governor of both IEEE ComSoc and of VTS. He has published 2000+ contributions at IEEE Xplore, 19 Wiley-IEEE Press books and has helped the fast-track career of 123 PhD students. Over 40 of them are Professors at various stages of their careers in academia and many of them are leading scientists in the wireless industry. Lajos Hanzo, FIEEE'04) received his Master degree and Doctorate in 1976 and 1983, respectively from the Technical University. TU) of BudapestHe was also awarded the Doctor of Sciences (DSc) degree by the University of Southampton (2004) and Honorary Doctorates by the TU of Budapest (2009) and by the University of Edinburgh. He is also a Fellow of the Royal Academy of Engineering (FREng), of the IET and of EURASIP. He is the recipient of the 2022 Eric Sumner Field AwardLajos Hanzo (http://www-mobile.ecs.soton.ac.uk, https://en.wikipedia.org/wiki/Lajos Hanzo) (FIEEE'04) received his Master degree and Doctorate in 1976 and 1983, respectively from the Technical University (TU) of Budapest. He was also awarded the Doctor of Sciences (DSc) degree by the University of Southampton (2004) and Honorary Doctorates by the TU of Budapest (2009) and by the University of Edinburgh (2015). He is a Foreign Member of the Hungarian Academy of Sciences and a former Editor-in-Chief of the IEEE Press. He has served several terms as Governor of both IEEE ComSoc and of VTS. He has published 2000+ contributions at IEEE Xplore, 19 Wiley-IEEE Press books and has helped the fast-track career of 123 PhD students. Over 40 of them are Professors at various stages of their careers in academia and many of them are leading scientists in the wireless industry. He is also a Fellow of the Royal Academy of Engineering (FREng), of the IET and of EURASIP. He is the recipient of the 2022 Eric Sumner Field Award.
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arxiv |
ON SETS OF MARKED ONCE-HOLED TORI ALLOWING HOLOMORPHIC MAPPINGS INTO RIEMANN SURFACES WITH MARKED HANDLE
18 Apr 2016
Makoto Masumoto
ON SETS OF MARKED ONCE-HOLED TORI ALLOWING HOLOMORPHIC MAPPINGS INTO RIEMANN SURFACES WITH MARKED HANDLE
18 Apr 2016arXiv:1604.05014v1 [math.CV]
In our previous work[10], for a given Riemann surface Y 0 with marked handle, we investigated geometric properties of the set of marked onceholed tori X allowing holomorphic mappings of X into Y 0 . It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.1991 Mathematics Subject Classification. Primary 30F99; Secondary 30F45, 30F60, 32G15.
Introduction
Let R 1 and R 2 be Riemann surfaces. It is a natural question whether there are holomorphic or conformal mappings of R 1 into R 2 with some geometric or analytic properties. In the present article we consider the problem in the case where R 1 is a once-holed torus, and look for handle-preserving mappings.
Since Riemann surfaces of genus zero are conformally equivalent to plane regions, Riemann surfaces of positive genus should play the leading character in Riemann surfaces theory. Once-holed tori are topologically the simplest among the nonplanar Riemann surfaces. They are building blocks of Riemann surfaces of positive genus; every Riemann surface of positive genus g is obtained from g once-holed tori by suitable identification. Open disks are one of the simplest plane domains, and studies of functions on open disks are of fundamental importance for local theory. Thus studies of holomorphic mappings on once-holed tori would be significant for "local theory" of holomorphic mappings between Riemann surfaces. While open disk are conformally equivalent to one another, once-holed tori are not. Hence we need to know which once-holed tori are included in a Riemann surface under consideration. This amounts to ask the existence of conformal mappings of onceholed tori into Riemann surfaces.
For the existence of conformal mappings of once-holed tori several results are known. In [11] and [12] Shiba investigated the set of tori into which a given open Riemann surface of genus one is conformally embedded. His results give solutions to our problem in the case where R 2 is a torus. Also, we gave a characterization for the existence of conformal mappings of a once-holed torus into another explicitly in terms of finitely many extremal lengths (see [7]). In [8] we examined the set of once-holed tori that can be conformally embedded into a given Riemann surface of positive genus. For topologically finite surfaces Kahn-Pilgrim-Thurston [4] has recently given a characterization for the existence of conformal embeddings in terms of extremal lengths.
On the other hand, few results are known for the existence of holomorphic mappings of once-holed tori. If R 2 is a torus, then the Behnke-Stein theorem yields that any once-holed torus allows handle-preserving holomorphic mappings into R 2 . If R 2 is not a torus, then it carries a hyperbolic metric. Since holomorphic mappings decrease hyperbolic lengths, we obtain necessary conditions for the existence of holomorphic mappings. However, they are not sufficient by a recent result of Bourque [1].
The space T of marked once-holed tori is a three-dimensional real-analytic manifold with boundary. In our previous work [10], for a given Riemann surface Y 0 with marked handle, we investigated the set T a [Y 0 ] of marked once-holed tori X such that there is a holomorphic mapping of X into Y 0 . We introduced a new condition called a handle condition to obtain the following two results.
Proposition 1 ([10, Theorem 1]). T a [Y 0
] is a closed domain with Lipschitz boundary, and is a retract of the whole space T.
The second result is expressed in terms of a specific coordinate system on T. Every once-holed torus is realized as a slit torus. For (τ, s) ∈ H × [0, 1) let X (s) τ denote the marked once-holed torus obtained from the marked torus X τ of modulus τ by deleting a horizontal segment of length s, where H is the upper half-plane. into Y 0 for some s.
Propositions 1 and 2 raise the following natural questions:
(1) Is the boundary of T a [Y 0 ] smooth? (2) What is the value of λ a [Y 0 ]? In the present paper we answer these questions. We first show that the boundary of T a [Y 0 ] is not smooth in most cases:
Theorem 1. If Y 0
is not a marked torus or a marked once-punctured torus, then the boundary of T a [Y 0 ] is not smooth.
We prove Theorem 1 in §3 after summarizing results of [10] in §2. In §4 we compare T a [Y 0 ] with the set T σ [Y 0 ] of marked once-holed tori having longer geodesics than corresponding geodesics on Y 0 . In the final section we give an answer to second question (2) (see Theorem 3).
Preliminaries
Let R be a Riemann surface of positive genus; it may be compact or of infinite genus. It has one or more handles. A handle of R is specified by a couple of loops on R. With this in mind we make the following definitions. A mark of handle of R is, by definition, an ordered pair χ = {a, b} of simple loops a and b on R whose geometric intersection number a×b is equal to one. A Riemann surface with marked handle means a pair (R, χ), where R is a Riemann surface of positive genus and χ is a mark of handle of R.
Let Y 1 := (R 1 , χ 1 ) and Y 2 := (R 2 , χ 2 ) be Riemann surfaces with marked handle, where χ j = {a j , b j } for j = 1, 2. If a continuous mapping f : R 1 → R 2 maps a 1 and b 1 onto loops freely homotopic to a 2 and b 2 on R 2 , respectively, then we say that f is a continuous mapping of Y 1 into Y 2 and use the notation f : Y 1 → Y 2 . If f : R 1 → R 2 possesses some additional properties, then f : Y 1 → Y 2 is said to possess the same properties. For example, if f : R 1 → R 2 is conformal, then f : Y 1 → Y 2 is called conformal. Here, by a conformal mapping we mean a holomorphic injection; we do not require conformal mappings to be surjective. We consider continuous mappings of Y 1 into Y 2 preserve the handles specified by χ 1 and χ 2 .
A once-holed torus is, by definition, a noncompact Riemann surface of genus one with exactly one boundary component in the sense of Kerékjártó-Stoïlow. For example, the Riemann surface obtained from a torus, that is, a compact Riemann surface of genus one, by removing one point is a once-holed torus, which will be referred to as a once-punctured torus. Note that once-holed tori are not bordered surfaces. A once-holed torus with marked handle is usually called a marked onceholed torus. The meaning of a marked once-punctured torus is obvious.
Let T denote the set of marked once-holed tori, where two marked once-holed tori are identified if there is a conformal mapping of one onto the other. As a set, it is the disjoint union of the Teichmüller space T ′ of a once-punctured torus and the reduced Teichmüller space T ′′ of a once-holed torus that is not a once-punctured torus.
There is a canonical injection Λ of T into R 3 + ; if X = (T, χ) with χ = {a, b}, then Λ(X) is the triplet of the extremal lengths of the free homotopy classes of a, b and ab −1 . We know that
L := Λ(T) = {x ∈ R 3 + | Q(x) + 4 ≦ 0}, where Q(x) = Q(x 1 , x 2 , x 3 ) = x 2 1 + x 2 2 + x 3 3 − 2(x 1 x 2 + x 2 x 3 + x 3 x 1 )
. Note that Λ maps T ′ and T ′′ onto the boundary ∂L and the interior L • := L \ ∂L, respectively. Moreover, the restrictions Λ| T ′ : T ′ → ∂L and Λ| T ′′ : T ′′ → L • are real-analytic diffeomorphisms. We regard T as a three-dimensional real-analytic manifold with boundary so that Λ : T → L is a real-analytic diffeomorphism. In the rest of the article we use the notations ∂T and T • instead of T ′ and T ′′ , respectively. For details, see [7, §7]. Now, fix a Riemann surface Y 0 = (R 0 , χ 0 ) with marked handle. For a given marked once-holed torus X there may or may not exist holomorphic mappings of X into Y 0 . We are interested in the set of marked once-holed tori which allow holomorphic mappings into Y 0 . We denote by T a [Y 0 ] (resp. T c [Y 0 ]) the set of X ∈ T such that there exists a holomorphic (resp. conformal) mapping of X into Y 0 . In our previous work [10] we introduced handle conditions to investigated geometric properties of T a [Y 0 ] and T c [Y 0 ]. We recall the definition of a handle condition.
For X, X ′ ∈ T we say that X is smaller than X ′ and write X X ′ if there is a conformal mapping of X into X ′ . The relation is then an order relation on T.
A mathematical statement P(X), where the free variable X ranges over T, is called a handle condition if P(X 1 ) implies P(X 2 ) for all X 1 , X 2 ∈ T with X 2 X 1 . Important examples are the statements "there is a holomorphic mapping of X into Y 0 " and "there is a conformal mapping of X into Y 0 ," which will be denoted by P a (X) and P c (X), respectively. For ν ∈ N the statement P ν (X) = "There is a holomorphic mapping f :
X → Y 0 with d(f ) < ν + 1" is another handle condition, where d(f ) is the supremum of the cardinal numbers of f −1 (p), p ∈ R 0 . Note that P 1 (X) = P c (X). Set T[P] = {X ∈ T | P(X)}.
Then we have the following proposition. Actually, we can show more. The eigenvalues of the coefficient matrix of the quadratic form Q(x) are −1 and 2. The corresponding eigenspaces V −1 and V 2 re, respectively, the line x 1 = x 2 = x 3 and the plane
x 1 + x 2 + x 3 = 0. Let e = 1/ √ 3 , 1/ √ 3 , 1/ √ 3 ∈ V −1 .
Proposition 4 ([10, Proposition 4]).
There is a Lipschitz continuous function Every marked once-holed torus is realized as a horizontal slit torus (see Shiba [11]). Specifically, for each point τ in the upper half-plane H, let G τ be the additive subgroup of C generated by 1 and τ . Then T τ := C/G τ is a torus. Let π τ : C → T τ be the natural projection, and set a τ = π τ ([0, 1]) and b τ = π τ ([0, τ ]), where [z 1 , z 2 ] stands for the oriented line segment joining z 1 with z 2 ; if z 1 = z 2 , then [z 1 , z 2 ] denotes the singleton {z 1 }. Then χ τ := {a τ , b τ } is a mark of handle of T τ , and we obtain a marked torus X τ : [10, §4]). In other words, its inverse Σ : X
e[P]( · ) on V 2 such that Λ(T • [P]) = {ζ + te | ζ ∈ V 2 [P], t > e[P](ζ)}, provided that T[P] = ∅. Since T a [Y 0 ] = T[P a ], Proposition 1 follows from Proposition 3 together with the fact that T a [Y 0 ] is closed. The set T c [Y 0 ] := T[P c ] possesses the same properties. In fact, setting T ν [Y 0 ] = T[P ν ],= (T τ , χ τ ). Now, for s ∈ [0, 1) set T (s) τ = T τ \ π τ ([0, s]); it is a once-holed torus. We choose a mark χ (s) τ = a (s) τ , b (s) τ of handle of T (s) τ so that the inclusion mapping of T (s) τ into T τ is a conformal mapping of X (s) τ := T (s) τ , χ (s) τ into X τ . Then the correspondence (τ, s) → X (s) τ is a homeomorphism of H × [0, 1) onto T, which is a real-analytic diffeomorphism on H × (0, 1) (seeH 1 λ[P] ⊂ H[P] ⊂H 1 λ[P] ,
where 1/0 = +∞ and 1/(+∞) = 0.
We set λ a [Y 0 ] = λ[P a ] and λ c [Y 0 ] = λ[P c ].
They are referred to as the critical extremal lengths for the existence of holomorphic and conformal mappings of marked once-holed tori into Y 0 , respectively. Most part of Proposition 2 follows from Proposition 6. Note, however, that Proposition 2 asserts that the identity H[P a ] = H(1/λ[P a ]) actually holds. On the other hand, in general, [10,Example 13]).
H(1/λ[P c ]) is a proper subset of H[P c ] (see
Non-smoothness of boundaries
Propositions 1 and 5 show that T a [Y 0 ] and T ν [Y 0 ] have Lipschitz boundaries.
It is then natural to ask whether the boundaries are smooth or not. [10,Example 10]). Our first result, Theorem 1, claims that the boundary of T a [Y 0 ] is not, either, for most cases.
As for T c [Y 0 ] = T 1 [Y 0 ] we know that the answer is negative in general. In fact, if Y 0 is a marked once-holed torus, then Λ(T c [Y 0 ]) is a cone with vertex at Λ(Y 0 ) and hence the boundary of T c [Y 0 ] is not smooth at Y 0 (see
For the proof of Theorem 1 we define the handle covering surface of a Riemann surface Y 0 with marked handle. There is a Riemann surfaceỸ 0 = (R 0 ,χ 0 ) with marked handle together with a holomorphic mapping π 0 :Ỹ 0 → Y 0 such that (i) the fundamental group ofR 0 is generated by the loops inχ 0 , and (ii) π 0 :R 0 → R 0 is a covering map. We callỸ 0 the handle covering surface of Y 0 . The following lemma is easily verified. Lemma 1. Let Y 0 be a Riemann surface of marked handle andỸ 0 its handle covering surface.
(i) If Y 0 is not a marked torus, thenỸ 0 is a marked once-holed torus. If Y 0 is a marked torus or a marked once-holed torus, thenỸ 0 = Y 0 . (ii) IfỸ 0 is a marked once-punctured torus, then so is Y 0 .
(iii)Ỹ 0 ∈ T a [Y 0 ]. (iv) T a [Ỹ 0 ] = T a [Y 0 ].
Proof of Theorem 1. By Lemma 1 (iv) we may assume from the outset that Y 0 is an element of T • . We employ Fenchel-Nielsen coordinates (see Buser [2]). To be specific, let X = (T, χ) ∈ T, where χ = {a, b}. The once-holed torus T carries a hyperbolic metric, whose curvature is normalized to be −1. Denote by l(X) the length of the hyperbolic geodesic α freely homotopic to a. Let θ(X) stand for the twist parameter along α. Also, let l ′ (X) be the infimum of hyperbolic lengths of loops freely homotopic to aba −1 b −1 . Clearly, l ′ (X) vanishes if and only if X is a marked once-punctured torus. Setting Φ(X) = l(X), l ′ (X), θ(X) , we obtain a homeomorphism of T onto (0, +∞) × [0, +∞) × R, which is a real-analytic diffeomorphism of T • onto (0, +∞) × (0, +∞) × R.
If X ∈ T a [Y 0 ], then l(X) ≧ l(Y 0 ) and l ′ (X) ≧ l ′ (Y 0 ) since holomorphic mappings decrease hyperbolic metrics. As Y 0 ∈ T a [Y 0 ], these inequalities imply that Y 0 lies on the boundary ∂T a [Y 0 ] and that ∂T a [Y 0 ] is not smooth at Y 0 , provided that Y 0 ∈ ∂T. Theorem 1 has been thus established.
Remark. If Y 0 is a marked torus, then T a [Y 0 ] coincides with the whole space T (see [10,Example 9]). Thus its boundary is an empty set. For marked once-punctured tori Y 0 the boundary of Φ(T a [Y 0 ]) is not smooth. However, we do not know whether the boundary of Λ(T a [Y 0 ]) is smooth or not.
It is obvious that
T c [Y 0 ] = T 1 [Y 0 ] ⊂ · · · ⊂ T ν [Y 0 ] ⊂ T ν+1 [Y 0 ] ⊂ · · · ⊂ T a [Y 0 ]. If Y 0 ∈ T • , then both of ∂T c [Y 0 ] and ∂T a [Y 0 ]
contains Y 0 and are not smooth at Y 0 . We thus have the following corollary to Theorem 1.
Corollary 1.
If Y 0 is a marked once-holed torus which is not a marked oncepunctured torus, then for any positive integer ν the boundary of T ν [Y 0 ] is not smooth.
Hyperbolic length spectra
Let W be a free group generated by two elements. We regard it as the set of reduced words w(u, v) of two letters u and v. The unit is the void word. We denote by W * the subset of non-unit elements.
In general, let Y = (R, χ), where χ = {a, b}, be a Riemann surface with marked handle. For w = w(u, v) ∈ W * the notation w(a, b) denotes a loop on R. We set w(Y ) = w(a, b). In particular, u(Y ) = a and v(Y ) = b. Let l(Y, w) be the infimum of the hyperbolic lengths of loops in the free homotopy class Γ(Y, w) of w(Y ) on R, provided that R is not a torus. In the case where R is a torus, we set l(Y, w) = 0 for convenience. Now, fix a Riemann surface Y 0 with marked handle, and let T σ [Y 0 ] be the set of X ∈ T for which l(X, w) ≧ l(Y 0 , w) for all w ∈ W * . Since holomorphic mappings decrease hyperbolic lengths, [10,Example 12]).
T a [Y 0 ] is included in T σ [Y 0 ]. It follows from Bourque [1] that T a [Y 0 ] is in general a proper subset of T σ [Y 0 ].(iii) T σ [Y 0 ] \ T a [Y 0 ] is homeomorphic to C * × [0, 1), where C * = C \ {0}. Remark. If Y 0 is a marked torus, then T σ [Y 0 ] = T a [Y 0 ] = T (see
For the proof of Theorem 2 we introduce some notations, and prepare several lemmas. We first remark the following lemma. Remark. If a is a simple loop on D separating the boundary components of D, then so is a −1 . Though the free homotopy classes Γ + and Γ − of a and a −1 , respectively, are disjoint, their extremal lengths are identical with each other. The common value is denoted by λ(D). In the sequel Γ(D) will represent one of Γ + and Γ − .
Let Y = (R, χ) be a Riemann surface with marked handle, and take w ∈ W * . Let D be the annular covering surface of R with respect to the loop w(Y ) (see [5, §3]). Thus D is a doubly connected Riemann surface, and there is a holomorphic covering map π of D onto R which maps Γ(D) into Γ(Y, w). Clearly, we have
(2) l(D) = l(Y, w).
Lemma 3 (Wolpert [14, Lemma 3.1]). Let Y 1 and Y 2 be Riemann surfaces with marked handle. If there is a K-quasiconformal mapping of Y 1 onto Y 2 , then
1 K l(Y 1 , w) ≦ l(Y 2 , w) ≦ Kl(Y 1 , w)
for all w ∈ W * .
In fact, let D j be the annular covering surface of Y j with respect to w(Y j ). If there is a K-quasiconformal mapping of Y 1 onto Y 2 , then it is lifted to a K-quasiconformal mapping D 1 onto D 2 which maps Γ(D 1 ) to Γ(D 2 ). Since extremal lengths are quasiinvariant under quasiconformal mappings, the lemma is an immediate consequence of (1) and (2).
Lemma 4.
For each w ∈ W * the function l( · , w) is continuous on T.
Proof. It follows from Lemma 3 that l( · , w) is continuous on T • . To show that it is also continuous at each point of ∂T, take an arbitrary marked once-punctured torus X τ whose maximal dilatation is equal to e d H (τ0,τ ) , where d H (τ 0 , τ ) is the distance between τ 0 and τ with respect to the hyperbolic metric on H. We apply Lemma 3 to obtain
l X (s) τ , w − l X (0) τ0 , w ≦ l X (s) τ , w − l X (s) τ0 , w + l X (s) τ0 , w − l X (0) τ0 , w ≦ e d H (τ0,τ ) − e −d H (τ0,τ ) l X (s) τ0 , w + l X (s) τ0 , w − l X (0) τ0 , w . Consequently, l X (s) τ , w → l X (0)
τ0 , w as (τ, s) → (τ 0 , 0), which means that the function l( · , w) is continuous at X (0) τ0 , as desired.
Corollary 2. T σ [Y 0 ] is a closed subset of T.
Proof of Theorem 2. Since conformal mappings decrease hyperbolic metrics, the statement P σ (X) := "l(X, w) ≧ l(Y 0 , w) for all w ∈ W * " is a handle condition (see [10,Example 8]). As T[P σ ] = T σ [Y 0 ], Proposition 3 together with Corollary 2 implies assertion (i).
In order to show assertion (ii), by Lemmas 1 (iv) and 2, we have only to consider the case where Y 0 is a marked once-holed torus, say, X (s0) τ0 . If there is a holomorphic mapping of a marked once-punctured torus X
(0) τ into Y 0 = X (s0)
τ0 , then it is extended to a holomorphic mapping between the marked tori X τ and X τ0 , which must be conformal. Consequently, Y 0 is also a marked once-punctured torus identical with X (0) τ . We have shown that T a [Y 0 ]∩∂T is empty or a singleton and that in the latter case T a [Y 0 ] ∩ ∂T consists only of Y 0 .
Take an arbitrary X ∈ T a [Y 0 ] \ {Y 0 }. We employ arguments in [1] to prove that X lies in the interior of T σ [Y 0 ]. Let f be a holomorphic mapping of X into Y 0 . Let ρ X = ρ X (z) |dz| and ρ Y0 = ρ Y0 (ζ) |dζ| denote the hyperbolic metrics on X and Y 0 , respectively. By Schwarz's lemma the continuous function (f * ρ Y0 )/ρ X is strictly less than one pointwise, where f * ρ Y0 stands for the pull-back of ρ Y0 by f . The convex core C of X is compact and hence there is c ∈ (0, 1) for which (f * ρ Y0 )/ρ X < c on C. For w ∈ W * let γ w be the closed geodesic on X freely homotopic to w(X). Since γ w lies in C, we have
l(Y, w) ≦ f * γw ρ Y0 = γw f * ρ Y0 ≦ c γw ρ X = cl(X, w).
There is a neighborhood U of X such that for any X ′ ∈ U there is a (1/c)quasiconformal mapping of X onto X ′ . Applying Lemma 3, we infer that
X ′ ∈ T σ [Y 0 ]. Thus U ⊂ T σ [Y 0 ], or, X is an interior point of T σ [Y 0 ]
, as claimed. We have proved assertion (ii).
Assertion (iii) is now an easy consequence of Proposition 4. This completes the proof.
Remark. We see from the proof that the element in T a [Y 0 ] ∩ ∂T σ [Y 0 ] is the handle covering surfaceỸ 0 of Y 0 . There is exactly one holomorphic mapping ofỸ 0 into Y 0 (see [9]). Therefore, if Y 0 is not a marked once-holed torus, then there are no holomorphic mappings f :Ỹ 0 → Y 0 with d(f ) < ∞.
Critical extremal lengths
The purpose of this section is to evaluate the critical extremal lengths for the existence of holomorphic and conformal mappings of marked once-holed tori into a Riemann surface with marked handle. Let Y = (R, χ), where χ = {a, b}, be a Riemann surface with marked handle. Recall that W is the free group generated by u and v. Set Γ(Y ) = Γ(Y, u) and l(Y ) = l(Y, u). Thus Γ(Y ) is the free homotopy class of a = u(Y ). Let λ(Y ) stand for its extremal length. Note that λ X (s) τ = 1/ Im τ . Now, fix a Riemann surface Y 0 with marked handle. We begin with evaluating the critical extremal length λ a [Y 0 ] for the existence of holomorphic mappings.
Theorem 3. λ a [Y 0 ] = 1 π l(Y 0 ).
Proof. If Y 0 is a marked torus, then T a [Y 0 ] = T (see the remark following Theorem 2) and hence λ a [Y 0 ] = 0. Since l(Y 0 ) = 0 by definition, we see that the theorem is valid in this case. Next suppose that Y 0 is not a marked torus. By Lemma 1 (iv) we have only to consider the case where Y 0 is a marked once-holed torus. Let X (s) τ be an arbitrary element of T a [Y 0 ]. The image D τ of the horizontal strip {z ∈ C | 0 < Im z < Im τ } by the projection π τ : C → T τ = C/G τ is a doubly connected domain on T τ and is included in T (s)
τ . Since λ X (s) τ = λ(D τ ) = 1/ Im τ , we see from (1) that λ X (s) τ = 1 π l(D τ ) > 1 π l X (s) τ
(cf. Maskit [6, Proposition 1]). As holomorphic mappings decrease hyperbolic lengths, we obtain λ X
(s) τ > (1/π)l(Y 0 ), which implies λ a [Y 0 ] ≧ 1 π l(Y 0 ).
To show the opposite inequality we employ the annular covering surface D 0 of R 0 with respect to the loop a 0 , where Y 0 = (R 0 , χ 0 ) and χ 0 = {a 0 , b 0 }. For any ε > 0 choose a doubly connected and relatively compact subdomain
D of D 0 with Γ(D) ⊂ Γ(D 0 ) so that l(D) < l(D 0 ) + ε = l(Y 0 ) + ε.
We further assume that the components of ∂D are simple loops on D 0 . Let π 0 : D 0 → R 0 be the covering map. Since the closure π 0 (D) is compact in R 0 , we can find a simple loop on R 0 which is freely homotopic to b 0 and meets R 0 \ π 0 (D). Lifting the loop and deforming the lift, we obtain a simple arcb on D 0 such that (i) the end points ofb are projected to the same point by π 0 , and the image loop π 0 (b) is freely homotopic to b 0 , (ii) the arcb crossesã 0 once transversely, (iii) one of the end pints ofb is on ∂D and the other lies outside of D, and (iv) the partb ′ :=b \ D is connected. We construct a marked once-holed torusX = (T ,χ) belonging to T a [Y 0 ] as follows. We start with D ∪b. By identifying the end points ofb and thickening b ′ slightly and appropriately, we obtain a once-holed torusT so that π 0 induces a holomorphic mapping ofT into R 0 . The curvesã 0 andb 0 together make a markχ of handle ofT . It is obvious thatX = (T ,χ) is an element of T a [Y 0 ]. As D is a doubly connected domain onT with Γ(D) ⊂ Γ(X), we have
λ a [Y 0 ] ≦ λ(X) ≦ λ(D) = 1 π l(D) < 1 π {l(Y 0 ) + ε}.
Since ε is arbitrary, we deduce that
λ a [Y 0 ] ≦ 1 π l(Y 0 ),
which completes the proof of Theorem 3.
Next we evaluate the critical extremal length λ c [Y 0 ] for the existence of conformal mappings. The following theorem was announced in [10].
Theorem 4. λ c [Y 0 ] = λ(Y 0 ).
Proof. If there is a conformal mapping f of a marked once-holed torus X into Y 0 , then the image family f * Γ(X) is included in Γ(Y 0 ). Since conformal mappings keep extremal lengths invariant, it follows that λ(X) ≧ λ(Y 0 ) and hence that
λ c [Y 0 ] ≧ λ(Y 0 ).
To show that the sign of equality actually occurs, we employ results on Jenkins-Strebel differentials, that is, holomorphic quadratic differentials with closed trajectories (see Strebel [13,Chapter 5]). There uniquely exists a doubly connected domain ∆ 0 on R 0 such that Γ(∆ 0 ) ⊂ Γ(Y 0 ) and λ(∆ 0 ) = λ(Y 0 ). It is dense in R 0 and is swept out by closed horizontal trajectories of a holomorphic quadratic differential on R 0 . Let δ 0 = π/λ(Y 0 ) and r 0 = e δ0 , and set A(δ) = {z ∈ C | e −δ r 0 < |z| < e δ r 0 } for δ ∈ (0, δ 0 ]. Then there is a conformal mapping F 0 of the annulus A 0 := A(δ 0 ) onto ∆ 0 , which is continuously extended to a union E 0 of open arcs on ∂A 0 . We assume that E 0 is maximal with this property. Thus F 0 is a continuous mapping of A 0 ∪ E 0 onto R 0 , and R 0 is obtained from A 0 ∪ E 0 by identifying points on E 0 in the obvious manner (see Jenkins-Suita [3, Corollary 1 to Theorem 2]). Let a ′ 0 be the loop on R 0 corresponding to the circle |z| = r 0 ; we orient it so that it is freely homotopic to a 0 . Take a piecewise analytic simple loop b ′ 0 on R 0 freely homotopic to b 0 such that the intersection of F −1 0 (b ′ 0 ) with the closure of a narrow annulus A ′ := A(δ 1 ) is a radial segment, where δ 1 is a sufficiently small positive number. By thickening b ′ 0 we obtain a doubly connected domain B ′ with b ′ 0 separating the boundary components of B ′ . We choose B ′ so that the union T ′ :
= F 0 (A ′ ) ∪ B ′ is a once-holed torus included in R 0 . Obviously, χ ′ := {a ′ 0 , b ′ 0
} is a mark of handle of T ′ and the inclusion mapping T ′ → R 0 is a conformal mapping of the marked once-holed torus (T ′ , χ ′ ) into Y 0 . For each δ ∈ (0, δ 0 ) take a homeomorphism h δ of the interval [1,
r 2 0 ] = [1, e δ0 r 0 ] onto itself such that h δ (1) = 1, h δ (e −δ1 r 0 ) = e −δ r 0 and h δ (e δ1 r 0 ) = e δ r 0 . It induces a homeomorphism H δ of R 0 onto itself satisfy- ing H δ • F 0 (re iθ ) = F 0 h δ (r)e iθ . Intuitively, H δ fattens F 0 (A ′ ) if δ > δ 1 . The marked once-holed torus X ′ δ := H δ (T ′ ), χ ′ δ , where χ ′ δ = {a ′ 0 , H δ (b ′ 0 )}, is confor- mally embedded into Y 0 . Thus X ′ δ ∈ T c [Y 0 ]. Since H δ (T ′ ) includes F 0 A(δ) , we have λ c [Y 0 ] ≦ λ(X ′ δ ) ≦ λ A(δ) = π δ .
Letting δ → δ 0 , we obtain
λ c [Y 0 ] ≦ π δ 0 = λ(Y 0 ).
This completes the proof.
Theorems 3 and 4 give a simple alternative proof of one of our previous results. The next corollary implies that T a [Y 0 ] \ T c [Y 0 ] has a nonempty interior since T a [Y 0 ] and T a [Y 0 ] are closed domains with Lipschitz boundary.
Corollary 3 ([10, Theorem 8]). λ a [Y 0 ] < λ c [Y 0 ].
Proof. Let ∆ 0 be the doubly connected domain as in the proof of Theorem 4. Then
λ a [Y 0 ] = 1 π l(Y 0 ) < 1 π l(∆ 0 ) = λ(∆ 0 ) = λ(Y 0 ) = λ c [Y 0 ],
where the inequality follows from the fact that ∆ 0 is a proper subdomain of R 0 .
Let T ∞ [Y 0 ] be the set of X ∈ T such that there is a holomorphic mapping does not belong to T a [Y 0 ] for any s ∈ [0, 1) (see [10,Theorem 6]). This is not always the case for the critical extremal lengths for the existence of conformal mappings. In fact, if Y 0 is a marked once-holed torus, then the strip Π • Σ(T c [Y 0 ]) and the line Im τ = 1/λ c [Y 0 ] intersect precisely at one point: there uniquely exists τ ∈ H with Im τ = 1/λ c [Y 0 ] such that X (s) τ belongs to T c [Y 0 ] for some s ∈ [0, 1). We show that there is also a Riemann surface Y 0 such that Π • Σ(T c [Y 0 ]) does not meet the horizontal line Im τ = 1/λ c [Y 0 ]. To construct an example we give a preparatory consideration. Let Y 0 = (R 0 , χ 0 ), where χ 0 = (a 0 , b 0 ), be a Riemann surface with marked handle which is not a marked torus, and let ∆ 0 be the (unique) doubly connected domain on R 0 with Γ(∆ 0 ) ⊂ Γ(Y 0 ) and λ(∆ 0 ) = λ(Y 0 ). Suppose that there is a conformal mapping f of a marked once-holed torus X (s)
f : X → Y 0 with d(f ) < ∞. Again, Π • Σ(T ∞ [Y 0 ])τ with Im τ = 1/λ c [Y 0 ] into Y 0 . Since λ f * (D τ ) = λ c [Y 0 ] = λ(Y 0 )
, we obtain f (D τ ) = ∆ 0 by uniqueness. The horizontal arc T
Proposition 3 ([ 10 ,
310Theorem 3]). If T[P] = ∅, then its interior T • [P] is a domain with Lipschitz boundary.Remark. In the case where T[P] = T, we consider T[P] to have a Lipschitz boundary though the boundary ∂T[P] is in fact an empty set.
we deduce the following proposition.
Proposition 5 ([ 10 ,
510Corollary 1]). The sets T ν [Y 0 ], ν ∈ N, are closed domains with Lipschitz boundary, and are retracts of T.
(τ, s) serves as a global topological coordinate system on T. Let Π : H × [0, 1) → H be the natural projection. Then for any handle condition P(X) the image H[P] := Π • Σ(T[P]) is a horizontal strip. To be more precise for t ∈R + := R + ∪ {+∞} = [0, +∞] set H(t) = {t ∈ C | 0 < Im z < t} and letH(t) denote its closure in H. Note that H(0) =H(0) = ∅ and H(+∞) =H(+∞) = H.
Proposition 6 (
6[10, Theorem 4]). For every handle condition P(X) there exists a constant λ[P] ∈R + such that
Lemma 2 .
2IfỸ 0 is the handle covering surface of Y 0 , then T σ [Ỹ 0 ] = T σ [Y 0 ]. Let D be a doubly connected Riemann surface. Denote by λ(D) the extremal length of the free homotopy class Γ(D) of a simple loop separating the boundary components of D. Unless D is conformally equivalent to the punctured plane C\{0}, it carries a hyperbolic metric. Let l(D) stand for the infimum of the hyperbolic lengths of loops in Γ(D). Define l(D) = 0 if D is conformally equivalent to C\{0} = 0. Note that the identity
equivalent to the annulus A(s) := z ∈ C | exp −2π 2 /l(X (s) τ0 , w) < |z| < 1 . The inclusion mapping ι s of X conformal mapping of A(s) into A(0). Observe that the function s → l(X (s) τ0 , w) is increasing. Since ι s tends to the identity mapping of X Now, for (τ, s) ∈ H×[0, 1) the R-linear mapping F τ of C onto itself with F τ (1) = 1 and F τ (τ 0 ) = τ induces a quasiconformal mapping of X
is a horizontal strip. In fact, there is a nonnegative number λ ∞ [Y 0 ] such that(i) if Im τ ≧ 1/λ ∞ [Y 0 ], then X (s) τ ∈ T ∞ [Y 0 ] for any s ∈ [0, 1), while (ii) if Im τ < 1/λ ∞ [Y 0 ], then X (s) τ ∈ T ∞ [Y 0 ] for some s ∈ [0that the horizontal strip Π • Σ(T a [Y 0 ])never meets the critical horizontal line Im τ = 1/λ a [Y 0 ]. In other words, if Im τ = 1/λ a [Y 0 ], then X (s) τ
D τ is mapped onto an arc γ 0 on the boundary ∂∆ 0 , and f T (s) τ is identical with ∆ 0 ∪ γ 0 . This imposes a condition on b 0 , for, it is freely homotopic to f * b (s) τ on R 0 .Example 1. Set R 0 = T τ0 \ π τ0 ({0, 1/2}), which is a twice-punctured torus. Let a 0 and b 0 be the projections of the segments [τ 0 /2, 1 + τ 0 /2] and [3/4, 3/4 + τ 0 ], respectively. They are simple loops on R 0 , and make a mark χ 0 of handle of R 0 . Let Y 0 = (R 0 , χ 0 ). Then λ c [Y 0 ] = λ(Y 0 ) = 1/ Im τ 0 . Since X (1/2) τ0 belongs to T c [Y 0 ], the strip Π • Σ(T c [Y 0 ]) and the line Im τ = 1/λ c [Y 0 ] meet at τ 0 . Example 2. Let R 0 and a 0 be as in the preceding example. Let b ′ 0 be the projection of the polygonal arc obtained by joining the segments [−1/4, τ 0 /4], [τ 0 /4, 1/2−τ 0 /4] and [1/2 − τ 0 /4, 3/4 + τ 0 ]. Set χ ′ 0 = {a 0 , b ′ 0 } and Y ′ 0 = (R 0 , χ ′ 0 ). Again, we have λ c [Y ′ 0 ] = 1/ Im τ 0 . However, Π • Σ(T c [Y ′ 0 ]) does not meet the critical horizontal line Im τ = 1/λ c [Y ′ 0 ].
Proposition 2 ([10, Theorem 2]). There exists a nonnegative number λ a [Y 0 ] such that (i) if Im τ ≧ 1/λ a [Y 0 ], then there are no holomorphic mappings of X for any s, while (ii) if Im τ < 1/λ a [Y 0 ], then there are holomorphic mappings of X(s)
τ
into
Y 0 (s)
τ
The following theorem claims more:Theorem 2. Let Y 0 be a Riemann surface with marked handle which is not a marked torus. Then (i) T σ [Y 0 ] is a closed domain with Lipschitz boundary, (ii) its boundary ∂T σ [Y 0 ] meets T a [Y 0 ] exactly at one point, and
The converse of the Schwarz lemma is false. M F Bourque, Ann. Acad. Sci. Fenn. 41M. F. Bourque, The converse of the Schwarz lemma is false, Ann. Acad. Sci. Fenn. 41 (2016), 235-241.
Geometry and spectra of compact Riemann surfaces. P Buser, Birkhäuser, Boston-Basel-BerlinP. Buser, Geometry and spectra of compact Riemann surfaces, Birkhäuser, Boston-Basel- Berlin, 1992.
On analytic self-mappings of Riemann surfaces II. J A Jenkins, N Suita, Math. Ann. 209J. A. Jenkins and N. Suita, On analytic self-mappings of Riemann surfaces II, Math. Ann. 209 (1974), 109-115.
J Kahn, K M Pilgrim, D P Thurston, arXiv:1507.05294Conformal surface embeddings and extremal length, preprint. J. Kahn, K. M. Pilgrim and D. P. Thurston, Conformal surface embeddings and extremal length, preprint, arXiv:1507.05294.
Analytic self-mappings of Riemann surfaces. A Marden, I Richards, B Rodin, J. Anal. Math. 18A. Marden, I. Richards and B. Rodin, Analytic self-mappings of Riemann surfaces, J. Anal. Math. 18 (1967), 197-225.
Comparison of hyperbolic and extremal lengths. B Maskit, Ann. Acad. Sci. Fenn. 10B. Maskit, Comparison of hyperbolic and extremal lengths, Ann. Acad. Sci. Fenn. 10 (1985), 381-386.
Conformal mappings of a once-holed torus. M Masumoto, J. Anal. Math. 66M. Masumoto, Conformal mappings of a once-holed torus, J. Anal. Math. 66 (1995), 117-136.
Once-holed tori embedded in Riemann surfaces. M Masumoto, Math. Z. 257M. Masumoto, Once-holed tori embedded in Riemann surfaces, Math. Z. 257 (2007), 453-464.
On critical extremal length for the existence of holomorphic mappings of once-holed tori. M Masumoto, J. Inequal. Appl. 282M. Masumoto, On critical extremal length for the existence of holomorphic mappings of once-holed tori, J. Inequal. Appl. 2013, 2013:282.
Holomorphic mappings of once-holed tori. M Masumoto, to appear in J. Anal. MathM. Masumoto, Holomorphic mappings of once-holed tori, to appear in J. Anal. Math.
The moduli of compact continuations of an open Riemann surface of genus one. M Shiba, Trans. Amer. Math. Soc. 301M. Shiba, The moduli of compact continuations of an open Riemann surface of genus one, Trans. Amer. Math. Soc. 301 (1987), 299-311.
The euclidean, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems. M Shiba, Kodai Math. J. 16M. Shiba, The euclidean, hyperbolic, and spherical spans of an open Riemann surface of low genus and the related area theorems, Kodai Math. J. 16 (1993), 118-137.
K Strebel, Quadratic Differentials. Berlin-Heidelberg-New York-TokyoSpringer-VerlagK. Strebel, Quadratic Differentials, Springer-Verlag, Berlin-Heidelberg-New York-Tokyo, 1984.
The length spectra as moduli for compact Riemann surfaces. S Wolpert, Ann. of Math. 109S. Wolpert, The length spectra as moduli for compact Riemann surfaces, Ann. of Math. 109 (1979), 323-351.
| {'fraction_non_alphanumeric': 0.08175734652313063, 'fraction_numerical': 0.03008437590922316, 'mean_word_length': 3.1286486486486487, 'pattern_counts': {'":': 0, '<': 18, '<?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': 15, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In our previous work[10], for a given Riemann surface Y 0 with marked handle, we investigated geometric properties of the set of marked onceholed tori X allowing holomorphic mappings of X into Y 0 . It turned out that it is a closed domain with Lipschitz boundary. In the present paper we show that the boundary is never smooth. Also, we evaluate the critical extremal length for the existence of holomorphic mappings in terms of hyperbolic lengths.1991 Mathematics Subject Classification. Primary 30F99; Secondary 30F45, 30F60, 32G15.', 'arxivid': '1604.05014', 'author': ['Makoto Masumoto '], 'authoraffiliation': [], 'corpusid': 119122691, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12579, 'n_tokens_neox': 10724, 'n_words': 6967, 'pdfsha': '749e977447b41dc729a9cfd838ac1a9848a357c7', 'pdfurls': ['https://export.arxiv.org/pdf/1604.05014v1.pdf'], 'title': ['ON SETS OF MARKED ONCE-HOLED TORI ALLOWING HOLOMORPHIC MAPPINGS INTO RIEMANN SURFACES WITH MARKED HANDLE', 'ON SETS OF MARKED ONCE-HOLED TORI ALLOWING HOLOMORPHIC MAPPINGS INTO RIEMANN SURFACES WITH MARKED HANDLE'], 'venue': []} |
arxiv |
A generalized Isserlis theorem for location mixtures of Gaussian random vectors
12 Jul 2011
C Vignat
E.P.F.L., L.T.H.I
LausanneSwitzerland
A generalized Isserlis theorem for location mixtures of Gaussian random vectors
12 Jul 2011Isserlis theoremnormal-variance mixturegeneralized hyperbolic distribution
In a recent paper, Michalowicz et al. provide an extension of Isserlis theorem to the case of a Bernoulli location mixture of a Gaussian vector. We extend here this result to the case of any location mixture of Gaussian vector; we also provide an example of the Isserlis theorem for a "scale location" mixture of Gaussian, namely the d−dimensional generalized hyperbolic distribution.
Introduction
Isserlis theorem, as discovered by Isserlis [1] in 1918, allows to express the expectation of a monomial in an arbitrary number of components of a zero mean Gaussian vector X ∈ R d in terms of the entries of its covariance matrix only. Before providing in Thm 1 the slightly generalized version of Isserlis theorem due to Withers [3], we introduce the following notations: for any set A = {α 1 , . . . , α N } of integers such that 1 ≤ α i ≤ d and any vector X ∈ R d , we use the multi-index notation and denote X A = αi∈A X αi with the convention that for the empty set
X ∅ = 1.
A pairing in a set A is a partition of A into disjoint pairs. We denote by Π (A) the set of all pairings σ in A: note that Π (A) is empty if A has an odd number of elements. For a given σ ∈ Π (A), we denote by A/σ the set {i; σ = (i, σ (i))} ; finally,
A E (X i X j ) denotes the sum σ∈Π(A) i∈A/σ E X αi X α σ(i) .
In other words, for a given pairing σ in the set A, we compute the product of all possible moments E (X i X j ) where i and j are paired by σ; then, A E (X i X j ) denotes the sum of these products over all possible pairings in A. As an example
{1,1,2,4} E (X i X j ) = E X 2 1 E (X 2 X 4 ) + 2E (X 1 X 2 ) E (X 1 X 4 ) .
A general form of Isserlis theorem, due to Withers, is as follows. 2N ] and X ∈ R d is a Gaussian vector with zero mean then
Theorem 1. If A = {α 1 , . . . , α 2N } is a set of integers such that 1 ≤ α i ≤ d, ∀i ∈ [1,EX A = A E (X i X j )(1)
Moreover, if A = {α 1 , . . . , α 2N +1 } then, under the same assumptions,
EX A = 0.
For example, choosing α i = i, 1 ≤ i ≤ 4 yields the well-known identity
E (X 1 X 2 X 3 X 4 ) = E (X 1 X 2 ) E (X 3 X 4 ) + E (X 1 X 3 ) E (X 2 X 4 ) + E (X 1 X 4 ) E (X 2 X 3 ) .
However, indices α i need not be distinct: for example, choosing α i = 1, 1 ≤ i ≤ 4 yields EX 4 1 = 3EX 2 1 . Several extensions of this result have been provided recently: in [3], Withers extends Isserlis theorem to the case of noncentral Gaussian vectors and relates the result with multivariate Hermite polynomials; in [4], a general formula for Gaussian scale mixtures, and more generally for elliptically distributed vectors is derived; it is applied to the computation of moments of the uniform distribution on the sphere. In [5], Isserlis theorem is extended to the computation of the moments of linear combinations of independent Student-t vectors. In [6], Isserlis theorem is extended to Gaussian matrix mixtures, i.e. random vectors of the form
X = AN
where N is a standard Gaussian vector in R d and A is a (d × d) random matrix. Let us also mention the reference [8] where the author tackles the computational complexity of formula (1), using Magnus lemma to replace a product of n variables by sums of polynomials of degree n in these variables.
Recently, Michalowicz et al. [2] addressed the case of Gaussian location mixtures: they provided an extension of Isserlis theorem to the case of a random vector X ∈ R d with probability density
f X (x) = 1 2 φ R (x + µ) + 1 2 φ R (x − µ) (2) where φ R (x) = 1 |2πR| 1 2 exp − 1 2 x t R −1 x
is the d−variate Gaussian density with zero mean and covariance matrix R.
In the following, we give a new and simple proof of the result by Michalowicz et al., adopting a formalism that allows us to extend their results to the general case of an arbitrary Gaussian location mixture. We also provide an extension of these results to the case of a scale-location mixture of Gaussian.
Extensions of the result by Michalowicz et al.
A key observation is that the random vector X with density (2) reads
X = ǫµ + ζ(3)
where ǫ is a Bernoulli random variable (Pr {ǫ = −1} = Pr {ǫ = 1} = 1 2 ), ζ ∈ R d is a zero mean Gaussian vector and equality is in the sense of distributions. This stochastic representation allows to prove easily a generalized version of the main result of [2], namely Theorem 2. If X ∈ R d is distributed according to (2) and
A = {α 1 , . . . , α 2N } with 1 ≤ α i ≤ d then EX A = N k=0 S⊂A |S|=2k µ S A\S E (ζ i ζ j )
.
If A = {α 1 , . . . , α 2N +1 } then EX A = 0.
The simplified proof we propose is as follows: by (3),
EX A = E (ǫµ + N ) A
and since the product (a + b) A can be expanded as
(a + b) A = 2N k=0 S⊂A |S|=k a S b A\S ,
we deduce that
EX A = 2N k=0 S⊂A |S|=k µ S E ǫ k E ζ A\S .
By Isserlis theorem, the expectation of the product of an odd number of centered Gaussian random variables ζ i is equal to zero so that this expression simplifies to N k=0 S⊂A |S|=2k
µ S E ǫ 2k E ζ A\S .
Since ǫ is Bernoulli distributed, all its even moments are equal to 1; moreover, since ζ has zero mean, by Isserlis theorem, E ζ A\S = A\S E (ζ i ζ j ) and we obtain
EX A = N k=0 S⊂A |S|=2k µ S A\S E (ζ i ζ j )
which is the desired result. The case where A has an odd number of elements is equally simple. We note that Theorem 2 can be also easily deduced using generating functions as done in [3, Theorem 1.1] who proves a version of Wick's theorem for a Gaussian vector with mean µ = 0 : choosing a Bernoulli randomized version of this mean as in (3) yields the result.
The general case of Gaussian location mixture
With the useful representation (3), we can generalize the preceding result to any kind of location mixture of Gaussian: namely, we consider a random vector X ∈ R d that reads
X = µ + ζ (4)
where ζ is a zero-mean Gaussian vector in R d , independent of the random vector µ ∈ R d with probability distribution F µ ; note that the vector µ may be discrete -taking values µ i with probabilities p i -or not, but we don't need to assume the existence of a density f µ . In the discrete case, the density of X reads
f X (x) = +∞ i=0 p i φ R (x − µ i ) ;
and in the most general case,
f X (x) = R d φ R (x − µ) dF µ (µ) .
We now state our main theorem.
Theorem 3. Assume that X ∈ R d follows model (4) and that all the first-order moments m k = Eµ k of µ exist. Then if A = {α 1 , . . . , α 2N +ǫ } , with ǫ ∈ {0, 1} ,
EX A = N k=0 S⊂A |S|=2k+ǫ E (µ S ) A\S E (ζ i ζ j )(5)
We remark that if all elements α i of A are different and if the vector µ has independent components, this expression can be further simplified to
EX A = N k=0 S⊂A |S|=2k+ǫ (Eµ) S A\S E (ζ i ζ j ) ,(6)
noting the difference between E (µ S ) = E αi∈S µ αi in (5) and (Eµ) S = αi∈S Eµ αi in (6). The proof is as follows.
Proof 1. From (4), we deduce
EX A = 2N +ǫ k=0 S⊂A |S|=k E (µ S ) E ζ A\S .
Since the cardinality of A\S is 2N + ǫ − k, E ζ A\S = 0 unless |S| = k has the same parity as ǫ, in which case it is equal to A\S E (ζ i ζ j ), hence formula (5). Formula (6) is easily deduced from formula (5) assuming that the components of µ are independent and that all elements of A are distinct.
We now provide a further generalization of Isserlis theorem by considering a Gaussian vector with both random scale and location parameters.
A Normal variance-mean mixture application
The generalized d−dimensional hyperbolic distribution was introduced by Barndorff-Nielsen in 1978 [7]. It is the distribution of a random vector that reads
X = µ + σ 2 ∆β + σ∆ 1 2 ζ(7)
where µ and β are two deterministic vectors in R d , ∆ is a deterministic (d × d) matrix with |∆| = 1, ζ is a standard Gaussian vector in R d and σ 2 is a scalar random variable that follows the Generalized Inverse Gaussian GIG (ψ, χ, λ) distribution
f ψ,χ,λ (x) = ψ χ λ 2 2K λ √ ψχ x λ−1 exp − 1 2 χx −1 + ψx , x > 0(8)
with parameters ψ > 0, χ > 0 and λ ∈ R. We note that in (7), the GIG random variable σ 2 appears both as a scale and location parameter of the Gaussian vector, hence the "normal variance-mean mixture" name. From the stochastic representation (7), we derive a version of the Isserlis theorem as follows.
Theorem 4. If X ∈ R d is a generalized hyperbolic vector as in (7) and A = {α 1 , . . . , α 2N +ǫ } with ǫ ∈ {0, 1} then
EX A = 0 ≤ l ≤ N 0 ≤ p ≤ 2l + ǫ T ⊂S⊂A |T |=p,|S|=2l+ǫ µ T γ S T m N +l−p+ǫ A\S EZ i Z j
where γ = ∆β, where Z is a centered Gaussian vector with covariance matrix ∆ and
m l = Eσ 2l = ψ χ − l 2 K λ+l √ ψχ K λ √ ψχ .
Proof 2. Assuming first ǫ = 0, we have
EX A = E N l=0 S⊂A |S|=2l µ + σ 2 γ S (σZ) A\S = N l=0 S⊂A |S|=2l E σ 2N −2l µ + σ 2 γ S EZ A\S with EZ A\S = A\S EZ i Z j and Eσ 2N −2l µ + σ 2 γ S = 2l p=0 T ⊂S |T |=p µ T γ S\T E σ 2 N +l−p .
The moment of order l of the GIG random variable σ 2 can be easily computed from (8) as
m l = ψ χ − l 2 K λ+l √ ψχ K λ √ ψχ ,
hence the result. The case ǫ = 1 follows the same steps.
On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables. L Isserlis, Biometrika. 12Isserlis L., On a formula for the product-moment coefficient of any order of a normal frequency distribution in any number of variables, Biometrika, 1918, 12, 134-139
A general Isserlis theorem for mixed-Gaussian random variables. J V Michalowicz, J M Nichols, F Bucholtz, C C Olson, Statistics & Probability Letters. Michalowicz J.V., Nichols J.M., Bucholtz F., Olson C.C., A general Isserlis theorem for mixed-Gaussian random variables, Statistics & Probability Letters, August 2011, 81-8,1233-1240
The moments of the multivariate normal. C S Withers, Bulletin of the Australian Mathematical Society. 32Withers C.S., The moments of the multivariate normal, Bulletin of the Australian Mathematical Society, 1985, 32, 103-107
An extension of Wick's theorem. C Vignat, S Bhatnagar, Statistics & Probability Letters. Vignat C. and Bhatnagar S., An extension of Wick's theorem, Statistics & Probability Letters, 2008, 78-15, 2404-2407
The Wick theorem for non-Gaussian distributions and its application for noise filtering of correlated q-Exponentially distributed random variables. P Repetowicz, P Richmond, arXiv:math-ph/0411020v1unpublishedRepetowicz P. and Richmond P., The Wick theorem for non-Gaussian distributions and its application for noise filtering of correlated q-Exponentially distributed random variables, unpublished, arXiv:math-ph/0411020 v1, Nov 2004
On the Wick theorem for mixtures of centered Gaussian distributions. B Grigelionis, Lithuanian Mathematical Journal. Grigelionis B., On the Wick theorem for mixtures of centered Gaussian distributions, Lithuanian Mathematical Journal, 2009, 49-4, 372-380
O Barndorff-Nielsen, Hyperbolic Distributions and Distributions on Hyperbolae. Barndorff-Nielsen O., Hyperbolic Distributions and Distributions on Hyperbolae, Scandinavian Journal of Statistics, 1978, 5-3, 151-157
From Moments of Sum to Moments of Product. R Kan, Journal of Multivariate Analysis. 99Kan R., From Moments of Sum to Moments of Product, Journal of Multivariate Anal- ysis, 99, 542-554, 2008.
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arxiv |
Crystallization dynamics of magnetic skyrmions in a frustrated itinerant magnet
25 May 2023
Kotaro Shimizu
Department of Applied Physics
The University of Tokyo
113-8656TokyoJapan
Department of Physics
University of Virginia
22904CharlottesvilleVirginiaUSA
Gia-Wei Chern
Department of Physics
University of Virginia
22904CharlottesvilleVirginiaUSA
Crystallization dynamics of magnetic skyrmions in a frustrated itinerant magnet
25 May 2023(Dated: May 26, 2023)
We investigate the phase ordering kinetics of skyrmion lattice (SkL) in a metallic magnet. The SkL can be viewed as a superposition of magnetic stripes whose periods are determined by the quasi-nesting wave vectors of the underlying Fermi surface. An effective magnetic Hamiltonian that describes the electron-mediated spin-spin interaction is obtained for a two-dimensional s-d model with the Rashba spin-orbit coupling. Large-scale Landau-Lifshitz-Gilbert dynamics simulations based on the effective spin Hamiltonian reveal a two-stage phase ordering of the SkL phase after a thermal quench. The initial fast crystallization of skyrmions is followed by a slow relaxation dominated by the annihilation dynamics of dislocations, which are topological defects of the constituent magnetic stripe orders. The late-stage phase ordering also exhibits a dynamical scaling symmetry. We further show that the annihilation of dislocations follows a power-law time dependence with a logarithmic correction that depends on magnetic fields. Implications of our results for SkL phases in magnetic materials are also discussed.Complex magnetic textures such as vortices and skyrmions are not only of great fundamental interest in magnetism but also have important implications in the emerging technology of spintronics[1][2][3][4]. Both vortices and skyrmions are nano-sized particle-like spintextures characterized by nontrivial topological invariants. The presence of such complex patterns in metallic magnets could give rise to intriguing electronic and transport properties due to a nontrivial Berry phase acquired by electrons when traversing over closed loops of noncollinear or noncoplanar spins[1,[5][6][7]. The well-studied topological Hall effects [8-13] and topological Nernst effects[14][15][16]are some of the representative examples. Also importantly, such topological electronic responses in metallic magnets can be controlled via the manipulation of magnetic textures.In magnetic materials, skyrmions are often stabilized in the form of a skyrmion lattice (SkL), which is a periodic array of such particle-like topological spin-textures. Indeed, SkL as a spontaneous ground state was already predicted in the pioneering work of Bogdanov and Yablonskii that later triggered the enormous interest in magnetic skyrmions[17,18]. SkLs have since been reported in several chiral magnets such as MnSi [19] and other B20 compounds, as well as centrosymmetric materials[20]. While the picture of SkL as an array of particle-like objects offers an intuitive framework to understand certain structural and dynamical aspects of skyrmion phases[21], recent studies have revealed the close connection between SkL and multiple-Q magnetic ordering driven by partial nesting of the electron Fermi surface[22][23][24]. Indeed, this mechanism has been conjectured to be ubiquitous for itinerant spin systems with a wide range of filling fractions.Despite the huge interest and extensive research on magnetic SkL over the past decades, the phase-ordering dynamics of SkLs remains an open subject. Specifically, here one concerns the dynamical evolution and potential universal behaviors of skyrmion crystallization when a magnet is quenched into a skyrmion phase. It is known that the kinetics of phase ordering depends crucially on topological defects of the symmetry-breaking phase. Several super-universality classes of domain-growth laws have been established over the years[25,26]. The fact that both magnetic and translational symmetries are broken in a skyrmion crystal indicates rich relaxational dynamics of SkL phases which has yet to be systematically investigated. Understanding the phase ordering of SkL is also crucial to the engineering and control of skyrmion phases in real materials.In this paper, we make an important step toward this goal by investigating the crystallization dynamics of a SkL in a realistic model of chiral metallic magnets. A minimum microscopic model for such itinerant spin systems is the s-d Hamiltonian H sd = k,σ ϵ k,σ c † k,σ c k,σ − J sd i S i · c † iσ σ σ,σ ′ c iσ ′ , where the first term describes electron hopping on a lattice with t ij the transfer integrals and the second term represents a local coupling between the itinerant s-electron and the local moments S i of d-electrons; J sd represents the coupling strength. Spin-orbit coupling (SOC) within such single-band models can be described by either Rashba or Dresselhaus hopping terms. Dynamical simulations based on such electron models requires solving a disordered electron tightbinding Hamiltonian at every time-step, which could be prohibitively expensive for large systems. Instead, here we consider an effective spin Hamiltonian, similar in spirit to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction[27][28][29], with SOC properly included[30]. At the leading second-order perturbation, the effective Hamiltonian has the general form(1) arXiv:2305.16182v1 [cond-mat.str-el]
We investigate the phase ordering kinetics of skyrmion lattice (SkL) in a metallic magnet. The SkL can be viewed as a superposition of magnetic stripes whose periods are determined by the quasi-nesting wave vectors of the underlying Fermi surface. An effective magnetic Hamiltonian that describes the electron-mediated spin-spin interaction is obtained for a two-dimensional s-d model with the Rashba spin-orbit coupling. Large-scale Landau-Lifshitz-Gilbert dynamics simulations based on the effective spin Hamiltonian reveal a two-stage phase ordering of the SkL phase after a thermal quench. The initial fast crystallization of skyrmions is followed by a slow relaxation dominated by the annihilation dynamics of dislocations, which are topological defects of the constituent magnetic stripe orders. The late-stage phase ordering also exhibits a dynamical scaling symmetry. We further show that the annihilation of dislocations follows a power-law time dependence with a logarithmic correction that depends on magnetic fields. Implications of our results for SkL phases in magnetic materials are also discussed.
Complex magnetic textures such as vortices and skyrmions are not only of great fundamental interest in magnetism but also have important implications in the emerging technology of spintronics [1][2][3][4]. Both vortices and skyrmions are nano-sized particle-like spintextures characterized by nontrivial topological invariants. The presence of such complex patterns in metallic magnets could give rise to intriguing electronic and transport properties due to a nontrivial Berry phase acquired by electrons when traversing over closed loops of noncollinear or noncoplanar spins [1,[5][6][7]. The well-studied topological Hall effects [8][9][10][11][12][13] and topological Nernst effects [14][15][16] are some of the representative examples. Also importantly, such topological electronic responses in metallic magnets can be controlled via the manipulation of magnetic textures.
In magnetic materials, skyrmions are often stabilized in the form of a skyrmion lattice (SkL), which is a periodic array of such particle-like topological spin-textures. Indeed, SkL as a spontaneous ground state was already predicted in the pioneering work of Bogdanov and Yablonskii that later triggered the enormous interest in magnetic skyrmions [17,18]. SkLs have since been reported in several chiral magnets such as MnSi [19] and other B20 compounds, as well as centrosymmetric materials [20]. While the picture of SkL as an array of particle-like objects offers an intuitive framework to understand certain structural and dynamical aspects of skyrmion phases [21], recent studies have revealed the close connection between SkL and multiple-Q magnetic ordering driven by partial nesting of the electron Fermi surface [22][23][24]. Indeed, this mechanism has been conjectured to be ubiquitous for itinerant spin systems with a wide range of filling fractions.
Despite the huge interest and extensive research on magnetic SkL over the past decades, the phase-ordering dynamics of SkLs remains an open subject. Specifically, here one concerns the dynamical evolution and potential universal behaviors of skyrmion crystallization when a magnet is quenched into a skyrmion phase. It is known that the kinetics of phase ordering depends crucially on topological defects of the symmetry-breaking phase. Several super-universality classes of domain-growth laws have been established over the years [25,26]. The fact that both magnetic and translational symmetries are broken in a skyrmion crystal indicates rich relaxational dynamics of SkL phases which has yet to be systematically investigated. Understanding the phase ordering of SkL is also crucial to the engineering and control of skyrmion phases in real materials.
In this paper, we make an important step toward this goal by investigating the crystallization dynamics of a SkL in a realistic model of chiral metallic magnets. A minimum microscopic model for such itinerant spin systems is the s-d Hamiltonian
H sd = k,σ ϵ k,σ c † k,σ c k,σ − J sd i S i · c † iσ σ σ,σ ′ c iσ ′ ,
where the first term describes electron hopping on a lattice with t ij the transfer integrals and the second term represents a local coupling between the itinerant s-electron and the local moments S i of d-electrons; J sd represents the coupling strength. Spin-orbit coupling (SOC) within such single-band models can be described by either Rashba or Dresselhaus hopping terms. Dynamical simulations based on such electron models requires solving a disordered electron tightbinding Hamiltonian at every time-step, which could be prohibitively expensive for large systems. Instead, here we consider an effective spin Hamiltonian, similar in spirit to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction [27][28][29], with SOC properly included [30]. At the leading second-order perturbation, the effective Hamiltonian has the general form Here we have included a Zeeman coupling to an external field H = (0, 0, H), and J(r) represents an effective 3 × 3 interaction matrix between two spins separated by r. Its Fourier transform is given bỹ
H = − ij S i · J(r i − r j ) · S j − i H · S i .(1J(q) = J 2 sd N k σ,σ ′ χ 0 σσ ′ (k, q) I + F σσ ′ (k, q) ,(2)
where
χ 0 σσ ′ = [f (ϵ k,σ ) − f (ϵ k+q,σ ′ )]/(ϵ k+q,σ ′ − ϵ k,σ )
is the spin-dependent susceptibility, ϵ k,σ is the electron band energy, f (ϵ) is the Fermi-Dirac function, I is the identity matrix, and the dimensionless matrix F σσ ′ accounts for the anisotropic spin interaction due to SOC [30,31].
The interaction matrixJ(q) is often dominated by a few wave vectors Q η when part of the electron Fermi surface is connected by them, i.e. ϵ k+Qη,σ ′ ≈ ϵ k,σ . Such partial nesting of the Fermi surface has been shown to be a primary mechanism for the stabilization of skyrmion or vortex lattices in metallic magnets [23]. An effective real-space Hamiltonian is then given by the inverse Fourier transformation ofJ(q), which likely can only be done numerically for general dispersion relation ϵ k,σ . As a first-order approximation, effective spin Hamiltonians can be obtained by keeping only contribution from the nesting wave vectors:
J(q) ≈ ηJ (Q η )δ(q − Q η )
. This approach has been employed to investigate complex spin textures in the ground state of itinerant magnets [23,24,[32][33][34]. However, this approach gives rise to unrealistic infinite-range spin interactions. A more realistic analytical approach, while preserving the correct form of the anisotropic interaction, is to replace the δ-function by a peak-shape function h(q) of a finite width, giving rise to spin-spin interactions which decay with distance in real space [35].
For concreteness, here we apply the procedure described above to a square-lattice s-d model with a Rashba SOC. A square array of skyrmions can be stabilized by partial nesting with two wave vectors Q 1 = (Q, 0) and Q 2 = (0, Q). The resultant real-space interaction matrix is given by
J(r) = η=1,2 g(r) [ReJ η cos(Q η ·r) − ImJ η sin(Q η ·r)] ,(3)
where
J 1 = J ⊥ 0 0 0 J ⊥ −iD 0 iD J zz , J 2 = J ⊥ 0 −iD 0 J ⊥ 0 iD 0 J zz . (4)
and J ⊥ , J zz describe electron-mediated exchange interactions, and D represents effective long-ranged Dzyaloshinskii-Moriya interaction (DMI) induced by SOC; both J and D are of the order of J 2 sd /W , where W is the electron bandwidth. In the following, we set 2J ⊥ + J z = 1 to serve as the unit for energy (and inverse time) and D = 0.3. The function g(r) describes the decay of spin-spin interaction with distance. For simplicity, we assume a Lorentzian function for h(q), which leads to an exponential decaying g(r) = A e −γ(|x|+|y|) , where A is a normalization constant [31]. A hard cutoff r c such that J(r) = 0 for |x| + |y| > r c is further introduced for large-scale simulations of N = 1000 2 spins. For results presented in the following, parameters γ = 0.3 and r c = 16 are used.
The dynamical evolution of the magnet is described by the Landau-Lifshitz-Gilbert (LLG) equation
dS i dt = 1 1 + α 2 ∂H ∂S i × S i + αS i × S i × ∂H ∂S i ,(5)
where α is the Gilbert damping coefficient, which is set to 0.1 in the following simulations. A fourth-order Runge-Kutta method is used to integrate the LLG equation with a time-step ∆t = 0.05. The phase diagram, shown in Fig. 1(a) in the plane of exchange anisotropy J zz versus field H, includes a 1Q-cycloidal order, a double-Q SkL, and a forced ferromagnetic state at high field. It is worth noting that our results are consistent with the phase diagram obtained from simulated annealing minimization of the infinite-range effective spin model [30]. The double-Q magnetic order at H = 0, shown in Fig. 1(b), exhibits a non-coplanar Néel-type vortex texture. Importantly, for small J zz , the SkL is stable for a wide range of magnetic field, allowing us to study the crystallization dynamics of skyrmions and the field effects. To this end, the LLG dynamics was employed to simulate thermal quenches of a spin system with the effective RKKY interaction in Eq. (3). An initial state of random spins, corresponding to equilibrium at high temperatures, is suddenly quenched to zero temperature at time t = 0. A typical example of the subsequent evolution of spins is shown in Fig. 2 for the case of zero-field quench. The color shows the scalar chirality which is defined as χ sc i = △i S 1 · S 2 × S 3 /2, where the summation is taken over four triplets of nearest spins on a triangle. As skyrmions, including the Néel vortex texture at zero field, are characterized by non-coplanar spins that wrap around a sphere, the emergence of a SkL domain is indicated by the staggered checkerboard-like arrangement of positive and negative scalar chirality. As shown in Figs. 2(a) and (b), small patches of skyrmion arrays quickly emerge after the thermal quench. However, longrange coherence is yet to be established between different patches, and large areas of vanishing scalar chirality mark the boundaries between skyrmion crystallites. As long-range crystallization order further develops, the incoherent regions quickly contract to particle-like objects of similar size as a skyrmion as shown in Figs. 2(c) and (d). These "particles", also characterized by a vanishing χ sc i , correspond to dislocation defects, which are topological defects associated with broken translational symmetries. The dislocations here correspond to a starting point of an extra row or column of skyrmion lines in the square lattice. These particle-like defects carry a topological charge corresponding to the so-called Burgers vector. Their topological nature also manifests itself in the fact that dislocations are created and annihilated in pairs. The phase-ordering of SkL is now dominated by the dynamics of dislocation defects.
The magnetic structure of the double-Q SkL can be approximated by an equal superposition of two spirals:
S(r i ) ∼ (cos Q 1 , cos Q 2 , b(sin Q 1 + sin Q 2 ) + m 0 ) ,(6)
where Q η = Q η · r i + const, and the coefficients b and m 0 depend on model parameters and magnetic field. Importantly, the x-and y-component of the spin field correspond to simple unidirectional stripes along the x and y directions, respectively. The fundamental defects of stripe order are also dislocations, as shown in Figs. 3(a) and (b). We can thus further classify the dislocation defects of SkL according to whether it is associated with the S x or S y component. Indeed, our simulations find that pair annihilations are possible only for dislocations of the same spin component.
To further quantify the phase ordering of SkL, we examine the time-dependent spin structure factor, defined as S(q, t) = 1
N 2 ⟨| i S i (t) exp(iq · r i )| 2 ⟩,
where ⟨· · ·⟩ denotes the average of independent initial conditions. Examples of the structure factor at the early and late stages of phase ordering are shown in Figs. 4(a) and (b), respectively. The structure factor exhibits four broad peaks at ±Q 1 and ±Q 2 , where Q 1 = (Q, 0), Q 2 = (0, Q), and Q = 0.785, quickly after the quench; see e.g. Fig. 4(a) for t = 40. These peaks at the nesting wave vectors become sharper as the system relaxes toward equilibrium, as shown in Fig. 4(b). Moreover, satellite peaks at n 1 Q 1 + n 2 Q 2 with n 1 + n 2 = 2n + 1 (n is an integer) start to emerge at late times, signaling the onset of higher harmonics of the constituent spiral orders.
An overall order parameter of the SkL phase can be defined as the sum of peak intensities M(t) = S(Q 1 , t) + S(Q 2 , t). As shown in Fig. 4(c), this SkL order parameter clearly exhibits a two-stage ordering discussed above: the fast development of quasi-long-ranged crystalline domains (t < ∼ 80), followed by a slow powerlaw growth dominated by the annihilation dynamics of dislocations (t > ∼ 80). To characterize the late-stage phase ordering, Fig. 4(c) also shows the growth of the correlation length defined as the inverse widths of the Fourier peaks: ξ Qη = 1/∆Q η , where ∆Q η = q∼Qη |q− Q η |S(q, t)/ q∼Qη S(q, t). Furthermore, the power-law growth of the correlation lengths is intimately related to that of SkL order parameter M. Indeed, we find that the late-stage ordering of SkL exhibits a dynamical symmetry. The structure factor at different times can be described by a universal function F with proper rescaling
S(q, t) = ξ 2 Qη (t) F |q − Q η | ξ Qη (t) .(7)
This is illustrated by the excellent data points collapsing shown in Fig. 4(d). The above late-time power-law behaviors are related to the dynamics of dislocations, which are topological defects of emergent square skyrmion arrays discussed above. As the long-range coherence of a crystalline order is disrupted by dislocations, the correlation length of SkL can be interpreted as the average distance ℓ between dislocations, which is related to their number density as ℓ ∼ ρ −1/2 d . The power-law growth of correlation length ξ ∼ t α thus implies a power-law decrease of dislocation density ρ d ∼ t −η with the exponent η = 2α. This is indeed confirmed in our large-scale LLG simulations summarized in Fig. 5, where a naive power-law fitting gives an exponent η that depends weakly on magnetic field H. For example, the zero-field exponent η ∼ 1.03 is consistent with the grow exponent of the correlation length shown in Fig. 4(c).
To understand the power-law annihilation of dislocations, we note that these topological defects can be mapped to vortices of effective 2D-XY models. As discussed in Eq. (6), the SkL can be viewed as comprised of two spiral spin orders. The x and y components of the coarse-grained spin field represent two orthogonal stripes S x (r) ∼ cos[Q 1 ·r + θ 1 (r)] and S y (r) ∼ cos[Q 2 ·r + θ 2 (r)]. The dislocations associated with the two stripes, shown in Figs. 3(a) and (b), correspond to vortex singularities of the phase fields θ 1,2 (r). At the leading-order approximation, our system can thus be described by two coupled XY models:
E = A {[(∇θ 1 ) 2 + (∇θ 2 ) 2 ] + 2κ(∇θ 1 · ∇θ 2 )}d 2 r,(8)
where A > 0 represents the stiffness of the phase fields, Real-time dependence of the density of dislocations defects ρ d (t) with J zz = 0 for varying magnetic field H, obtained by averaging over 32 independent runs. The positions of defects are detected from the energy density [38]. The pale lines represent the fitting by the formula
ρ d (t)/(log(ρ0/ρ d (t)) − 1) = a(t − t0) −1 .
and κ denotes their coupling. A large κ could induce a bound state of vortices from the two XY fields. Nonetheless, it is straightforward to show that the above action is equivalent to two independent XY fields θ ± = θ 1 ± θ 2 with different stiffness: A ± = A(1 ± κ)/2. Importantly, it has been shown that the phase ordering of the 2D XY model is governed by the annihilation of vortices, whose number density follows a power-law ρ v ∼ t −1 time dependence, up to a logarithmic correction [36,37]. Our results thus strongly indicate that the ordering kinetics of the SkL phase belongs to the same dynamical universality class of 2D XY model. In fact, while the extracted exponent is weakly dependent on H, the fact that all η values lie in the vicinity of unity suggests a universal ρ d ∼ t −1 behavior, yet with a field-dependent logarithmic correction. We have checked that the various curves in Fig. 5 indeed can be well described by the formula ρ d /(log(ρ 0 /ρ d ) − 1) = a(t − t 0 ) −1 , where ρ 0 , t 0 , and a are fitting parameters [36].
To conclude, we have presented a comprehensive study on the ordering dynamics of SkL phases in chiral metallic magnets, which is important to the design and engineering of skyrmion-based spintronics devices. Fundamentally, magnetic skyrmion lattices also provide another platform to study crystallization phenomena in two dimensions, a field which has yet to be systematically investigated. An intriguing fast crystallization process has recently been observed in the "square ice" formed by water molecules locked between two graphene sheets [39]. While our work here sheds new light on this fundamental subject of 2D crystallization dynamics, several important issues remain to be addressed. For example, although the long-range nature of electron-mediated interactions might be crucial to the initial fast crystallization, a detailed study on the effect of interaction range and its interplay with other factors is desired. The built-in chirality due to SOC is another crucial component for the fast establishment of coherent lattices. The recent experimental observation of square SkL in centrosymmetric magnets such as GdRu 2 Si 2 [40] and EuAl 4 [41] thus calls for theoretical studies of skyrmion crystallization dynamics in non-chiral itinerant magnets [23,42]. Yet another interesting question is the effects of crystal geometry. More complicated annihilation dynamics of dislocations might arise in the triangular SkL observed in many skyrmion materials.
The authors thank Y. Kato
The ground state phase diagram includes a square Skyrmion lattice (SkL), an 1Q state, and a forced ferromagnetic state (FFM). (b) The real-space spin texture for J zz = 0 and H = 0. The color of the arrows represents the out-of-plane component of spins.
FIG. 2 .
2Spatial distribution of the scalar spin chirality χ sc i obtained from spin configurations at different times after a thermal quench. Parameters J zz = 0, and H = 0 are used in the LLG simulation of a 1000 2 system. Shown here are a selected 200 × 200 region of the lattice.
FIG. 3 .
3Spatial profile of the three components of spin field Si at late stage of the phase ordering of SkL. Parameters are the same as those inFig. 2.
factor S(q, t) with J zz = 0 and H = 0 at (a) t = 40 and (b) t = 400. (c) The SkL order parameter M(t) = S(Q1, t) + S(Q2, t) and the correlation length ξ Qη versus time. Both exhibit a power-law growth t α with exponent αM ≈ 1.046 and α ξ ≈ 0.476, respectively. (d) Scaling plot of the structure factor around the peak at Q1 along the qx axis.
FIG. 5 .
5FIG. 5. Real-time dependence of the density of dislocations defects ρ d (t) with J zz = 0 for varying magnetic field H, obtained by averaging over 32 independent runs. The positions of defects are detected from the energy density [38]. The pale lines represent the fitting by the formula ρ d (t)/(log(ρ0/ρ d (t)) − 1) = a(t − t0) −1 .
and Y. Motome for fruitful discussions. This work was supported by JSPS KAK-ENHI Grant Number No. JP21J20812. K.S. was supported by the Program for Leading Graduate Schools (MERIT-WINGS). Parts of the numerical calculations were performed in the supercomputing systems in ISSP, the University of Tokyo. G.W.C was partially supported by the US Department of Energy Basic Energy Sciences under Contract No. DE-SC0020330.
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| {'fraction_non_alphanumeric': 0.061626844235539886, 'fraction_numerical': 0.04201592897245071, 'mean_word_length': 4.319627726073065, 'pattern_counts': {'":': 0, '<': 1, '<?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': 'We investigate the phase ordering kinetics of skyrmion lattice (SkL) in a metallic magnet. The SkL can be viewed as a superposition of magnetic stripes whose periods are determined by the quasi-nesting wave vectors of the underlying Fermi surface. An effective magnetic Hamiltonian that describes the electron-mediated spin-spin interaction is obtained for a two-dimensional s-d model with the Rashba spin-orbit coupling. Large-scale Landau-Lifshitz-Gilbert dynamics simulations based on the effective spin Hamiltonian reveal a two-stage phase ordering of the SkL phase after a thermal quench. The initial fast crystallization of skyrmions is followed by a slow relaxation dominated by the annihilation dynamics of dislocations, which are topological defects of the constituent magnetic stripe orders. The late-stage phase ordering also exhibits a dynamical scaling symmetry. We further show that the annihilation of dislocations follows a power-law time dependence with a logarithmic correction that depends on magnetic fields. Implications of our results for SkL phases in magnetic materials are also discussed.Complex magnetic textures such as vortices and skyrmions are not only of great fundamental interest in magnetism but also have important implications in the emerging technology of spintronics[1][2][3][4]. Both vortices and skyrmions are nano-sized particle-like spintextures characterized by nontrivial topological invariants. The presence of such complex patterns in metallic magnets could give rise to intriguing electronic and transport properties due to a nontrivial Berry phase acquired by electrons when traversing over closed loops of noncollinear or noncoplanar spins[1,[5][6][7]. The well-studied topological Hall effects [8-13] and topological Nernst effects[14][15][16]are some of the representative examples. Also importantly, such topological electronic responses in metallic magnets can be controlled via the manipulation of magnetic textures.In magnetic materials, skyrmions are often stabilized in the form of a skyrmion lattice (SkL), which is a periodic array of such particle-like topological spin-textures. Indeed, SkL as a spontaneous ground state was already predicted in the pioneering work of Bogdanov and Yablonskii that later triggered the enormous interest in magnetic skyrmions[17,18]. SkLs have since been reported in several chiral magnets such as MnSi [19] and other B20 compounds, as well as centrosymmetric materials[20]. While the picture of SkL as an array of particle-like objects offers an intuitive framework to understand certain structural and dynamical aspects of skyrmion phases[21], recent studies have revealed the close connection between SkL and multiple-Q magnetic ordering driven by partial nesting of the electron Fermi surface[22][23][24]. Indeed, this mechanism has been conjectured to be ubiquitous for itinerant spin systems with a wide range of filling fractions.Despite the huge interest and extensive research on magnetic SkL over the past decades, the phase-ordering dynamics of SkLs remains an open subject. Specifically, here one concerns the dynamical evolution and potential universal behaviors of skyrmion crystallization when a magnet is quenched into a skyrmion phase. It is known that the kinetics of phase ordering depends crucially on topological defects of the symmetry-breaking phase. Several super-universality classes of domain-growth laws have been established over the years[25,26]. The fact that both magnetic and translational symmetries are broken in a skyrmion crystal indicates rich relaxational dynamics of SkL phases which has yet to be systematically investigated. Understanding the phase ordering of SkL is also crucial to the engineering and control of skyrmion phases in real materials.In this paper, we make an important step toward this goal by investigating the crystallization dynamics of a SkL in a realistic model of chiral metallic magnets. A minimum microscopic model for such itinerant spin systems is the s-d Hamiltonian H sd = k,σ ϵ k,σ c † k,σ c k,σ − J sd i S i · c † iσ σ σ,σ ′ c iσ ′ , where the first term describes electron hopping on a lattice with t ij the transfer integrals and the second term represents a local coupling between the itinerant s-electron and the local moments S i of d-electrons; J sd represents the coupling strength. Spin-orbit coupling (SOC) within such single-band models can be described by either Rashba or Dresselhaus hopping terms. Dynamical simulations based on such electron models requires solving a disordered electron tightbinding Hamiltonian at every time-step, which could be prohibitively expensive for large systems. Instead, here we consider an effective spin Hamiltonian, similar in spirit to the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction[27][28][29], with SOC properly included[30]. At the leading second-order perturbation, the effective Hamiltonian has the general form(1) arXiv:2305.16182v1 [cond-mat.str-el]', 'arxivid': '2305.16182', 'author': ['Kotaro Shimizu \nDepartment of Applied Physics\nThe University of Tokyo\n113-8656TokyoJapan\n\nDepartment of Physics\nUniversity of Virginia\n22904CharlottesvilleVirginiaUSA\n', 'Gia-Wei Chern \nDepartment of Physics\nUniversity of Virginia\n22904CharlottesvilleVirginiaUSA\n'], 'authoraffiliation': ['Department of Applied Physics\nThe University of Tokyo\n113-8656TokyoJapan', 'Department of Physics\nUniversity of Virginia\n22904CharlottesvilleVirginiaUSA', 'Department of Physics\nUniversity of Virginia\n22904CharlottesvilleVirginiaUSA'], 'corpusid': 258888105, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12583, 'n_tokens_neox': 10511, 'n_words': 5977, 'pdfsha': '5d3888e7dff5d7ed8937206c3f011bfe07f93715', 'pdfurls': ['https://export.arxiv.org/pdf/2305.16182v1.pdf'], 'title': ['Crystallization dynamics of magnetic skyrmions in a frustrated itinerant magnet', 'Crystallization dynamics of magnetic skyrmions in a frustrated itinerant magnet'], 'venue': []} |
arxiv |
END-TO-END NEURAL SPEAKER DIARIZATION WITH SELF-ATTENTION
Yusuke Fujita
Hitachi, Ltd. Research & Development Group
Japan
Center for Language and Speech Processing
Johns Hopkins University
USA
Naoyuki Kanda
Hitachi, Ltd. Research & Development Group
Japan
Shota Horiguchi
Hitachi, Ltd. Research & Development Group
Japan
Yawen Xue
Hitachi, Ltd. Research & Development Group
Japan
Kenji Nagamatsu
Hitachi, Ltd. Research & Development Group
Japan
Shinji Watanabe
Center for Language and Speech Processing
Johns Hopkins University
USA
END-TO-END NEURAL SPEAKER DIARIZATION WITH SELF-ATTENTION
Index Terms-speaker diarizationneural networkend-to-endself-attention
Speaker diarization has been mainly developed based on the clustering of speaker embeddings. However, the clustering-based approach has two major problems; i.e., (i) it is not optimized to minimize diarization errors directly, and (ii) it cannot handle speaker overlaps correctly. To solve these problems, the End-to-End Neural Diarization (EEND), in which a bidirectional long short-term memory (BLSTM) network directly outputs speaker diarization results given a multi-talker recording, was recently proposed. In this study, we enhance EEND by introducing self-attention blocks instead of BLSTM blocks. In contrast to BLSTM, which is conditioned only on its previous and next hidden states, self-attention is directly conditioned on all the other frames, making it much suitable for dealing with the speaker diarization problem. We evaluated our proposed method on simulated mixtures, real telephone calls, and real dialogue recordings. The experimental results revealed that the self-attention was the key to achieving good performance and that our proposed method performed significantly better than the conventional BLSTM-based method. Our method was even better than that of the state-of-the-art x-vector clustering-based method. Finally, by visualizing the latent representation, we show that the self-attention can capture global speaker characteristics in addition to local speech activity dynamics. Our source code is available online at https://github.com/hitachi-speech/EEND. Index Termsspeaker diarization, neural network, end-to-end, self-attention arXiv:1909.06247v1 [eess.AS] 13 Sep 2019 SAD MFCC X-vector extraction PLDA scoring AHC SAD neural network X-vector neural network Same/Diff covariance matrices Diarization result (a) X-vector clustering-based method Log-Mel Joint speech activity detection of all speakers EEND neural network
INTRODUCTION
Speaker diarization is the process of partitioning an audio recording into homogeneous segments according to the speaker's identity. The speaker diarization has a wide range of applications, such as information retrieval from broadcast news, generating minutes of meetings, and a turn-taking analysis of telephone conversations [1,2]. It also helps automatic speech recognition performance in multispeaker conversation scenarios in meetings (ICSI [3,4], AMI [5,6]) and home environments (CHiME-5 [6][7][8][9][10]).
Typical speaker diarization systems are based on the clustering of speaker embeddings [11][12][13][14][15][16][17][18]. For instance, i-vectors [12,13,17,19], d-vectors [18,20], and x-vectors [16,21] are commonly used in speaker diarization tasks. These embeddings of short segments are partitioned into speaker clusters by using clustering algorithms, such as Gaussian mixture models [11,12], agglomerative hierarchical clustering [11,13,16,17], mean shift clustering [14], k-means clustering [15,18], Links [18,22], and spectral clustering [18]. These clustering-based diarization methods have shown themselves to be The first author performed the work while at Center for Language and Speech Processing, Johns Hopkins University as a Visiting Scholar. effective on various datasets (see the DIHARD Challenge 2018 activities, e.g., [23][24][25]).
However, such clustering-based methods have a number of problems. First, they cannot be optimized to minimize diarization errors directly, because the clustering procedure is a type of unsupervised learning methods. Second, they have trouble handling speaker overlaps, since the clustering algorithms implicitly assume one speaker per segment. Furthermore, they have trouble adapting their speaker embedding models to real audio recordings with speaker overlaps, because the speaker embedding model has to be optimized with single-speaker non-overlapping segments. These problems hinder the speaker diarization application from working on real audio recordings that usually contain overlapping segments.
To solve these problems, we propose Self-Attentitive End-to-End Neural Diarization (SA-EEND). Different from most of the other methods, our proposed method does not rely on clustering. Instead, a self-attention-based neural network directly outputs the joint speech activities of all speakers for each time frame, given an input of a multi-speaker audio recording. Our method can naturally handle speaker overlaps during the training and inference time by exploiting a multi-label classification framework. The neural network is trained in an end-to-end fashion using a recently proposed permutation-free objective function that provides minimal diarization errors [26].
This paper shows that our method achieves a significant performance improvement over end-to-end neural diarization (EEND) [26], for which promising but preliminary results were reported with a bidirectional long short-term memory (BLSTM) [27]. In particular, it shows that the self-attention mechanism [28,29] is the key to achieving good speaker-diarization performance in this paper. We demonstrate that the self-attention mechanism gives significantly better results for multiple datasets compared with the BLSTMbased method [26] and the state-of-the-art x-vector-based speaker diarization method. In contrast to BLSTM, which is conditioned only on its previous and next hidden states, the self-attention layer is conditioned on all the other input frames by computing the pairwise similarity between all frame pairs. We believe that this mechanism is the key to speaker diarization since it can capture global speaker characteristics in addition to local speech activity dynamics. By visualizing the learned representation, we show that some selfattention heads capture speaker-dependent global characteristics, while the remaining heads represent temporal features.
RELATED WORK
Clustering-based methods
The x-vector clustering-based system is commonly used for speaker diarization [23,24,30]. A diagram of the system is depicted in Fig. 1(a). To build the system, one has to prepare three independent models: (i) a speech activity detection (SAD) neural network, (ii) x-vector extraction neural network, and (iii) PLDA model including the same/different speaker covariance matrices. None of these models can be trained to directly minimize the diarization errors. Joint modeling methods have been studied in an effort to alleviate the complex preparation process and take into account the dependencies between these models. They include, for example, joint modeling of x-vector extraction and PLDA scoring [16,31] and joint modeling of SAD and speaker embedding [32]. However, the clustering process has remained unchanged because it is an unsupervised process.
In contrast to these methods, the EEND method uses only one neural network model, as depicted in Fig. 1(b). This method does not rely on clustering, and the model can be directly optimized with the reference diarization results of the training data.
This neural-network-based end-to-end approach, in which only one neural network model directly computes the final outputs, has been successfully applied in a variety of tasks, including neural machine translation [33,34], automatic speech recognition [35][36][37], and text-to-speech [38,39].
Direct optimization minimizing diarization errors
A fully supervised diarization method has been proposed for optimization based on a diarization error minimization objective [40]. This is the first successful approach that does not cluster speaker embeddings. The method formulates the speaker diarization problem on the basis of a factored probabilistic model, which consists of modules for determining speaker changes, speaker assignments, and feature generation. These models are jointly trained using input features and corresponding speaker labels. However, the SAD model and their speaker embedding (d-vector) model have to be trained separately in their method. Moreover, their speaker-change model assumes one speaker for each segment, which hinders its application to speaker-overlapping speech.
In contrast to their method, the EEND method uses an end-toend neural network that accepts audio features as input and outputs the joint speech activities of multiple speakers. The network is optimized using the entire recording, including non-speech and speaker overlaps, with a diarization-error-oriented objective. This end-toend model was first introduced in [26]; this paper describes an extension of the model that includes a self-attention mechanism.
Self-attention mechanism
The self-attention mechanism was originally proposed for extracting sentence embeddings for text processing [28]. Recently, the self-attention mechanism has shown superior performance in a variety of tasks, including machine translation [29], video classification [41], and image segmentation [42]. For audio processing, a self-attention mechanism has been incorporated in acoustic modeling for ASR [43,44], sound event detection [45], and speaker recognition [46]. For speaker diarization, the self-attention mechanism has been applied to the speaker embedding extraction model [25] and the scoring model [31] of clustering-based methods. This study describes a self-attention mechanism for clustering-free speaker diarization.
PROPOSED METHOD: SELF-ATTENTIVE
END-TO-END NEURAL DIARIZATION 3.1. End-to-end neural diarization: review
Here, we describe the EEND method proposed in [26]. The speaker diarization task can be formulated as a multi-label classification problem, as follows. Given a T -length observation sequence X = (xt ∈ R F | t = 1, · · · , T ) from an audio signal, speaker diaization problem tries to estimate the corresponding speaker label sequence Y = (yt | t = 1, · · · , T ). Here, xt is a F -dimensional observation feature vector at time index t. Speaker label yt = [yt,c ∈ {0, 1} | c = 1, · · · , C] denotes a joint activity for multiple (C) speakers at time index t. For example, yt,c = 1 and y t,c = 1 (c = c ) represent an overlap situation in which speakers c and c are both present at time index t. Thus, determining Y is a sufficient condition to determine the speaker diarization information.
The most probable speaker label sequenceŶ is selected from among all possible speaker label sequences Y, as follows:
Y = arg max Y ∈Y P (Y |X).
(1) P (Y |X) can be factorized using the conditional independence assumption as follows:
P (Y |X) = t P (yt|y1, · · · yt−1, X),(2)≈ t P (yt|X) ≈ t c P (yt,c|X).(3)
Here, we assume that the frame-wise posterior is conditioned on all inputs, and each speaker is present independently. The frame-wise posterior P (yt,c|X) can be estimated using a neural-network-based model.
Self-attention-based neural network
In [26], a BLSTM based neural network was used for estimating the frame-wise posteriors P (yt,c|X). In this paper, we propose selfattentive end-to-end neural diarization (SA-EEND), which uses selfattention-based encoding blocks instead of BLSTMs, as depicted in Fig. 2. The input features are transformed as follows:
e (0) t = W0xt + b0 ∈ R D ,(4)e (p) t = Encoder (p) t (e (p−1) 1 , · · · , e (p−1) T ) (1 ≤ p ≤ P ). (5)
Here, W0 ∈ R D×F and b0 ∈ R D project an input feature into Ddimensional vector. Encoder (p) t (·) is the p-th encoder block which accepts an input sequence of D-dimensional vectors and outputs a D-dimensional vector e (p) t at time index t. We use P encoder blocks followed by the output layer for frame-wise posteriors.
The architecture of the encoder block is depicted in Fig. 2. This configuration of the encoder block is almost the same as the one in the Speech-Transformer introduced in [44], but without positional encoding. The encoder block has two sub-layers. The first is a multi-head self-attention layer, and the second is a position-wise feed-forward layer.
Multi-head self-attention layer
The multi-head self-attention layer transforms a sequence of input vectors as follows. The sequence of vectors (e (p−1) t |t = 1, · · · , T ) is converted into a R T ×D matrix, followed by layer normalization [47]:Ē
(p−1) = LayerNorm([e (p−1) 1 · · · e (p−1) T ] ) ∈ R T ×D .(6)
Then, for each head, a pairwise similarity matrix A
(p) h is computed using the dot products of query vectorsĒ (p−1) Q (p) h ∈ R T ×d and key vectorsĒ (p−1) K (p) h ∈ R T ×d : A (p) h =Ē (p−1) Q (p) h (Ē (p−1) K (p) h ) ∈ R T ×T (1 ≤ h ≤ H), (7) where, Q (p) h , K (p) h ∈ RA (p) h = Softmax A (p) h √ d ∈ R T ×T .(8)
Then, using the attention weight matrix, context vectors C
(p) h are computed as a weighted sum of the value vectorsĒ (p−1) V (p) h ∈ R T ×d : C (p) h =Â (p) h (Ē (p−1) V (p) h ) ∈ R T ×d ,(9)
where V h ∈ R D×d is the value projection matrix. Finally, the context vectors for all heads are concatenated and projected using the output projection matrix O (p) ∈ R D×D :
E (p,SA) = [C (p) 1 · · · C (p) H ]O (p) ∈ R T ×D .(10)
Following the self-attention layer, a residual connection and layer normalization is applied:
E (p,SA) = LayerNorm(Ē (p−1) + E (p,SA) ) ∈ R T ×D .(11)
Position-wise feed-forward layer
The position-wise feed-forward layer transformsĒ (p,SA) as follows:
E (p,FF) = ReLU(Ē (p,SA) W (p) 1 + b (p) 1 1)W (p) 2 + b (p) 2 1 ∈ R T ×D ,(12)where W (p) 1 ∈ R D×d ff and b (p) 1
∈ R d ff are the first linear projection matrix and bias, respectively, 1 ∈ R 1×T is an all-one row vector, and ReLU(·) is the rectified linear unit activation function.
E (p,SA) E (p,FF) E (p,SA) Fig. 2. Two-speaker SA-EEND model trained with permutation-free loss. d ff is the number of internal units in this layer. W (p) 2 ∈ R d ff ×D and b (p) 2
∈ R D are the second linear projection matrix and bias, respectively.
Finally, the output of the encoder block e (p) t for each time frame is computed by applying a residual connection as follows:
[e (p) 1 · · · e (p) T ] = (Ē (p,SA) + E (p,FF) )(13)
Output layer for frame-wise posteriors
The frame-wise posteriors zt are calculated from e (P ) t (in Eq. 5) using layer normalization and a fully-connected layer as follows:
E (P ) = LayerNorm([e (P ) 1 · · · e (P ) T ] ) ∈ R T ×D ,(14)
[z1 · · · zT ] = σ(Ē (P ) W3 + b31) ,
where W3 ∈ R D×C and b3 ∈ R C are the linear projection matrix and bias, respectively, and σ(·) is the element-wise sigmoid function.
Permutation-free training
The difficulty of training the model described above is that the model must deal with speaker permutations: changing the order of speakers within a correct label sequence is also regarded as correct. An example of permutations in a two-speaker case is shown in Fig. 2. In this paper, we call this problem "label ambiguity." This label ambiguity obstructs the training of the neural network when we use a standard binary cross entropy loss function.
To cope with the label ambiguity problem, the permutationfree training scheme considers all the permutations of the reference speaker labels. The permutation-free training scheme has been used in research on source separation [48][49][50]. Here, we apply the permutation-free loss function to a temporal sequence of speaker labels. The neural network is trained to minimize the permutationfree loss between the output zt predicted in Eq. 15 and the reference speaker label lt, as follows: where perm(C) is the set of all the possible permutations of (1, . . . , C), and l φ t is the φ-th permutation of the reference speaker label, and BCE(·, ·) is the binary cross entropy function between the label and the output.
J PF = 1 T C min φ∈perm(C) t BCE(l φ t , zt),(16)
EXPERIMENTAL SETUP
Data
To verify the effectiveness of the SA-EEND method for various overlap situations, we prepared two training sets and five test sets, including simulated and real datasets. The statistics of the training and test sets are listed in Table 1. The overlap ratio is computed as the ratio of the audio time during which two or more speakers are active, to the audio time during which one or more speakers are active.
Note that training data for the EEND method is different from those for the x-vector clustering-based method. Whereas the xvector clustering-based method uses single-speaker segments for training their x-vector neural network, the EEND method uses audio mixtures of multiple speakers. Such mixtures can be simulated infinitely with a combination of single-speaker segments. Moreover, the EEND model can be trained with not only simulated mixtures but real audio mixtures with speaker overlaps.
Simulated mixtures
Each mixture was simulated by Algorithm 1. Unlike the mixture simulation of source separation studies [48], we consider a diarization-style mixture: each speech mixture should have dozens of utterances per speaker with reasonable silence intervals between utterances. The silence intervals are controlled by the average interval of β. Larger values of β generate speech with less overlap.
The set of utterances used for the simulation was comprised of the Switchboard-2 (Phase I, II, III), Switchboard Cellular (Part 1, Part2), and NIST Speaker Recognition Evaluation datasets (2004,2005,2006,2008). All recordings are telephone speech sampled at 8 kHz. There are 6,381 speakers in total. We split them into 5,743 speakers for the training set and 638 speakers for the test set. Note that the set of utterances for the training set is identical to that of the Kaldi CALLHOME diarization v2 recipe [53] 1 , making it fair comparison with the x-vector clustering-based method.
Since there are no time annotations in these corpora, we extracted utterances using speech activity detection (SAD) on the basis The set of background noises was from the MUSAN corpus [54]. We used 37 recordings that are annotated as "background" noises. The set of 10,000 room impulse responses (RIRs) was from the Simulated Room Impulse Response Database used in [55]. The SNR values were sampled from 10, 15, and 20 dBs. These sets of non-speech corpora are also used for training the x-vector and SAD models in the x-vector clustering-based method.
We generated two-speaker mixtures for each speaker with 10-20 utterances (Nspk = 2, Numin = 10, Numax = 20). For the simulated training set, 100,000 mixtures were generated with β = 2. For the simulated test set, 500 mixtures were generated with β = 2, 3, and 5. The overlap ratios of the simulated mixtures are ranging from 19.5 to 34.4%.
Real datasets
We used real telephone speech recordings as the real training set. A set of 26,172 two-speaker recordings were extracted from the recordings of the Switchboard-2 (Phase I, II, III), Switchboard Cellular (Part 1, Part 2), and NIST Speaker Recognition Evaluation datasets. The overlap ratio of the training data was 3.7%, far less than that of the simulated mixtures.
We evaluated the proposed method on real telephone conversations in the CALLHOME dataset [51]. We randomly split the twospeaker recordings from the CALLHOME dataset into two subsets: an adaptation set of 155 recordings and a test set of 148 recordings. The average overlap ratio of the test set was 13.0%.
In addition, we conducted an evaluation on the dialogue part of the Corpus of Spontaneous Japanese (CSJ) [52]. The CSJ con-tains 54 two-speaker dialogue recordings 3 . They were recorded using headset microphones in separate soundproof rooms. The average overlap ratio of the CSJ test set was 20.1%, larger than the CALL-HOME test set.
Model configuration
Clustering-based systems
We compared the proposed method with two conventional clusteringbased systems [23]: the i-vector system and x-vector system were created using the Kaldi CALLHOME diarization v1 and v2 recipes.
These recipes use agglomerative hierarchical clustering (AHC) with the probabilistic linear discriminant analysis (PLDA) scoring scheme. The number of clusters was fixed to 2. Though the original recipes use oracle speech/non-speech marks, we used the SAD model with the same configuration as described in Sec. 4.1.
BLSTM-based EEND system
We configured a BLSTM-based EEND method (BLSTM-EEND), as described in [26]. The input features were 23-dimensional log-Melfilterbanks with a 25-ms frame length and 10-ms frame shift. Each feature was concatenated with those from the previous seven frames and subsequent seven frames. To deal with a long audio sequence in our neural networks, we subsampled the concatenated features by a factor of ten. Consequently, a (23 × 15)-dimensional input feature was fed into the neural network every 100 ms.
We used a five-layer BLSTM with 256 hidden units in each layer. The second layer of the BLSTM outputs was used to form a 256-dimensional embedding; we then calculated the deep clustering loss in this embedding to discriminate different speakers. We used the Adam [56] optimizer with a learning rate of 10 −3 . The batch size was 10. The number of training epochs was 20.
Because the output of the neural network is the probability of speech activity for each speaker, a threshold is required to obtain the decision of speech activity for each frame. We set the threshold to 0.5. Furthermore, we applied 11-frame median filtering to prevent production of unreasonably short segments.
For domain adaptation, the neural network was retrained using the CALLHOME adaptation set. we used the Adam optimizer with a learning rate of 10 −6 and ran 5 epochs. For the postprocessing, we adjusted the threshold to 0.6 so that the DER of the adaptation set has the minimum value.
Self-attentive EEND system
Here, we used the same input features as were input to the BLSTM-EEND system. Note that the sequence length at the training stage was limited to 500 (50 seconds in audio time) because our system uses more memory than the BLSTM-based network does. Therefore, we split the input audio recordings into non-overlapping 50-second segments. At the inference stage, we used the entire sequence for each recording.
We used two encoder blocks with 256 attention units containing four heads (P = 2, D = 256, H = 4). We used 1024 internal units in a position-wise feed-forward layer (d ff = 1024). We used the Adam optimizer with the learning rate scheduler introduced in [29]. The number of warm-up steps used in the learning rate scheduler was 25,000. The batch size was 64. The number of training epochs was 100. After 100 epochs, we used an averaged model obtained by averaging the model parameters of the last 10 epochs. As with the BLSTM-EEND system, we applied 11-frame median filtering. For domain adaptation, the averaged model was retrained using the CALLHOME adaptation set. We used the Adam optimizer with a learning rate of 10 −5 and ran 100 epochs. After 100 epochs, we used an averaged model obtained by averaging the model parameters of the last 10 epochs.
Performance metric
We evaluated the systems with the diarization error rate (DER) [57]. Note that the DERs reported in many prior studies did not include misses or false alarm errors due to their using oracle speech/nonspeech labels. Overlapping speech segments had also been excluded from the evaluation. For our DER computation, we evaluated all of the errors, including overlapping speech segments, because the proposed method includes both the speech activity detection and overlapping speech detection functionality. As is done typically, we used a collar tolerance of 250 ms at the start and end of each segment.
RESULTS
Evaluation on simulated mixtures
DERs on various test sets are shown in Table 2. The clustering-based systems performed poorly on heavily overlapped simulated mixtures. This result is within our expectations, because the clusteringbased systems did not consider speaker overlaps; there are more misses when the overlap ratio is high.
The BLSTM-EEND system trained with the simulated training set showed a significant DER reduction compared with the clustering-based systems on the simulated mixtures. Among the differing overlap ratios, it showed the best performance on the highest Fig. 3. Attention weight matrices at the second encoder block. The input was the CALLHOME test set (recording id: iagk). The model was trained with the real training set followed by domain adaptation. The top two rows show the reference speech activity of two speakers. overlap ratio condition (β = 2). The BLSTM-EEND system worked well on the overlapping condition matched with training data.
The proposed system, SA-EEND, trained with the simulated training set had significantly fewer DERs compared with the BLSTM-EEND system on every test set. As well as the BLSTM-EEND system, it showed the best performance on the highest overlap ratio condition (β = 2). However, the DER degradation on the less overlapping conditions was smaller than that of the BLSTM-EEND system, which indicated that the self-attention blocks improved robustness to variable overlapping conditions.
Evaluation on real test sets
In contrast to the good performance on the simulated mixtures, the BLSTM-EEND system had inferior DERs to those of the clusteringbased systems evaluated on the real test sets. Although the BLSTM-EEND system showed performance improvements when the training data were switched from simulated to real data, its DERs were still higher than those of the clustering-based systems.
The proposed system, SA-EEND, trained with the simulated training set showed remarkable improvements on real datasets of the CALLHOME and CSJ, which indicates the strong generalization capability of the self-attention blocks. For the CSJ, even without domain adaptation, the proposed system performed better than the x-vector clustering-based method.
The SA-EEND system trained with the real training set performed the best on the real test sets, however, it had poor DERs on the simulated mixtures. We expected that the result was due to the small number of mixtures and low overlap ratio of the real training set. It would be much improved by feeding more real data with more speaker overlaps, or by combining with simulated training data.
Effect of domain adaptation
The EEND models trained with simulated training set were overfitted to the specific overlap ratio of the training set. We expected that the overfitting would be mitigated by using domain adaptation. DERs on the CALLHOME with and without domain adaptation are shown in Table 3. As expected, the domain adaptation significantly reduced the DER; our system thus achieved even better results than those of the x-vector-based system.
A detailed DER comparison on the CALLHOME test set is shown in Table 4. The clustering-based systems had few SAD errors thanks to the robust SAD model trained with various noiseaugmented data. However, there were numerous misses and confusion errors due to its lack of handling speaker overlaps. Compared with clustering-based systems, the proposed method produced significantly fewer confusion and miss errors. The domain adaptation reduced all error types except confusion errors.
Visualization of self-attention
To analyze the behavior of the self-attention mechanism in our diarization system, Fig. 3 visualizes the attention weight matrix at the second encoder block, corresponding to (p=2) h in Eq. 8. Here, head 1 and head 2 have vertical lines at different positions. The vertical lines correspond to each speaker's activity. The attention weight matrix with these vertical lines transformed the input features into the weighted mean of the same speaker frames. These heads actually captured the global speaker characteristics by computing the similarity between distant frames. Interestingly, heads 3 and 4 look like identity matrices, which results in position-independent linear transforms. These heads are considered to work for speech/non-speech detection. We conclude that the multi-head self-attention mechanism captures global speaker characteristics in addition to local speech activity dynamics, which leads to a reduction in DER. Experiments on various combinations of the number of heads and the number of speakers would be an interesting future work.
CONCLUSION
We incorporated a self-attention mechanism in the end-to-end neural diarization model. We evaluated our model on simulated mixtures and two real datasets. Experimental results showed that the self-attention mechanism significantly reduced DERs and showed higher generalization quality compared with a BLSTM-based neural diarization system. The self-attention based systems even outperformed x-vector clustering-based systems. We also showed that the self-attention blocks actually captured global speaker characteristics by visualizing the latent representation.
Fig. 1 .
1System diagrams for speaker diarization
1Sample
Sample a set of Nspk speakers S from S Nu from {Numin, . . xs ⊕ 0 (δ) ⊕ Us [u] * i 10 X .add (xs) 11 Lmax = maxx∈X |x| 12 y ← x∈X x ⊕ 0 (Lmax−|x|) 13 Sample n from N // Background noise 14 Sample r from R // SNR 15 Determine a mixing scale p from r, y, and n 16 n ← repeat n until reach the length of y 17 y ← y + p · n of time-delay neural networks and statistics pooling 2 .
D×d are query and key projection matrices for the h-th head, respectively. d = D/H is a dimension of each head, and H is the number of heads. The pairwise similarity matrix A (p) h is scaled by 1/ √ d and a softmax function is applied to form the attention weight matrixÂ(p)
h :
Table 1 .
1Statistics of training and test sets.# mixtures avg. duration
overlap
(sec) ratio (%)
Traning sets
Simulated (β = 2)
100,000
87.6
34.4
Real (SWBD+SRE)
26,172
304.7
3.7
Test sets
Simulated (β = 2)
500
87.3
34.4
Simulated (β = 3)
500
103.8
27.2
Simulated (β = 5)
500
137.1
19.5
CALLHOME [51]
148
72.1
13.0
CSJ [52]
54
766.3
20.1
Table 2 .
2DERs (%) on various test sets. For EEND systems, the CALLHOME (CH) results are obtained with domain adaptation.Simulated
Real
β = 2 β = 3 β = 5
CH
CSJ
Clustering-based
i-vector
33.74
30.93
25.96 12.10 27.99
x-vector
28.77
24.46
19.78 11.53 22.96
BLSTM-EEND
trained with sim. 12.28
14.36
19.69 26.03 39.33
trained with real
36.23
37.78
40.34 23.07 25.37
SA-EEND
trained with sim.
7.91
8.51
9.51
13.66 22.31
trained with real
32.72
33.84
36.78 10.76 20.50
Table 3. DERs (%) on the CALLHOME with and without domain
adaptation.
w/o adaptation with adaptatation
x-vector clustering
11.53
N/A
BLSTM-EEND
trained with sim.
43.84
26.03
trained with real
31.01
23.07
SA-EEND
trained with sim.
17.42
13.66
trained with real
12.66
10.76
Table 4 .
4Detailed DERs (%) evaluated on the CALLHOME. DER is composed of Misses (MI), False alarms (FA), and Confusion errors (CF). The SAD errors are composed of Misses (MI) and False alarms (FA) errors.DER breakdown
SAD errors
Method
DER
MI
FA
CF
MI
FA
i-vector
12.10 7.74 0.54 3.82 1.4
0.5
x-vector
11.53 7.74 0.54 3.25 1.4
0.5
SA-EEND
no-adapt 12.66 7.42 3.93 1.31 3.3
0.6
adapted
10.76 6.68 2.40 1.68 2.3
0.5
https://github.com/kaldi-asr/kaldi/tree/master/ egs/callhome_diarization
The SAD model: http://kaldi-asr.org/models/m4
We excluded four out of 58 recordings that contain speakers in the official speech recognition evaluation sets.
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| {'fraction_non_alphanumeric': 0.06988881778915373, 'fraction_numerical': 0.03129499280115182, 'mean_word_length': 4.211442267611504, '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': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Speaker diarization has been mainly developed based on the clustering of speaker embeddings. However, the clustering-based approach has two major problems; i.e., (i) it is not optimized to minimize diarization errors directly, and (ii) it cannot handle speaker overlaps correctly. To solve these problems, the End-to-End Neural Diarization (EEND), in which a bidirectional long short-term memory (BLSTM) network directly outputs speaker diarization results given a multi-talker recording, was recently proposed. In this study, we enhance EEND by introducing self-attention blocks instead of BLSTM blocks. In contrast to BLSTM, which is conditioned only on its previous and next hidden states, self-attention is directly conditioned on all the other frames, making it much suitable for dealing with the speaker diarization problem. We evaluated our proposed method on simulated mixtures, real telephone calls, and real dialogue recordings. The experimental results revealed that the self-attention was the key to achieving good performance and that our proposed method performed significantly better than the conventional BLSTM-based method. Our method was even better than that of the state-of-the-art x-vector clustering-based method. Finally, by visualizing the latent representation, we show that the self-attention can capture global speaker characteristics in addition to local speech activity dynamics. Our source code is available online at https://github.com/hitachi-speech/EEND. Index Termsspeaker diarization, neural network, end-to-end, self-attention arXiv:1909.06247v1 [eess.AS] 13 Sep 2019 SAD MFCC X-vector extraction PLDA scoring AHC SAD neural network X-vector neural network Same/Diff covariance matrices Diarization result (a) X-vector clustering-based method Log-Mel Joint speech activity detection of all speakers EEND neural network', 'arxivid': '1909.06247', 'author': ['Yusuke Fujita \nHitachi, Ltd. Research & Development Group\nJapan\n\nCenter for Language and Speech Processing\nJohns Hopkins University\nUSA\n', 'Naoyuki Kanda \nHitachi, Ltd. Research & Development Group\nJapan\n', 'Shota Horiguchi \nHitachi, Ltd. Research & Development Group\nJapan\n', 'Yawen Xue \nHitachi, Ltd. Research & Development Group\nJapan\n', 'Kenji Nagamatsu \nHitachi, Ltd. Research & Development Group\nJapan\n', 'Shinji Watanabe \nCenter for Language and Speech Processing\nJohns Hopkins University\nUSA\n'], 'authoraffiliation': ['Hitachi, Ltd. Research & Development Group\nJapan', 'Center for Language and Speech Processing\nJohns Hopkins University\nUSA', 'Hitachi, Ltd. Research & Development Group\nJapan', 'Hitachi, Ltd. Research & Development Group\nJapan', 'Hitachi, Ltd. Research & Development Group\nJapan', 'Hitachi, Ltd. Research & Development Group\nJapan', 'Center for Language and Speech Processing\nJohns Hopkins University\nUSA'], 'corpusid': 202572979, 'doi': '10.1109/asru46091.2019.9003959', 'github_urls': ['https://github.com/hitachi-speech/EEND.', 'https://github.com/kaldi-asr/kaldi/tree/master/'], 'n_tokens_mistral': 16204, 'n_tokens_neox': 14169, 'n_words': 7647, 'pdfsha': 'c340b89e7b7fa84fac85cdcf38ba7007e2e71930', 'pdfurls': ['https://arxiv.org/pdf/1909.06247v1.pdf'], 'title': ['END-TO-END NEURAL SPEAKER DIARIZATION WITH SELF-ATTENTION', 'END-TO-END NEURAL SPEAKER DIARIZATION WITH SELF-ATTENTION'], 'venue': []} |
arxiv |
Time variability of ultra fast BAL outflows using SALT: C iv equivalent width analysis
2015
P Aromal
IUCAA
Postbag 4411007Ganeshkind, PuneIndia
†
R Srianand
IUCAA
Postbag 4411007Ganeshkind, PuneIndia
P Petitjean
Institut d'Astrophysique de Paris
Sorbonne Université and CNRS
98bis boulevard Arago75014ParisFrance
Time variability of ultra fast BAL outflows using SALT: C iv equivalent width analysis
MNRAS
0002015Accepted XXX. Received YYY; in original form ZZZPreprint 5 May 2023 Compiled using MNRAS L A T E X style file v3.0galaxies:active -quasars: absorption lines -quasars: general
We study the time variability (over 7.3 yrs) of ultra fast outflows (UFOs) detected in a sample of 64 C iv broad absorption line (BAL) quasars (with 80 distinct BAL components) monitored using the Southern African Large Telescope. By comparing the properties of the quasar in our sample with those of a control sample of non-BAL quasars we show that the distributions of black hole mass are different and the bolometric luminosities and optical photometric variations of UFO BAL quasars are slightly smaller compared to that of non-BAL quasars. The detection fraction of C iv equivalent width (W) variability (∼95%), the fractional variability amplitude ( ∆W W ) and the fraction of "highly variable" BAL (i.e., ∆W W >0.67) components (∼ 33%) are higher in our sample compared to the general BAL population. The scatter in ∆W W and the fraction of "highly variable" BALs increase with the time-scale probed. The ∆W W distribution is asymmetric at large time scales. We attribute this to the BAL strengthening time scales being shorter than the weakening time scales. The BAL variability amplitude correlates strongly with the BAL properties compared to the quasar properties. BALs with low W, high-velocity, shallow profiles, and low-velocity width tend to show more variability. When multiple BAL components are present a correlated variability is seen between low-and high-velocity components with the latter showing a larger amplitude variations. We find an anti-correlation between the fractional variations in the continuum flux and W. While this suggests photoionizationinduced variability, the scatter in continuum flux is much smaller than that of W.
INTRODUCTION
In the spectra of 10%-20% of optically selected QSOs, strong absorption features, so-called Broad Absorption Lines (BAL), of Si iv, C iv and N v (occasionally of Fe ii and Mg ii, Wampler et al. 1995) are observed blueshifted up to 0.2c from the corresponding rest-frame emission lines (Weymann et al. 1991). It has been suggested that the intrinsic BAL fraction, after accounting for dust and other observational biases, could be as high as ∼40% (Dai et al. 2008;Allen et al. 2011). Rare examples of QSO spectra exhibiting broad absorption lines redshifted with respect to the emission lines † E-mail: [email protected] (PA) are also known (e.g., Hall et al. 2013). Based on large velocity widths and blue shifts it is believed that the absorbing gas is related to the central regions of quasars. The BAL fraction among quasars reflects either the covering factor of outflows that prevail in all quasars (e.g., Elvis 2000) or a precise AGN evolutionary phase (e.g., Wang et al. 2013;Chen et al. 2022). Based on the measured properties like blackhole mass (MBH ), Eddington ratio (λ Edd ) etc., hot dust emission, and overall optical-IR spectral energy distribution (SED), it appears that both BAL and non-BAL QSOs are drawn from the same parent population (Reichard et al. 2003;Rankine et al. 2020;Yi et al. 2020;Temple et al. 2021). While this may favor the first possibility, it does not rule out the BAL being part of an evolutionary sequence. It is also widely believed that these outflows could play an important role in the central black hole growth, the host galaxy evolution (Ostriker et al. 2010;Kormendy & Ho 2013), and the chemical enrichment of the intergalactic medium (Wampler et al. 1995;Dunn et al. 2010;Capellupo et al. 2012;Borguet et al. 2013). Very large velocity outflows (vout ∼ 0.1 − 0.2c) are seen both in the form of X-ray (e.g., Ultra fast outflows (UFOs) studied by Tombesi et al. 2010) and UV (e.g., Extremely high-velocity outflows, EHVOs, studied by Rodríguez Hidalgo et al. 2020) absorption. BAL outflows with such large velocities are interesting as to achieve large terminal velocities, the launching power (∝ v 3 out ) must be very large and the wind must be launched very close to the central engine (see for example, Murray et al. 1995). Hence these outflows that carry large momentum can significantly influence the properties of gas in host galaxies and be part of what is known as AGN feedback (Hopkins & Elvis 2010). Also, the origin and evolution of the UFO BALs can be considerably different from their lower velocity counterparts, hence a study which focuses on absorption as a function of its velocity will provide additional clues on their launching and acceleration mechanism. A similar study done by Tombesi et al. (2013) for X-ray outflows concludes that the X-ray UFOs and the comparatively lower velocity warm absorbers could actually represent parts of a single large-scale stratified outflow observed at different locations from the black hole where the X-ray UFOs are likely launched from the inner accretion disc and the WAs at larger distances, such as the outer disc and/or torus.
In particular, the exact mechanism that can accelerate the absorbing gas to such large velocities is still debated. Line-driven radiative acceleration is generally preferred based on the evidence found through observations of line-locking (e.g., Srianand et al. 2002) and the Lyα ghost signature (e.g., Arav 1996). Also, note that Bowler et al. (2014) found line-locking in both BAL and non-BAL populations with similar frequency suggesting line-driving is important in both populations. Given the high luminosity of the quasar, an extended uniform-density absorber will be too highly ionized to produce C iv absorption out to ∼1 kpc (Baskin et al. 2014). Therefore it will be very difficult to accelerate the gas through line-driven acceleration as it is expected to be highly ionized. In the framework of the traditional disk-wind model, Murray et al. (1995) suggested that this over-ionization problem can be avoided by introducing a highly ionized gas component (i.e., the so-called shielding gas) located close to the ionizing source that acts as a screen to prevent the high energy photons from heating the gas producing BAL (see also Proga et al. 2000;Higginbottom et al. 2013;Matthews et al. 2016). Such a scenario can also be useful in understanding the X-ray weakness of BAL quasars (Gibson et al. 2009;Stalin et al. 2011). Alternatively, magnetic driving can also explain high velocity and highly ionized gas (de Kool & Begelman 1995).
The validity of a given outflow model and/or constraints on the parameters of models can be obtained using highresolution spectroscopic observations and/or spectroscopic monitoring with adequate time sampling. While the former can be used to constrain density, covering factor, chemical enrichment, and location of the gas the latter is useful in constraining variability time scales that can be linked to density and or time scales of either the gas ejection event and/or transverse motions. The distance measurement obtained for some of the BALs using fine-structure excitation tend to be very large (see for example, Arav et al. 2018). While large distances make BAL outflows an important contributor to the AGN feedback, it poses a problem to simulations as distance scales are much smaller in simulations. Time dependence, velocity dependence, and correlated velocity scale of absorption line variability can also provide interesting constraints for the disk wind models. Here we mainly focus on the variability of C iv BALs having high ejection velocities.
The existence of BAL variability has been known for over three decades (e.g., Foltz et al. 1987; Barlow et al. 1992) and surveys of increasing number of objects have become common (Lundgren et al. 2007;Gibson et al. 2008Gibson et al. , 2010Filiz Ak et al. 2012;Capellupo et al. 2011Capellupo et al. , 2012Vivek et al. 2014;Filiz Ak et al. 2013;McGraw et al. 2017;Rogerson et al. 2018;Vivek et al. 2018;De Cicco et al. 2018;Aromal et al. 2022). In general, the BAL variability includes extreme optical depth variations like emergence, disappearance, and kinematic shifts. Possible origins of BAL variability are: (i) large variations in the quasar ionizing flux, (ii) changes in the covering factor (fc) of the outflow with respect to the background source, (iii) the transverse motion of the outflow perpendicular to our line of sight. Detailed investigation of the time variability of BAL profiles can yield tight constraints on the lifetime and the location of the outflow, and provides significant insights on the origin and physical mechanisms driving the flow. If the BALs are formed in the vicinity of the launching region, then the timescale for wind material to cross the region of interest is about 1-10 yr, and this is a reasonable characteristic timescale over which flow structures are expected to change (Capellupo et al. 2013). This is also the characteristic timescale for significant angular rotation of the accretion disk at the wind-launching radius.
For the past several years, we have been carrying out a low-resolution spectroscopic monitoring of a sample of 64 quasars that show C iv BALs with outflow velocities (vout) greater than 15000 km s −1 (refer to as UFO BAL in this work) using the Southern African Large Telescope (SALT). Detailed analysis of two of these quasars was presented in our previous papers (Aromal et al. 2021(Aromal et al. , 2022. In this paper, we focus on the statistical analysis of the full sample. In particular, we study the C iv equivalent width variability of UFO BALs over time scales up to ∼7.3 yrs. Our main aim is to quantify, the C iv equivalent width variability, its dependence on time-scale, physical parameters of quasar and the BAL absorption profile, emergence/disappearance and acceleration signatures and their correlation to broad emission lines and continuum variability.
This article is arranged as follows. In Section 2 we present our sample of UFO BALs, methods for identifying BAL regions and their variations, and also construct a control sample of non-BAL quasars matched in r-band magnitude and redshift. Section 3 provides details of spectroscopic and photometric data used in this study and compares certain photometric and spectral properties of UFO BALs to that of the quasars in the control sample. Section 4 presents our results on BAL variability and its dependence on timescale, quasar and BAL properties, properties of C iv emission line and photometric variability nature of the quasars. In section 5, we discuss our main results and their implications. Throughout this paper we use the flat ΛCDM cosmology with H0 = 70 km s −1 Mpc −1 and Ωm,0 = 0.3.
UFO BAL SAMPLE AND CONTROL SAMPLES
We have constructed a sample of C iv UFO BALs from the SDSS data release 12 quasar population (Pâris et al. 2017) by applying the following criteria. The BAL parameter in the catalog of Pâris et al. (2017) should be set to 1 and the Balnicity index (BI) should be greater than 0 km s −1 . The observed maximum outflow velocity at the time of observations should be v outflow, max > 15000 km s −1 . We restrict our sample to quasars having zem > 2.0 to ensure that C iv (also Si iv in most cases) absorption falls in the optical region and in the most sensitive wavelength range of the spectra in the case of both SDSS and SALT observations. We also restrict our sample to objects with declination < +10 deg and magnitude brighter than mr= 18.5 mag. The former is to ensure that the source is accessible to SALT and the latter is to ensure that we get a sufficiently high spectroscopic signalto-noise ratio (SNR) even if the observing conditions are sub-optimal with SALT.
We end up with a sample of 66 sources. From these sources, we remove 2 sources (namely, J021119.65-042158.2 and J110100.38+092314.30), where wrong identifications of BELs have led to inaccurate redshifts and the correct redshifts are found to be zem < 2. After visual inspection of the spectra, we remove another source, namely J002248.46-044510.3, where the identification of BAL is suspected to be wrong. Hence, we find a total of 63 UFO BAL sources from the SDSS DR12 catalog. We add to this list another UFO BAL source namely J132216.25+052446.3, an interesting BAL quasar we have been monitoring for the past 7 years using SALT. This source is identified as a BAL QSO in the SDSS DR12 BAL quasar sample, but the UFO BAL is absent in the SDSS spectrum and has only emerged during our other observing programme (Aromal et al. 2022). Our final sample consists of 64 UFO BAL quasars which are studied in detail in this paper. The median values of the rband magnitude and zem are 18.25 and 2.3437 respectively for quasars in our sample. The list of sources, log of observations, and details of spectra obtained at different epochs are given in Table B1 in the online material.
In Table 1 we provide some physical characteristics of the quasars in our sample. Columns 8, 9 and 10 of Table 1 give, respectively, the mass of the central black hole (MBH ), the bolometric luminosity (L Bol ) and the Eddington ratio (λ Edd ). The emission redshift (zem, given in column 2) is taken from Hewett & Wild (2010) (which derives the systemic redshift from the fit of the C iii] emission line) whenever available and if not from Pâris et al. (2017).
BAL identification
We used the publicly available multi-component spectral fitting code PyQSOFit 1 (Guo et al. 2018) to fit the continuum and broad emission lines (BEL) of all quasar spectra in our sample. We visually inspected each spectrum and identified wavelength ranges devoid of any absorption lines. We then fitted these line-free regions with a power-law + multiple Gaussian (for BELs) model which provided fairly good fits using χ 2 minimization as shown in Fig 1 for a Figure 1. Examples of observed spectra and the best fits to the continuum and broad emission lines are shown in blue and orange respectively. Each panel shows absorption profiles of UFO BALs (green shaded regions) belonging to different classes as defined later in Section 4.6. The rest wavelength (bottom) and velocity (top) scales are defined with respect to zem given in Table 1. The gray-shaded region in the top panel corresponds to the CCD gap in SALT spectra. in our sample. For sources where C iv BEL is not severely contaminated by absorption lines, we typically needed two Gaussians to fit the line whereas for all other cases, a single Gaussian was used. Similarly, for other prominent BELs in the spectra like Si iv, N v etc., we used single Gaussian to fit the lines. We then identified BAL troughs using the conventional definition given by Weymann et al. (1991) where a BAL is defined as a continuous absorption wider than 2000 km s −1 below 90 % of the continuum level. Note that a spectrum can contain several distinct BAL troughs. Here, we searched for C iv BALs in the region between 3,000 to 30,000 km s −1 from the emission redshift. We do not consider BALs beyond 30,000 km s −1 as they could be contaminated by the Si iv absorption from lower velocities. We consider only BALs beyond 3,000 km s −1 to avoid contamination by narrow associated absorption systems (that appear broad due to blending and moderate resolution spectra used here) that are not part of the BAL flow.
For the identification of BAL complexes, we adopt the same method as in Filiz Ak et al. (2013) (refer to their Section 3.2) as this method helps in efficiently quantifying the BAL variability across different spectroscopic epochs. In short, we define the BAL regions for each source after assigning a minimum and maximum velocity associated with these BALs considering all epochs. These velocities (denoted as vmin and vmax) for each source are listed in columns 5 and 6 of Table 1. After visually inspecting the results, we clearly identify 80 distinct BAL complexes in our UFO BAL sam- Column 3 : The absorption redshift of the BAL calculated using the optical depth weighted velocity centroid; Columns 4, 5 and 6 : The minimum velocity (v min ), maximum velocity (vmax) and balnicity index (BI) of the identified BAL region respectively; Column 7 : The class in which the source belongs according to the shape of the BAL profile as mentioned in Section 4.6. The BAL that could not be fitted into the classification scheme is indicated with '999'; Columns 8, 9 and 10 : The estimated quasar properties black hole mass (M BH ), bolometric luminosity (L bol ) and Eddington luminosity (λ Edd ) respectively; Column 11 : The total number of spectroscopic epochs for the source; Column 12 : The minimum and maximum rest-frame time separation between two spectroscopic epochs for the source.
ple. The redshifts of individual BAL complexes and the minimum and maximum velocity spanned by them are provided in columns 3, 4 and 5 respectively in Table 1. The measured maximum Balnicity Index (BI) for individual BAL components are provided in column 6 of the same table.
BAL variations
Once the BAL troughs were identified, we estimated different C iv BAL properties such as equivalent width (W) and the maximum depth of the BAL (dBAL). The BAL properties at different epochs including W, dBAL etc. are given in Table B2 in the online material. Using the equivalent width as a measure of the strength of the BAL, we quantified the BAL variability by calculating the variations and the fractional variations in W as defined below (see also Filiz Ak et al. 2013) :
∆W = W2 − W1, σ∆W = σ 2 W 1 + σ 2 W 2 (1) ∆W W = W2 − W1 (W1 + W2) × 0.5 (2) σ ∆W W = 4 × (W1σW 2 + W2σW 1 ) (W1 + W2) 2(3)
where W1 and W2 are equivalent widths measured at t1 and t2 respectively with t1 < t2. Thus an increase (or decrease) of W with time results in a positive (or negative) ∆W . Based on the above equation, variations in equivalent width by a factor of 2 and 3 will correspond to ∆W/W of 0.67 and 1.0 respectively. An emerging (disappearing) absorption will also correspond to a ∆W/W of +2 (respectively −2). σ∆W and σ ∆W/W are errors in ∆W and ∆W W respectively.
Control sample
For comparing different physical properties of UFO BAL quasars in our sample with that of non-BAL quasars, we made a control sample of non-BAL quasars having similar emission redshift (∆zem 0.2) and r-band magnitude (∆mr 0.3 mag) distributions. For Non-BAL quasars, we selected sources that are included in both SDSS DR7 (Shen et al. 2011) and SDSS DR12 catalogs since Shen et al. (2011) provides several quasar parameters such as black-hole mass (MBH ), bolometric luminosity of the quasar (L bol ) and Eddington ratio (λ Edd ). Hence, we build a sample of 320 non-BAL sources (i.e., 5 non-BAL quasars for each quasar in our UFO sample) following the above criteria. For these objects, we used the parameters provided by Shen et al. (2011) and for the UFO BAL sample, we derived the corresponding quantities from the results of the fits performed with PyQSOFit and listed them in columns 8-10 of Table 1. Details of how we measure these quantities are provided in the Appendix-A. Yi et al. (2020) reported a correlation between λ Edd and MBH for their high-z (i.e., 3 z 5) BAL as well as non-BAL comparison samples. However, the correlation is found to be only tentative in the case of the BAL sample. They attribute this possible difference to the presence of substantial outflows in BAL QSOs as testified by large blueshift in the C iv emission lines. In Fig. 2, we plot MBH vs. λ Edd for our UFO sample as well as for the non-BAL control sample. A strong anti-correlation is apparent in this figure. Visually, the UFO BAL sample traces the same region spanned by the data from the control sample. The Spearman's coefficients are found to be r = −0.68 and r = −0.82 for UFO BAL and non-BAL samples respectively. In both cases, the p-values are < 10 −3 indicating the correlation to be significant.
In the top panel of Fig. 2, we compare the probability distribution function (PDF) for MBH between the UFO BAL and non-BAL control samples. These PDFs were estimated using a non-parametric kernel density estimation (KDE, Silverman 1986) for a fixed bandwidth Gaussian kernel. The median MBH for the UFO BAL and control samples are 10 9.47 M and 10 9.62 M respectively. The KS test confirms the MBH distribution in the two samples to be significantly different with a p-value of 6×10 −3 . This difference comes from the fact that there is a clear lack of objects with log MBH 10.0 (alternatively quasars with FWHM of C iv BEL more than 10,000 km s −1 ) and an excess of objects with log MBH ∼9.3 in our UFO BAL sample. The lack of objects with large MBH (or FWHM of the C iv BEL) may be attributed to the presence of absorption features biasing our Gaussian fits towards lower values.
In the right panel of Fig. 2 we compare the λ Edd PDFs of the UFO BAL and non-BAL control samples. We find the distribution of λ Edd to be similar for both samples. The KStest also confirms the same with a p-value of 0.60. This lack of difference between the UFO BAL and non-BAL samples is consistent with the finding of Yi et al. (2020), for their high-z (3 z 5) BAL QSO sample where the MBH measurements are based on rest-frame optical emission lines. Interestingly, we notice that for a given MBH the λ Edd values observed in our UFO BAL and control samples are lower than that found for the sample of Yi et al. (2020). Two main differences are (i) QSOs in the sample of Yi et al. (2020) are from higher redshifts and (ii) their MBH measurements are based on rest-frame optical lines that are more reliable compared to C iv emission line used in our study.
As a consequence of the above results we find the distributions of Bolometric luminosity are different for non-BAL and the UFO BAL sample (i.e., a p-value of 2.8 × 10 −4 for the KS test). This is surprising as we have matched the r-band magnitudes while constructing the control sample. As the bolometric luminosity is obtained using the rest frame 1350Å continuum luminosity, the above result suggests a possible difference in the colour distribution of the non-BAL and UFO BAL sub-samples. This may indicate a redder distribution of color for the UFO BAL sample which is consistent with the general understanding that BAL quasars are redder on average than non-BAL quasars (Brotherton et al. 2001;Reichard et al. 2003). However, we do find the median Bolometric luminosity to be similar i.e., 10 47.22 erg s −1 for the non-BAL control sample and 10 47.13 erg s −1 for the UFO BAL sample.
DETAILS OF OBSERVATIONS AND DATA USED IN THIS STUDY
For the detailed analysis presented in this work, we have used available spectra from the Sloan Digital Sky Survey-I/II (SDSS), Baryon Oscillation Spectroscopic Survey (BOSS) and supplemented them with spectra from our own ongoing spectroscopic monitoring using the Southern African Large Telescope (SALT) (Buckley et al. 2005).
For the SALT observations, we used the Robert Stobie Spectrograph (RSS, Burgh et al. 2003;Kobulnicky et al. 2003) in the long-slit mode with a 1.5" wide slit and the PG0900 grating. This combination gives a typical spectral resolution of ∼300 km s −1 . For each SALT/RSS observation, the GR angle was chosen such that the CCD gaps do not fall in the expected wavelength range of the broad absorption lines. In cases where this was not possible, we tried to observe the target with different GR angles to cover the full wavelength range without any gap. A detailed log of observations for all the objects in our sample is provided in Table B1. In the case of J1621+0758, we also have NTT/ESO observations taken in the year 2014 (details of which can be seen in, Aromal et al. 2021) We have, in total 375 spectra out of which 211 spectra are taken from SDSS and 164 spectra from our SALT observations. For our SALT observations, the preliminary processing of raw CCD frames were carried out using the SALT data reduction pipeline (Crawford et al. 2010). We used the standard iraf 2 procedures to reduce the resulting 2D spectra. Flat-field corrections and cosmic ray zapping were applied to all science frames. We extracted the one-dimensional quasar spectrum from the background subtracted 2D science frames from each epoch using the iraf task "apall". Wavelength calibration was performed using different standard lamp spectra like Ar, ThAr, HgAr and Xe. In addition, skylines from the wavelength calibrated spectrum were matched with the sky line atlas provided by SALT and, if needed, corrections were applied to increase the wavelength accuracy. Similarly, flux calibration was performed using standard reference stars observed close to our observing nights. We performed continuum fitting to our SALT spectra using the same method described above in Section 2.1.
One of the most important aspects of our study is to
Photometric variability
As in Aromal et al. (2022), we look for correlations between the absorption line variability and the continuum variability using available broad-band photometric light curves. We have obtained publicly available photometric light curves of almost all the UFO BAL sources from the Panoramic Survey Telescope and Rapid Response System (Pan-STARRS; Chambers et al. 2016), the Palomar Transient Factory (PTF; Law et al. 2009) and the Zwicky Transient Facility (ZTF; Bellm et al. 2019a,b) surveys. Pan-STARRS provides photometric data of quasars in five broad band filters, i.e., g, r, i, z and y whereas ZTF gives the same for the g, r and i bands. While PTF and Pan-STARRS provide sparsely sampled light curves ZTF provides much better sampling since the year 2018 (i.e., MJD 58200) for all the sources in our UFO BAL QSO sample.
We also obtained all the available ZTF photometric measurements for all quasars in the "non-BAL" control sample. From the ZTF light curves, we first estimate the variability amplitude (σ 2 rms ) and its error (S 2 D ) using the normalized excess variance method described in Vaughan et al. (2003). The variability amplitude strength (VAS) is then defined as VAS = σ 2 rms / S 2 D . We show the distributions of VAS in gand r-bands for the two samples in Fig 4. It is interesting to note (from the indicated median values) that the VAS distribution measured for the g-band is wider compared to that of the r-band. This indicates larger amplitude variations at lower wavelengths for quasars in both samples. We also notice that in both the bands the VAS for the UFO BAL sample is smaller than that of the control sample. The KStest confirms this with a p-value of < 0.1. Therefore, purely based on optical photometric variability we do not find the . Comparison between C iv BEL blue-shifts in spectra from the non-BAL (blue) and UFO QSO BAL (orange) samples. quasars in our UFO BAL sample to be more variable compared to the general population of quasars.
Properties of the C iv emission line
In this section, we mainly focus on the C iv BEL properties of the UFO BALs compared to that of the non-BAL control sample. We recognize that the BEL of UFO quasars may be affected by strong BALs close to the emission redshift. Therefore, we only consider sources having no BALs at velocities less than 6000 km s −1 with respect to zem. We also visually inspected the double gaussian fits to the C iv BEL profiles in order to check and remove certain epochs that are severely affected by either narrow absorption lines or CCD gaps in the case of SALT epochs. Thus, we made a sub-sample of 41 sources satisfying the above criteria to compare the emission line properties and study the relationship between UFO BAL and BEL properties.
We estimate two quantities namely, the equivalent width (WBEL) of the emission line and its blueshift (BSBEL) for the C iv BEL. We follow the method by Rankine et al. (2020) in order to calculate BSBEL, i.e.,
BSBEL = c × (λr − λ half )/λr (4)
where c is the velocity of light, λr is the rest-frame wavelength of the emission line, 1549.48Å for the CIV doublet and λ half is the wavelength which bisects the cumulative total line flux. As shown in Fig 5, we confirm the findings by Rankine et al. (2020) that C iv BELs in sources with BALs reaching high velocities such as our UFO BAL sample tend to show more blue-shifted emission compared to the same of its non-BAL counterparts. But, it is good to keep in mind that the emission-line outflow and physical properties of BAL and non-BAL quasars overlap, hence demonstrating that (highionization) BALs and non-BALs represent different views of the same underlying quasar population.
RESULTS ON BAL VARIABILITY
Frequency of BAL variability
Following Filiz Ak et al. (2013), we identify the variable BALs when the C iv BAL equivalent width variation are > 3σ significance level. Using all possible pairs of epochs for each object, we find that 95.3 +14 −12 % 3 of UFO BAL QSOs in our sample show at least one variable trough. Among the 80 UFO BALs identified in Table 1, 95 +12 −11 % of them show >3σ variability. This confirms that ∼95% of UFO BAL quasars (and UFO BAL components) in our sample show significant variability at least once between possible pairs of epochs probed. Our results are consistent with that of Gibson et al. (2008) who have found 12/13 of the BALs in their sample to show significant equivalent width variability over 3-6 yrs.
However, smaller variability fractions have also been reported in the literature. Only 11/32 BAL trough in the sample of Lundgren et al. (2007) show more than 3σ equivalent width variability over a time-scale of 102 days. In their sample of 24 bright BAL quasars Capellupo et al. (2011) have found that 39% and 65% were varied in the short-term (i.e., 4-9 months) and long-term (i.e., 3.8-7.7 yr), respectively. Filiz Ak et al. (2013) have found that 62.2% of BAL quasars and 57.9% of the BAL trough in their sample show significant C iv equivalent width variation when they considered a pair with the shortest time interval for each quasar. The shortest time interval varies for each source due to the non-uniform SDSS observations and ranges from as small as 5.9 hr ( 10 −3 yrs) to 3.7 yr with a median of 2.1 yr. If we apply to our sample the same criteria as Filiz Ak et al. (2013), i.e., consider only the pair with the shortest timescale for each object (see last column in Table 1), we get ∼70% of UFO BAL QSOs and UFO BAL components to show variability. We also consider the epoch pairs with the longest time separation (i.e., ∆t in the range 2.0−7.3 yr) for each object and found that ∼89% and ∼87% of UFO BAL QSOs and UFO BAL components show variability respectively. Recall the shortest time intervals probed in our sample (see the middle panel in Fig. 3) are shorter than that of Filiz Ak et al. (2013). As the variability amplitudes are larger for longer time scales (see below), this may indicate that our UFO BALs are showing more variability compared to the general population of BAL quasars at all time scales (section 4.4 provides further discussions on this). Hemler et al. (2019), studied the very short time-scale (<10 quasar rest frame days) BAL variability towards 27 BAL QSOs observed as a part of reverberation mapping studies with a median of 58 spectral epochs per quasar. Using all possible combinations of epochs and the C iv equivalent widths given in Table 4 of Hemler et al. (2019), we find that the variable fraction of sources is 96% and the variable fraction of BALs is 92%. This is consistent with what we find in our sample.
To quantify the fraction of variable objects we consider three time bins corresponding to short (0.0-0.5 yrs), intermediate (0.5-2.0 yrs) and long (> 2.0 yrs) time-scales. The difference between the two samples is interesting. This could be either due to a difference in the properties of the quasars or to the more frequent sampling at smaller time scales in the case of Hemler et al. (2019). The typical sampling time scale between consecutive epochs in this study varies between less than a day to a few days where we expect the absorption line variation to be less.
In Fig. 6, we plot the fraction of sources showing C iv equivalent width variability between two epochs larger than a certain threshold ∆W W as a function of ∆W W . This fraction decreases with increasing ∆W W threshold. The percentage of sources showing ∆W W greater than 0.67 and 1.0 (corresponding to equivalent width variations by a factor of 2 and 3 respectively) are 33% and 11% respectively as shown in Fig 6. In the sample of Hemler et al. (2019) 26% of the BAL quasars show ∆W W >0.67. When we consider only BAL quasars satisfying our definition of UFOs the fraction increases to 40%. In the following discussions, we refer to BAL components with ∆W W >0.67 as "highly variable" BALs. In the case of Lundgren et al. (2007) ∼ 9% of the BALs are "highly variable" (at time-scales of 102 days) and this becomes ∼17% if we consider the BALs that satisfy our condition to be an UFO BAL. In the sample of Gibson et al. (2008) ∼ 8% of BALs are "highly variable" over a time scale of 3-6 yrs. Even in the sample of (Filiz Ak et al. 2013) we find only ∼8% of the BAL studied are highly variable. Thus it appears that UFO BALs tend to show equivalent width variations more frequently and with large amplitudes compared to the general BAL population.
Since most of the BALs in our sample show significant C iv equivalent width variations, it is interesting to study BALs that show roughly stable general profiles with insignificant equivalent width variations. It is possible that the nondetection of significant equivalent width variability in these types of sources can be attributed to the unavailability of a sufficient number of spectroscopic epochs. Alternatively, the absorption profile variation may be complex (i.e., uncorrelated over the full profile) and not perfectly captured by the equivalent width variations. This may be the case for some of the BALs which are spread over a few ten thousand km s −1 and consist of multiple narrow variable regions varying in an uncorrelated manner. To address such cases, we should consider pixel-based analysis which we plan to present in an upcoming paper. In our sample, the most dramatic BAL profile variation which resulted in the largest W change of 35Å is observed in J1156+0856 (3 epochs).
Time dependence of C iv BAL variability
Next, we study the variability of the C iv BAL equivalent width and its dependence on the time interval probed. In the top and middle panels of Fig. 7, respectively, we plot ∆W W and ∆W as a function of the quasar rest frame time interval for all possible pairs of epochs (see Eqs. 1 and 2 for definitions). In the two panels the best-fit relationship obtained by Filiz Ak et al. (2013) is over-plotted as red solid curves. We can see that a significant number of our data points lie beyond these curves. In the bottom panel of Fig. 7, we show ∆W W as a function of the maximum C iv rest equivalent width between the two epochs considered. The two horizontal dashed lines in the top and bottom panels indicate a fractional variability of ±0.67 (i.e., "highly variable" BALs). We note that the scatter in both ∆W W and ∆W is larger at longer time scales. It is also evident that the C iv equivalent width variations by more than a factor of 2 (or ∆W W 0.67) occur only on large time-scales (i.e., > 0.2 yr) in our sample.
From the bottom panel of Fig. 7, it is evident that there is very little scatter in ∆W W when the maximum C iv rest equivalent width is more than 55Å. On the other hand, a large scatter in ∆W W is seen at small C iv equivalent widths (i.e <5Å). We do find a nearly uniform scatter in ∆W W in the middle range of maximum C iv equivalent widths. Also, we note that the occurrence of epoch pairs with ∆W W > 0.67 (i.e., "highly variable" BAL) is independent of C iv rest equivalent width in this middle W range.
Overall, we have 42 (52), 32 (40), 64 (80), and 36 (46) individual sources (or BALs) contributing 131, 287, 291, and 192 pairs of epochs respectively in 0.1-0.5, 0.5-2.0, 2.0-3.5, and 3.5-7.5 yrs time bins. In these 4 time bins the fraction of UFO BAL quasars having at least one BAL component that is "highly variable" are 0.12, 0.16, 0.23 and 0.28. The same for the BAL components are 0.14, 0.18, 0.20 and 0.24 respectively. Thus there is a clear indication in our sample that the fraction of "highly variable" BALs increases with increasing time intervals. For the first three time-bins the fraction of "highly variable" BALs are 0.01, 0.08 and 0.11 for the BALs studied in Filiz Ak et al. (2013). This once again confirms our finding that UFO BALs are more variable com- Note that the apparent "track" of points located close to each other mostly arises when the max(W 1 ,W 2 ) remains the same but ∆W W keeps changing between various epochs for a single object.
pare to the general BAL population. In the time bin 2.0−3.5 yrs (where all quasars in our sample contribute) the number of UFO BAL quasars showing ∆W W > +0.67 (6 BALs) and ∆W W < −0.67 (7 BALs) are nearly identical. Three BALs show both positive and negative variations with ∆W W >0.67 in this time bin. However, in the 3.5−7.5 yrs time bin, we find a low fraction of UFO BAL quasars and BAL components showing ∆W W > 0.67 (3 BAL components) compared to that showing ∆W W < −0.67 (7 BAL components). Interestingly, for the 2.0−3.5 yrs time-bin, the number of "highly variable" BALs showing a negative trend is roughly a factor two higher than those showing positive trend in the sample of Filiz Ak et al. (2013). This may indicate that the growth and decay of C iv equivalent width may have different characteristic time scales. We will investigate this in more detail below.
We quantify the time dependence of C iv equivalent width variability using the inter-quantile range (IQR) as an indicator of the strength of variability for different time scales. We consider the same four time bins discussed above for the four time bins in the ascending order. This indicates that the scatter in ∆W W increases with time (as found in past studies e.g. Capellupo et al. 2011;Filiz Ak et al. 2013).
As shown in the top panel of Fig. 3, the number of spectroscopic epochs varies strongly from one source to the other in our sample (see Table 1). In order to probe any bias in the ∆W W distribution due to this non-uniform time sampling, we use two alternate methods to estimate the IQR. In the first method (hereafter method I), for each time bin, we randomly choose one measurement of ∆W W per BAL and measure the quantiles at 0.25, 0.50, and 0.75 and IQR for the derived ∆W W distribution. We repeat this procedure 1000 times and calculate the mean and σ of the resulting IQR distribution. Results using this method are shown in panel (b) of Fig. 8. The final IQR values are 0.18±0.03, 0.22±0.04, 0.26 ± 0.03, and 0.39 ± 0.07 for 0.1-0.5, 0.5-2.0, 2.0-3.5, and 3.5-7.5 yrs time bins respectively.
In the second method (hereafter method II), we randomly choose 50 out of 64 sources and re-sample 64 sources from this sub-sample with replacement. We then calculate IQR for each time bin using all points from the randomly selected sources. We repeat this procedure 100 times and the mean and σ of the resulting IQR distribution are estimated.
The final values of quantiles at 0.25, 0.50, and 0.75 and the IQR estimates are shown in panels (c) and (d) of Fig. 8. The final IQR values are 0.24 ± 0.04, 0.23 ± 0.07, 0.42 ± 0.05 and 0.53 ± 0.06 for 0.1-0.5, 0.5-2.0, 2.0-3.5 and 3.5-7.5 yrs time bins respectively. These values are even closer (i.e well within 1σ) to the ones obtained using the full sample. This exercise confirms that the increase in ∆W W with time is not biased by non-uniform time sampling in our sample.
In addition, we looked at the width of the ∆W W distribution after removing BALs having either very small or large W. For this, we considered only BALs with 5 < Wmax < 55 A which corresponds to 69 out of the 80 BALs in our sample and carried out the same analysis to derive IQR values. Using method I, we obtained IQR values of 0.25 ± 0.03, 0.36 ± 0.08, 0.37 ± 0.04, 0.61 ± 0.07 for 0.1-0.5, 0.5-2.0, 2.0-3.5, and 3.5-7.5 yrs time bins respectively. This confirms that the increase in the C iv equivalent width variability over time is not dominated by C iv BALs with large or small rest equivalent widths. In a similar way, we calculated the IQR for all the sources in the sample of Hemler et al. (2019) which turned out to be 0.11, 0.20, and 0.25 for 0-0.1, 0.1-0.5, and 0.5-2.0 years time bins respectively. This again suggests an increase with time of the scatter in the ∆W W distribution as shown in Fig 9. There are two time bins that are common to our sample for which the IQR measured are consistent within a 1.5σ range.
Is the ∆W
W variation symmetric? From Fig 8, we can see that the absolute value of the ∆W W distribution quantiles at 0.25 and 0.75 are not the same in the 3.5-7.5 yrs time bin. This might suggest a possible asymmetry in our sample. This is what we study in this section. A symmetric distribution will indicate a statistically similar characteristic time-scale for increase/decrease in the C iv BAL absorption. Interestingly, while studying emerging/disappearing BAL components, several authors have noted (see for example, McGraw et al. 2017;Mishra et al. 2019) that the average time scales probed are higher for BAL disappearance events compared to the emergence events. Discussions presented above also indicate that highly variable UFO BAL components tend to show a more negative trend in the 3.5−7.5 yrs time bin.
First, we carefully looked at the C iv rest equivalent width as a function of time for all objects in our sample to identify the presence of any monotonous trends either in the positive (i.e., increasing) or in the negative (decreasing) direction. Based on visual inspection, we classified the 80 BAL troughs in our sample into four classes: the ones showing (i) both considerable negative and positive ∆W variations, (ii) broadly positive ∆W variations (in these sources, there may be a few epochs with negative EW variations which are not significant compared to the general trend of variations), (iii) broadly negative ∆W variations and (iv) no systematic trend. The number of BALs in each category is 38, 16, 18, and 8 respectively. Thus, our sample does have 34 objects that show predominantly increasing or decreasing trends in the rest equivalent width as a function of time.
First, we consider the 3.5−7.5 yrs time bin, for which the quantiles plotted in the top 3 panels of Fig. 8 suggest a possible asymmetry towards the negative direction (i.e decreasing W). We measure the absolute difference in the values at 0.25 and 0.75 quantiles to be 0.25±0.09, 0.17±0.07 and 0.25±0.09 respectively for the values obtained using the full sample, method I and method II respectively. This suggests an asymmetry towards negative (decreasing W) at 2.4σ to 2.8σ level. We note however that 28% of the UFO BALs contributing to this time bin are "highly variable" BALs with the tendency to show negative variations. We thus probe the symmetry of the distribution after removing the 10 UFO BAL quasars in this bin that show "highly variable" BAL components. We find the distribution to be more symmetric with an absolute difference in the values at 0.25 and 0.75 quantiles to be 0.02 ± 0.04 when we use "method I". The KS-test returns a p-value of 0.71. Thus the large asymmetry seen in this time bin is largely driven by "highly variable" BAL sources.
Next, we consider the 2.0−3.5 yrs time bin where all the 64 UFO BAL quasars in our sample contribute. We measure the absolute difference in the values at 0.25 and 0.75 quantiles to be 0.05±0.06, 0.01±0.06 and 0.00±0.06 respectively for the values obtained using the full sample, method I and method II respectively. This indicates a symmetric distribution in the positive and negative directions. The KS-test results confirm the same with a p-value of 0.36. Next, we ask whether the distribution remains symmetric when we avoid the 10 UFO BAL QSOs that show "highly variable" BALs in the 3.5−7.5 yrs time-bin. In this case we get the absolute difference in the values at 0.25 and 0.75 quantiles to be 0.07 ± 0.08 when we use "method I" and the KS-test p-values of 0.41. The distribution is found to be symmetric in any case.
Since the asymmetry seems to arise from highly variable BALs on longer timescales, we focus on the time evolution of 21 BAL components from the 20 quasars in our sample that show high variability ( ∆W W > 0.67) for ∆t > 2 yr. To begin with, for each highly variable BAL in this sub-sample, we plot the rest equivalent width, W, normalized by the maximum W (i.e., Wmax) observed for that BAL against the time difference from the epoch where it attained the maximum W value (see Fig 10). Hence, the positive (negative) values in the abscissa correspond to epochs after (before) the maxi- Figure 10. The C iv rest equivalent width (W) normalized by the maximum observed W (i.e., W Wmax ) is plotted against the time difference (in year) from the epoch where it attained the maximum W value (i.e., t-t Wmax ) for each BAL in the "highly variable" sub-sample. The blue points correspond to BALs which had the maximum change in W Wmax in the increasing direction whereas the red correspond to the same in the decreasing direction. In the top panel, we plot the histogram of shortest t-t Wmax at which W Wmax crossed the 0.5 line with the median shown in dotted lines. mum W is achieved. Here, we clearly see that there are more BALs that show large variability on the decreasing side compared to the increasing side. This has been discussed in detail in Section 4.2. Interestingly, we also observe that many BALs on the decreasing side (i.e., positive t-tW max ) reach values less than Wmax 2 (as indicated by the red dashed lines in Fig 10) at larger time scales of more than 4 yr. But, on the increasing side (i.e., negative t-tW max ), among the comparatively lower number of BALs showing high variability, most of them increase to Wmax from values less than Wmax 2 at rather smaller time scales of ∼ 3 yr. We notice that the time scales for the equivalent width to decrease by a factor of 2 are on average higher than that of increasing equivalent width cases (look at the histogram in the top panel of Fig. 10). As asymmetry seen in ∆W W for this time bin is dominated by the "highly variable" BAL components this could be driven by the fact that the decreasing/disappearing time scale is generally longer than the increasing/appearing time scale. But it is good to keep in mind that when we study the appearing and disappearing BAL time scales, there can be a certain bias in the results due to the fact that all our BALs are fairly strong in the first epoch itself. This may affect the study of appearing time scales since we have missed the epochs when the BAL was formed.
Comparison with Filiz Ak et al. (2013) results
Filiz Ak et al. (2013) have obtained a relationship between the relative W variation ∆W W and the minimum sampled rest-frame timescale ∆tmin from the 428 BAL troughs identified in their sample (see equation 7 in their paper). As we noted in Fig. 7, our sample seems to show more scatter compared to this relationship. However, in order to perform a significant comparison we need to account for the differences in the distribution of minimum sampling time intervals (∆tmin) between their sample and our UFO sample (see Fig. 3). For this, we first calculate ∆W W for all possible combinations of epochs for each source in the UFO sam- Figure 11. The fractional variation in W of C iv UFO BAL observed in our sample (circles, see Section 4.4) is plotted versus the minimum sampled rest-frame timescale ∆t min . The best-fit relation obtained in Filiz Ak et al. (2013) is overplotted as a blue curve and the shaded region shows its uncertainty. The empty circles represent the same comparison after removing the "highly variable" sources (as explained in Section 4.3) from the sample.
ple and populate the ∆W W vs. ∆t plane. Next, we resample 80 points from this plane with the time sampling consistent with a normalized ∆tmin distribution as given in Filiz Ak et al. (2013) for the 428 distinct BAL troughs in their sample. Following Filiz Ak et al. (2013), we also do an average over 4 time-ordered data points from the above sampled points. This results in a total of 21 points. We repeat the above steps 1000 times and take the mean of each point from the total number of iterations. Now we compare these points (filled blue circles) with the best-fit relation from Filiz Ak et al. (2013) in Fig. 11. We observe that most of the points are located at a few years time-scale and almost all of them show much higher ∆W W values than what is expected from the fit. This confirms that the UFO BALs in our sample are highly variable compared to the fit from Filiz Ak et al. (2013). This difference can not be attributed to the differences in the sampling of ∆t. If we remove the 20 quasars with "highly variable" BALs and repeat the procedure, we derive measurements shown as open circles in Fig. 11 which are more consistent withe the Filiz Ak et al. (2013) relation. This confirms that the behavior of ∆W W is significantly influenced by the "highly variable" BALs.
What drives the BAL variability ?
In this section, we correlate ∆W W with different BAL and quasar properties to investigate the physical conditions in UFO BALs and the possible origin of the equivalent width variations.
Dependence on the BAL properties
We consider four properties of the observed BAL troughs: the C iv equivalent width W , the optical depth weighted velocity centroid (vcent), the maximum relative depth dBAL and the velocity width (∆V = vmax − vmin) of the BAL trough. For each of these parameters, we divide the BAL sample into two sub-samples of equal number including the BALs with parameter values larger and smaller than their For each source in these sub-samples, whenever available (as it is not necessary that all objects contribute to all the time bins considered) we randomly select a pair of epochs (and corresponding ∆W W ) that falls in the four time bins considered here. We then measure the average ∆W W for all sources in a sub-sample for each time bin. We repeat the above process 100 times and obtain the mean and standard deviation of the average ∆W W distribution of the subsamples for each time bin. Results are presented in Fig 12. In the first three time bins (i.e., ∆t 3.5 yrs) we note that the average ∆W W is higher for the sub-sample with lower equivalent width (i.e., Wmax < 21.2Å). The difference is larger than the 3.5 σ level. This is not the case for larger time scales (>3.5 yrs). This is consistent with what we find in the bottom panel of Fig. 7, where a large scatter in ∆W is seen for Wmax < 5. In panel (a) of Fig. 13, we plot ∆W W as a function of Wmax for all the observed pairs in the time bin 2.0 ∆t 3.5 years (where all the 64 UFO BALs contribute). It is evident that the scatter in ∆W W is larger when Wmax is below the median value indicated by the vertical dashed line. A similar trend is seen for dBAL (see panel (c) in Fig. 12 and Fig. 13). It is interesting to note that the trend is the same for the velocity widths (∆v). Note that, in Fig. 13, the points look discretized due to the relatively small changes in these parameter values plotted in the x-axis for a given object for all the epoch pairs considered. This means that absorption lines with low W having narrow and shallow absorption profiles tend to show large variability. Some of the above findings are consistent with those of Filiz Ak et al. (2013) (refer to subsection 4.5 therein). They find that shallower BALs show more variability and report a highly significant correlation (>99 %), between ∆W , ∆W W and average W in their sample (see also, Capellupo et al. 2011;Vivek et al. 2014;Hemler et al. 2019). However, contrary to our results, they find that wider BALs vary more than narrower ones and state that this is expected since wider BAL troughs might have a better chance of containing variable regions. This difference could be related to the way our UFO BALs sample is constructed which avoids BALs that only have strong narrow absorption at low velocities (i.e., vmax < 15000 km s −1 ). We address this point in detail below. Also, recall Filiz Ak et al. (2013) have used one measurement per quasar obtained at the lowest time scale probed. Here, we use multiple epoch measurements for each quasar but sampled once for each time bin.
In low-resolution spectra, like the one we consider here, the equivalent width variations can be driven by (i) optical depth variations, (ii) variations in the covering factor (fc; see for example, Muzahid et al. 2016) and/or (iii) changes in the line of sight density and velocity field due to transverse velocity component (see for example, Aromal et al. 2022). At this stage, there is no obvious way to disentangle these possibilities which are all reasonably possible as it is easier to change the property of weak components compared to strongly saturated ones.
In panel (b) of Fig 12 we notice that the sub-sample with large vcent shows a larger variability (significant at >3 σ level) compared to the sub-sample with smaller vcent for all time-scales ∆t > 0.5 yrs. In the lowest time bin, 0.1 ∆t(yrs) 0.5, the difference between the two subsamples is not statistically significant. In panel (b) of Fig. 13, we plot ∆W W as a function of maximum vcent for all the observed pairs in the time bin 2.0 ∆t(yrs) 3.5 years. It is evident from this plot that the scatter in ∆W W is larger for a higher value of maximum vcent. Similarly highly variable UFO BALs are more frequent when vcent is larger. This is consistent with the finding of Filiz Ak et al. (2013) that the high-velocity BALs vary more than lower-velocity ones. They interpret this as a secondary effect based on the fact that higher-velocity absorption has lower W . However, in our sample, we do not find any significant anti-correlation between Wmax and vcent (with a Spearman's correlation coefficient of −0.02 and p-value of 0.85). Thus it is not obvious that the dependence of vcent is a secondary effect. To further investigate this, we note that ten out of 80 BAL components have vcent < 8000 km s −1 and tend to have low equivalent widths (i.e., <20Å). If we do not consider these 10 lowvelocity components having low Wmax we do find indeed a moderate anti-correlation between Wmax and vcent (with a Spearman correlation coefficient of −0.32 and p-value of 0.01). We come back to this in a bit more detail below.
Note that in the case of simple disk wind models (as well as when gas motions are assumed to be nearly Keplerian), large velocities are related to gas components ejected closer to the accretion disk (Arav et al. 1994). In this kind of scenario vcent could well be the primary parameter of large ∆W W . Further Proga et al. (2012) demonstrated that C iv absorption produced by disk wind models from hydrodynamical simulations is quite complex and the simulated C iv absorption profiles show large variability, especially at high velocities. They attribute this excess variability at large velocities to the emergence of very fast mass ejections from relatively large distances, where the gas is well shielded from X-ray radiation. As we do not have independent distance measurements for different velocity components we will not be able to test such a scenario using our data. To summarize the results presented here demonstrate that, in general, more fractional equivalent width variability is observed in weak, high-velocity, and shallow BALs.
Dependence on the quasar properties
In this section, we explore whether the variations in ∆W W are related to any of the following quasar properties: MBH , λ Edd and bolometric luminosity L bol . For this, we again compare the ∆W W distributions of two sub-samples defined as the objects having one of the above parameters smaller vs. larger than the median value of the whole UFO sample. Fol-lowing the same procedure as described above, we obtain the mean and standard deviation of ∆W W in four time bins for each sub-sample. Results are summarized in Fig. 14. In the largest time bin (i.e., 3.5 t(yrs) 7.5) the variations of ∆W W are clearly independent of the quasar properties considered here. When we combine this with the discussions presented in the previous sub-section, it appears that for the largest time-scales only vcent seems to be related to the relative W variations indicating that this may be an important parameter for the interpretation of these flows at large time scales (i.e., t > 3.5 yrs).
In the case of L bol and λ Edd , ∆W W is found to be higher for low L bol and low λ Edd compare to high L bol and high λ Edd in all the three short time bins (i.e., for ∆t < 3.5 yrs in Fig. 14). In particular for 0.5 ∆t < 2.0 yrs, ∆W W shows a large difference between sub-samples defined based on all three properties considered here (see Fig. 14). As mentioned before, this time bin gets contributions from only 32 quasars with 40 UFO BALs in our sample. To explore this further, we plot ∆W W vs. L bol (top panel) and λ Edd (bottom panel) in Fig. 15 for the time bin 0.5 ∆t(yrs) 2.0. We do see the scatter to be larger towards lower L bol and λ Edd . However, we find (unlike in the case of vcent or Wmax) no clear tendency for the "highly variable" BAL components to have any preference for low L bol or λ Edd . Also due to the limiting magnitude used for defining our sample quasar properties of the objects in our sample may not span the full range spanned by the general population of quasars (see Fig. 2). Therefore, it would be good to confirm the differences seen in ∆W W for different sub-sample using more measurements. For the sake of comparison, we use an approach similar to that of Filiz Ak et al. (2013) and consider ∆W W for the shortest time scale for each source. Except for L bol which shows a moderate anti-correlation (Spearman's coefficient = −0.31, p-value=0.01), both MBH and λ Edd show no evidence of correlation. Similarly, when we consider the ∆W W over the longest time scale for each source we do not find any correlation with any of these quasar parameters. Filiz Ak et al. (2013) found a large scatter in the ∆W distribution for sources with low L bol on moderate (1-2.5 yr) and long (>2.5 yr) time scales compared to high L bol sources. However, they could not find any significant correlation using rank-correlation analysis except for moderate time scales (1-2.5 yrs). They also did not find evidence for a correlation between λ Edd and scatter in either ∆W or ∆W W . In the case of MBH significant (i.e., at 3.6σ level) difference is seen between our two sub-samples only in the 0.5 ∆t(yrs) 2.0 time bin. Filiz Ak et al. (2013) did not find any evidence for the presence of correlation between MBH and ∆W W in their sample. On the other hand, He et al. (2015) found that there is a medium strong negative correlation (r=−0.537, p=0.003) between ∆W and Mg ii based MBH estimates in 28 BAL QSOs with variable BAL regions.
In summary, in our sample, there is no strong evidence of any correlation between the BAL variations and the characteristics of the quasars.
Do highly variable UFO BAL quasars have different properties?
As mentioned before, 24 BAL components in 21 UFO BAL quasars in our sample tend to be "highly variable". In this section, we compare the properties of these "highly variable" UFO BALs with those of the rest of the UFO BALs in our sample. In Fig. 16 we compare the distributions of quasar properties (MBH , λ Edd and VASr as introduced in Section 3.1) and BAL properties (Wmax, maximum of vcent and maximum of dBAL) for "highly variable" BALs and the rest of our sample. The median value of each distribution is shown as a vertical (gray or yellow) dashed line in each panel.
For quasar properties, we find that the MBH and λ Edd distributions are similar for both sub-samples (p-value is >0.1 from the KS test) whereas for VASr the p-value is 0.03 for the KS test. This indicates that occurrence of "highly variable" BAL is not strongly coupled to MBH or λ Edd . However, there is an indication that the UFO BAL quasars showing "highly variable" BALs tend to show slightly larger photometric variability compared to the rest of the UFO quasars. This could either mean the large equivalent width variabilities are driven by photoionization or disk instabilities that also results in photometric variability. We come back to this discussion in more detail in section 4.8.
For BAL properties, the distributions of Wmax are identical as indicated by p-value > 0.1 from the K-S test (see also Fig. 16). However a larger fraction of "highly variable" BALs have comparatively higher values of vcent,max and lower values of dBAL,max. K-S test p-values are 0.01 and 0.03 for vcent,max and dBAL,max respectively. In summary, we find that "highly variable" BALs tend to have comparatively shallower absorption at higher velocities. A considerable fraction of UFO BALs with "highly variable" components tend to show excess variability in their r-band light curves.
Different classes of UFOs
Here we investigate whether there is a relation between the scatter in ∆W W and the overall BAL profile. For this, we classify quasars in our UFO sample into three classes based on the global absorption profile structure. Typical examples are shown in Fig. 1. Class-1: Includes sources with one or more C iv UFO BALs having vmin > 8000 km s −1 without any C iv broad absorption at lower velocities (top panel in Fig. 1). There are 33 sources in our sample that were classified as Class-1. In three of these cases (J0224-0528, J1054+0150 and J1317+0100) there are two UFO BALcomponents with vmin > 8000 km s −1 . In the remaining 30 sources there is only one C iv BAL complex. Basically, BALs in this class are well detached from the C iv emission line without any low-velocity absorption. Class-2: Includes only sources with multiple BAL troughs having at least one UFO BAL with vmin > 8000 km s −1 and one non-UFO BAL with vmin < 8000 km s −1 . In this case, the UFO BAL and non-UFO BAL components are distinguishable (middle panel of Fig. 1). There are 13 objects in our sample that belongs to this class. In three cases (J0046+0104, J0242+0049 and J2352+0105) the low-velocity BAL complex has vmax < 3000 km s −1 and is not listed in Table 1. In two sources (J2310+0746 and J1322+0524) we have identified three distinct BAL complexes with one of them being a UFO BAL and the other two being present at lower velocities. Class-3: Contains sources with a single UFO BAL trough having vmin < 8000 km s −1 (bottom panel of Fig. 1). This class, by definition, has absorption spread over a wide range of velocities and absorption from distinct components are merged into a single BAL component when we follow the definition of Filiz Ak et al. (2013). There are 17 sources in our sample that are classified as Class-3. Only one UFO BAL quasar, J0216+0115, could not be classified as the absorption profile does not fit with any of the definitions of the three classes mentioned above. Column 8 in Table 1 provides the classification for each quasar in our sample.
Note, the classification scheme discussed here is motivated by the two-component (polar and equatorial) wind models discussed in Borguet & Hutsemékers (2010). In this model, absorption profiles consistent with that of Class-3 will have contributions from both fast-moving polar and slow-moving equatorial winds seen at low inclination angles. Detached profiles as those of Class-1 will mainly come from fast-moving polar components seen at high inclination angles to the disk plane. Thus the classification we adopt may reflect a classification based on the inclination angle with respect to the disk. While this gives a motivation for the above classification scheme, we are aware that a given absorption profile can be produced by models with a wide range of parameters.
The fraction of "highly variable" sources are 0.30 (10/33), 0.54 (7/13), and 0.18 (3/17) for Class-1, Class-2 (considering only the UFO BAL components), and Class-3 respectively. This indicates the possible connection between profile shapes and variability of C iv absorption. To explore this further, we study how UFO BAL quasars belonging to different classes populate the Wmax vs. vcent plane (see Fig. 17). The low-velocity absorption components of Class-2 populate mostly the bottom left part of the plane whereas the high-velocity components populate the bottom right part. Class-3 populates mostly the upper left part of the plane as expected because the BAL extends continuously to large velocities with vcent being typically in the lower side. Class-1 objects populate both the upper and lower quarters of the upper half of vcent. The size of each point in Fig. 17 is proportional to the amplitude of ∆W W . The fraction of UFO BALs showing "highly variable" BALs are indicated in each quadrant in Fig. 17. Nearly 50% of the BALs in the high vcent and low Wmax show ∆W W > 0.67. This quadrant is mainly populated by Class-1 and high-velocity component of Class-2 BALs in our sample. This implies that the relatively weaker and detached high-velocity BALs have a tendency to show more variability. On the other hand, a lower fraction (i.e., 14%) of UFO BALs seems to be "highly variable" in the low vcent and high Wmax quadrant. This quadrant is predominantly occupied by Class-3 BALs. In the quadrant with low Wmax and low vcent we find only 19% of the BALs are "highly variable". This quadrant is predominantly occupied by low-velocity components of Class-2 BALs in our sample. On the other hand, in the quadrant with high vcent and high Wmax nearly 29% of BALs are "highly variable". All this clearly imply that high C iv BAL variability is associated with both large vcent and Wmax and Wmax alone is not a primary driver of the large variability.
Next, we study the ∆W W distribution of UFO BALs in different classes as a function of time (see Fig. 18). The histograms in the right panels of Fig. 18 show the distributions of ∆W W in each case. It is apparent that the scatter in the distribution is decreasing from Class-1 to Class-2 followed by Class-3. This result may be affected by the different number of sources in each class. To confirm the trend seen in Fig. 18, we measure IQR for different classes using method-I (see Section 4.2) for the four time bins (i.e., 0.1-0.5, 0.5-2.0, 2.0-3.5, and 3.5-7.5 yrs). IQR values obtained for Class-1 are 0.27±0.03, 0.45±0.10, 0.40±0.05, and 0.60±0.08 respectively. For Class-2, IQR values for the UFO BALs (i.e only for the high-velocity components) are 0.29±0.12, 0.46±0.15, 0.49±0.11, and 0.43±0.13 respectively. The IQR values for Class-1 and Class-2 objects are consistent within errors. However, for Class-3, IQR values are 0.06±0.01, 0.07±0.02, 0.22±0.06, and 0.61 ± 0.01 respectively. Clearly, Class-3 objects show consistently lower IQR values (compared to the other two classes) except for the longest-time bin of 3.5-7.5 years.
Next, we probe the possible correlation between the ∆W W distributions of the distinct low-and high-velocity BAL troughs seen in Class-2 sources (i.e., UFO and non-UFO BALs). From the top panel of Fig 19, it is clear that the variations in high-and low-velocity BAL troughs are correlated. Also, using the linear regression, we find a slope of 0.95 considering all the class 2 sources whereas if we exclude BAL components towards J1322+0524 that contribute an appreciable number of points to this plot, the slope turns out to be 1.45. This indicates that the UFO BALs show more variability compared to non-UFO BALs in general, but in the case of J1322+0524, the UFO and non-UFO BALs vary with similar strength leading to a slope close to 1. Also, from the bottom panel of Fig 19, we see that the ∆W W distributions of high-and low-velocity BALs are considerably different (pvalue = 0.04) with high-velocity BALs showing more scatter than the low-velocity ones. This is consistent with what was shown before: high-velocity BALs are more variable.
To extend this analysis to Class-3 objects, we divide the continuous absorption profile into two regions above and below the velocity mid point, v mid = v min +vmax 2 . Even in this case (see Fig. 20), we find a correlated variability between high-and low-velocity regions as in the case of Class-2 sources, but with a much steeper slope (1.95) indicating that high-velocity regions are much more variable compared to low-velocity ones. We should remember that this is due in part to the fact that low-velocity regions are more saturated.
C iv emission lines and BAL connection
As discussed in Section 3.2, we measured the equivalent width (WBEL) and the blueshift (BSBEL) of the C iv emission line in a sub-sample of 41 UFO BALs with vmin larger than 6000 km s −1 so that the C iv emission profile is not affected by the absorption. By definition, all 33 objects in Class-1 are part of this sample. In the remaining 8 sources, 2 and 6 sources belong to Class-2 and -3 respectively. As seen in Section 3.2 and Fig. 5, BSBEL is much larger for UFO BAL quasars compared to non-BAL quasars. The measured large values of BSBEL in BAL quasars are assumed to be a consequence of outflowing gas in the BLR. We searched for correlations by calculating Spearman's correlation coefficient and corresponding p-values (see Ta-ble 2) between BSBEL or max(∆WBEL/WBEL) with the BAL parameters Wmax, vcent,max, vmax, dBAL,max and ∆vmax. From Table 2, we observe a strong correlation between BSBEL and Wmax, vmax and dBAL,max whereas a moderate correlation is reported for ∆vmax. These strong correlations are consistent with recent studies including Rankine et al. (2020) who found that BAL quasars with the highest WBEL and lowest BSBEL tend to show weaker, narrower, and comparatively low-velocity BAL troughs and vice-versa. Rodríguez Hidalgo & Rankine (2022) also reported that the maximum velocity of EHVO increases with blueshift as seen in our sample. This seems to suggest that the high velocity BAL phenomenon is intimately related to the blueshift of the C iv emission line.
For each pair of epochs, we measured the fractional equivalent width changes in C iv BAL and BEL. The Spearman rank correlation test (with a coefficient of −0.05 and p-value of 0.47) confirms the lack of any correlation between the two quantities. Note that for a couple of individual cases with enough observations, we do find a possible correlation between the two quantities (see Aromal et al. 2021Aromal et al. , 2022. The lack of a similar correlation for the full sample is probably related to the insufficient time sampling that fails to detect the delayed response from the BEL regions. Establishing such a correlation is important to confirm the ionizationinduced BAL variations.
Next, we divide our sample into two around the median BSBEL. Then compared the mean value of ∆W W in the four time-bins considered in this study. We find a significant difference only for the third time-bin, i.e., 2.0-3.5 yrs. Table 2 combined with the fact that BALs with low W show more variability. It is difficult to confirm these trends given the limited size of the sample, but future studies with larger sample sizes and better time sampling can help in reaching stronger conclusions.
Are continuum variations related to BAL variability ?
As mentioned before, we have obtained publicly available light curves to see the trends in continuum variations during our spectroscopic monitoring period. From Section 3.1, it is clear that the variability amplitude strength (VAS) follows the same distribution irrespective of the presence of BALs. This implies that the continuum variations of our UFO BAL quasars are not very different from the general quasar population. However, we also notice that the distribution of VAS in the r-band is different for "highly variable" UFO BALs and the rest of the UFO BALs in our sample (see Fig. 16).
Here, we explore the possible connection between the continuum variability and the variability of C iv equivalent width in our UFO BAL sample. We consider photometric epochs in g-and r-bands which are within 10 days in the observer's frame (roughly 3 days in the rest-frame) to our spectroscopic epochs whenever available and convert them to flux units. Using these, we compare the fractional variations in total g-and r-band flux (Fg and Fr respectively) to that of the rest equivalent width of the C iv BAL. As shown in Fig 22, we see moderate anti-correlation with high significance between ∆Fg Fg and ∆W W with spearmann coefficient, r = −0.48, p-value = 3.50 × 10 −6 and weak anti-correlation with lesser significance between ∆Fr Fr and ∆W W with r = −0.25, pvalue = 1.34 × 10 −2 . In order to demonstrate that this effect is not dominated by the sources having a large number of epochs, we remove sources having more than 8 epochs of spectroscopic observations and carry out the same analysis to find even stronger anti-correlation signatures with r = −0.62, p-value = 1.27 × 10 −5 for g-band and r = −0.34, pvalue = 1.92 × 10 −2 for r-band. While the anti-correlation is statistically significant, we do see the spread in the fractional variation in the continuum flux is smaller than that of C iv rest equivalent width.
The analysis indicates that the continuum flux variations may be responsible for the observed BAL variability where the W of the C iv BAL decreases as the continuum increases. This is in agreement with several other studies. Lu et al. (2018) found the same trend with similar r-coefficient, r=-0.43 with p-value < 1e-44 between the fractional variations of the absorption C iv equivalent width and that of the monochromatic continuum luminosity at 1450Å calculated from the spectra. Mishra et al. (2019) showed that the dimming in the continuum is associated with the appearance of new BALs. Horiuchi et al. (2020) also looked at a few of the SDSS RM sources and found a possible presence of correlation between the BAL variability and photometric variability using iPTF and PanSTARRS surveys.
To ascertain this tendency, we also divided the sources into two sub-samples based on the VAS values in r-band (r-band is preferred over g-band since most BALs coincide with g-band region whereas r-band is little affected by the same) and looked at the average of ∆W W in different time bins similarly to the analysis performed in Section 4.5.3. The results are shown in Fig. 23. It is clear that except for the 4th time bin, sources with large photometric variability also show consistently higher BAL variability up to 3.5 years in rest-frame time scales. Aromal et al. (2022) while analyzing the multiple epoch spectroscopic observations of three BAL components in J1322+0524 found that the amplitude of optical continuum flux variations is much smaller than what is needed to produce the observed equivalent width variations. Based on this and a large scatter in W found for a given continuum flux they argued that C iv ionizing continuum and optical continuum need not vary in a correlated manner and the amplitude of ionizing continuum variation has to be much higher than what we see in the optical light curves. From Fig. 22, we see scatter in the equivalent width fractional variations are much larger than that seen in the continuum. This implies the conclusions drawn in the case of J1322+0524 may still be valid for most of the UFO BALs in our sample. Therefore, it will be interesting to probe the rest frame FUV variability of our UFO BAL quasars.
RESULTS AND DISCUSSIONS
In this work we have presented an analysis of the C iv absorption variability of 80 distinct BAL components observed in the spectra of 64 UFO BAL quasars. We used spectra from SDSS together with our own data from SALT. Our monitoring time-scale spans between a few months to ∼ 7 yrs in the quasar rest frame. Here, we mainly focus on the variability of the C iv rest equivalent width. We will present details of pixel-based optical depth analysis and a detailed study of interesting sources (as in Aromal et al. 2021Aromal et al. , 2022 in future papers. The main results from this study are : 1) BAL variability fraction: We find the C iv absorption in ∼ 95% of UFO BALs vary (by 3σ) at least once during our monitoring. Roughly 33% of quasars in our sample show "highly variable" (i.e., ∆W W >0.67) BAL components. Also, independent of the source and time scale considered, ∼ 80% of epoch pairs show significant variations. These percentages are higher than what is reported for the general BAL population in the literature. For example, when we consider only the pair of observations separated by the shortest time-scale for each object, ∼70% of UFO BAL QSOs and UFO BAL components show significant variability which is a higher occurrence than that found by Filiz Ak et al. (2013). The shortest time intervals probed here being shorter than that of Filiz Ak et al. (2013) one may have expected less variability in our sample. All this indicates that UFO BALs are more variable compared to the normal BAL population.
2) Time dependence of BAL variability: We study the variability of the C iv rest equivalent width in four different time-bins (i.e., 0.1-0.5, 0.5-2.0, 2.0-3.5 and 3.5-7.5 yrs) in the quasar rest frame. The inter quartile range (IQR) of ∆W W increases with increasing time scale which means that the variations become larger with increasing time scale. This result is not influenced by the different time-sampling achieved for different sources and is not related to any particular behavior of systems with low (W < 5Å) or high (W > 55Å) C iv rest equivalent widths. We also note that the fraction of "highly variable" BALs increases with increasing time scale. Such "highly variable" C iv BALs dominate the large variability seen at long time scales. By comparing our data (with appropriate time sampling) with the ∆W W vs. t relationship of Filiz Ak et al. (2013) we show our UFO BALs are more variable compared to the overall BAL population.
A larger fraction of UFO BALs in our sample show a negative ∆W W in the largest time-bin (3.5-7.5 yrs). In addition, using the time evolution of C iv rest equivalent width in 21 BAL components that show large variability over a time-scale > 2 yrs, we find that a decrease in rest equivalent width by a factor of two occurs over a longer time-scale (i.e., a median of ∼3.5 yrs) compared to an increase by the same amount (i.e., a median of ∼2.8 yrs). This is consistent with the finding that the average time scales are higher for BAL disappearance events compared to the emergence events (see, McGraw et al. 2017;Mishra et al. 2019). Confirming our results for the full sample through our ongoing monitoring programme will provide a possible link between the large equivalent width variability and extreme events like emergence/disappearance of BALs.
3) Dependence of BAL variability on quasar properties : We find no significant dependence of BAL variability on quasar properties like MBH , L bol and λ Edd even though a moderate correlation is observed for L bol on time scales less than 2 yrs. This is consistent with what has been found in the overall population of C iv BALs (see for example, Filiz Ak et al. 2013). We do notice that the properties of QSOs in our sample do not probe the same range as what we see in the general population of quasars. This may have limited our ability to detect any weak trend. While some quasars in our sample show large C iv BEL variations insufficient time-sampling related to the expected reverberation timescales prevents us from correlating the C iv BEL and UFO BAL variabilities. In cases where such detailed investigation is possible we do see large BEL variations in quasars that show large UFO BAL variations (see Aromal et al. 2021Aromal et al. , 2022. Spectroscopic monitoring, at shorter time intervals, of a sub-sample of quasars in our sample that show significant C iv BEL variations would be very useful.
We find that on an average the optical photometric variability of quasars in our sample are not statistically distinguishable from that of our control sample. However, we do find an anti-correlation between the optical continuum flux variations (in both g-and r-band) and variations in C iv BAL equivalent width. In addition, the scatter in ∆W W is more for objects showing VAS value larger than the median of our sample compared to the rest of the objects. All these are consistent with the absorption variability being somehow linked to the continuum flux variations. However, the spread in the fractional flux variations is much smaller than that of W . In a simple photo-ionization scenario this would imply much larger amplitude for the variability of the C iv ionizing flux compared to that in the rest frame NUV range probed by the optical variability. It will be interesting to check the relationship between the variability NUV flux and C iv ionizing flux using direct measurements 4) Dependence of BAL variability on BAL properties : By studying the variability nature of different sub-samples based on the C iv absorption properties, we find that weak, high-velocity, shallow, and low-width BALs tend to show more variability than others. Similarly the "highly variable" BAL components tend to be shallow having large velocities and narrow widths. We find that detached C iv BAL components with large ejection velocities show larger equivalent width variations. In general C iv equivalent width variations are by and large correlated across the velocity profile. However, the amplitude of ∆W W is larger in the case of high velocity components compared to their low-velocity counterparts. Based on this we suggest that both low-equivalent width and high-velocity are equally important to determine the strength of variability in a BAL.
5)
Reconciling with physical models? : Proga et al. (2000); Proga & Kallman (2004) use axisymmetric, time-dependent hydrodynamical simulations of radiation-driven disk winds in AGN to conclude that radiative line driving is efficient enough to accelerate disk winds to high velocities as seen in BALs. However the variable C iv absorption produced by such simulated disk winds depends on a complex velocity field and covering factor (Proga et al. 2012). We note that the simulated C iv absorption profiles in their study show high variability, especially at high velocities. They attribute this excess of variability at large velocities to the emergence of very fast mass ejections from relatively large distances, where the gas is well shielded from X-ray radiation.
The best way to constrain such models is to compare the gas density, covering factor and distance of the absorbing cloud from the continuum source in the model with the constrains coming from observation. For this we need high resolution spectroscopic observations of our objects covering a larger rest wavelength range (Srianand & Petitjean 2000). Alternatively, it will be interesting to ask whether observed properties like (i) the ∆W W vs. t relationship, (ii) the timescale of decreasing equivalent width being higher than that of increasing equivalent width, (iii) the lack of strong correlations with quasar properties, (iv) the connection between BEL blueshift and C iv absorption line variability, (v) the observation of large equivalent variability when there is no large optical continuum variations and (vi) the importance of low equivalent width and high velocity to produce high amplitude variation etc., can be reproduced by hydrodynamical simulations. Additional constrains for models can be obtained from pixed optical depth analysis. We will be presenting results of such an analysis for our sample in our upcoming paper.
APPENDIX A: ESTIMATION OF QUASAR PROPERTIES
In this section, we explain how we estimate different quasar properties and compare the same to Shen et al. (2011). We obtain MBH using the FWHM of the C iv emission measured from the spectrum using the empirical mass-scaling relationship given by Vestergaard & Peterson (2006) and the bolometric luminosity using bolometric correction to the monochromatic luminosity at 1350Å derived from the composite SED from Richards et al. (2006). Note that for these measurements, we used the SDSS spectrum with the highest SNR available for each object. Now, we compare our estimated values with that of Shen et al. (2011) for both MBH and L bol for 38 sources in our UFO sample and 20 non-BAL quasars from our control sample as shown in Fig A1. It is clear that Shen et al. (2011) tends to over-predict MBH and under-predict L bol for BAL quasars compared to their non-BAL counterparts as shown by the shift in respective distributions. We find that this may be due to the continuum fits used in Shen et al. (2011) which uses conventional line-free regions for the fitting that may still include BAL regions, leading to incorrect estimation of quasar properties. This mostly lead to overestimation C iv BEL FWHM and underestimation of L1350 which leads to similar probelms in the calculation of MBH and L bol respectively. Hence, we did a detailed visual inspection to select line-free regions for each source in our sample and used PyQSOFit to calculate the quasar properties mentioned above.
APPENDIX B: UFO BAL SAMPLE
This paper has been typeset from a T E X/L A T E X file prepared by the author.
Based on observations collected at Southern African Large Telescope (SALT; Programme IDs 2015-1-SCI-005, 2018-1-SCI-009, 2019-1-SCI-019 and 2020-1-SCI-011) and the European Organisation for Astronomical Research in the Southern Hemisphere under ESO programme 093.A-0255.
Figure 2 .
2M BH vs. λ Edd for the UFO BAL (yellow points and curves) and non-BAL control sample (green points and curves). A strong anti-correlation is evident for both samples. Top and right-hand side panels: PDFs of M BH and λ Edd respectively for the two samples. Dashed lines indicate the median values.
Figure 3 .Figure 4 .
34Top panel: Distribution of total number of epochs (combining both SDSS and SALT observations) probed for each UFO BAL (blue histogram) with the median of the distribution marked with a red dashed line. Middle panel: Distribution of minimum sampled rest-frame time scales, ∆t min , for the UFO sample (blue histogram) along with the same distribution for the SDSS sample studied by Filiz Ak et al. (2013) (orange histogram). Bottom panel: Time-scales probed by all possible combinations of SDSS and SALT epochs (blue) and only SDSS epochs (red) for each individual source in the sample.probe both the short-and long-time scale variability of the UFO BAL sample. Our SALT observations, in addition to the SDSS epochs, have brought great improvement to the time-sampling of these UFO BAL sources as demonstrated inFig 3.The top panel shows the distribution of the total number of spectroscopic epochs for each source with a median value of 4. For 62 out of 64 sources, at least one epoch was added by SALT observations. The middle panel shows the fraction of objects in different minimum sampled rest-frame time scale (∆tmin). It can be seen that roughly 70 % of the sources have a minimum separation between the spectroscopic epochs less than 0.5 yr. This is mostly because of the multiple SALT observations performed within a rest frame year which is crucial for characterizing the short-time scale variability. This is a great improvement compared to the study by FilizAk et al. (2013) which had only ∼20% of such sources (orange histogram in the figure). The bottom panel of Fig 3 shows the histogram of all possible time-scales as probed by both SALT and SDSS (blue) and SDSS alone (red). Again this clearly shows the improvement brought by SALT epochs at different time-scales including ∆t > 5.5 yr. Comparison of variability amplitude strength (VAS) between non-BAL (red) and UFO BAL (blue) samples. Results for the g-and r-bands (VASg and VASr respectively) are shown in the top and bottom panels respectively. Vertical dashed lines indicate the median values.
Figure 5
5Figure 5. Comparison between C iv BEL blue-shifts in spectra from the non-BAL (blue) and UFO QSO BAL (orange) samples.
Figure 6 .
6Fraction of sources showing C iv absorption variability larger than a certain threshold value of ∆W W vs the threshold value ∆W W . Vertical dashed lines mark ∆W W corresponding to equivalent with variations by a factor 2 and 3 respectively. Roughly 33% of quasars in our sample have UFO BALs showing W changing by a factor of 2 during our monitoring period. cient number of measurements, C iv BALs tend to show significant equivalent width variability at all time scales. Next, we consider the fraction of epoch pairs in a given time bin that is showing significant equivalent width variability. For our UFO BAL sample we find 0.78±0.08, 0.81±0.05, 0.85±0.05, 0.89±0.07 for the time intervals 0.1−0.5, 0.5−2.0, 2.0−3.5, >3.5 yrs respectively. This once again confirms the highly variable nature of BAL troughs in our UFO BAL sample. When we repeat the same exercise for the full sample of Hemler et al. (2019) we find 0.27±0.
Figure 7 .
7Top panel : This figure shows the ∆W W distribution for the full sample as explained in Section 4.2 along with the histogram (right panel). Horizontal dotted lines mark ∆W W = 0, ±0.67 (i.e a factor 2 variation in W CIV . Middle panel : This figure shows the ∆W distribution for the full sample with histogram (right panel). In both panels the red curves are the best fit relationship obtained by Filiz Ak et al. (2013). Bottom panel : This figure shows the ∆W W vs ∆W distribution for the full sample where horizontal dotted lines mark ∆W W = 0, ±0.67.
Figure 8 .
8In the top three panels, we show the mean and sigma of quantiles at 0.25, 0.5 and 0.75 of ∆W W distribution using all the points in ∆W W distribution (panel a), using method I (panel b) and method II (panel c) as described in Section 4.2, for four different time bins. In panel d, we show the IQR of ∆W W distribution as calculated by different methods using quantiles shown in the top 3 panels. In each panel, the vertical red dashed lines mark the four ∆t bins used for the analysis.
Figure 9 .
9This figure shows the ∆W W as a function of time for UFO (blue) and non-UFO (green) sources. For comparison, points from our UFO sample (red) is also shown. and compute the quantiles at 0.25, 0.50, and 0.75 of the ∆W W distribution for each time bin considering all points in the ∆W W distribution shown inFig. 7. The quantiles and the IQR estimated from these are shown in panels (a) and (d) ofFig. 8,respectively, for the four time bins. We observe an increasing trend in IQR with increasing time intervals with values of 0.23 +0.04 −0.04 , 0.20 +0.07 −0.06 , 0.41 +0.04 −0.05 , and 0.55 +0.04 −0.05
Figure 12 .
12The average ∆W W distribution of different sub-samples (Wmax in panel a, optical depth weighted velocity centroid in panel b, maximum depth in panel c, and velocity width in panel d) defined below and above the median (see section 4.5.1) of different BAL properties for different time bins. Errors correspond to the standard deviation. The number of UFO BALs used for each measurement is also indicated in the figure. median value. The median values are 21.2Å, 16434 km s −1 , 0.71 and 13939 km s −1 for maxima of W , vcent, dBAL and ∆v of the BAL respectively.
Figure 13 .
13All the measured ∆W W for each BAL component as a function of their properties for the time bin, 2.0 ∆t 3.5 yrs. The vertical dashed line in each panel indicates the median abscissa value. The horizontal line corresponds to ∆W W =0.67.
Figure 14 .Figure 15 .
1415Edd ) < median log(λ Edd ) > median Same as Fig. 12 for M BH (panel a), L bol (panel b), and λ Edd (panel c). ∆W W is shown for each UFO BAL as a function of L bol (top panel) and λ Edd (bottom panel) for the time bin 0.5< ∆t <2.0 yrs. The vertical dashed line indicates the median value of the quantity in the abscissa and the horizontal line indicates ∆W W = 0.68.
Figure 16 .
16Distribution of quasar (left panels) and BAL (right panels) properties of "highly variable" UFO BALs (shown in yellow) and the rest of the UFO sample (gray). Vertical dashed lines indicate the median values for each histogram.
Figure 17 .
17Maximum W (Wmax) as a function of maximum vcent (vcent,max) for all the UFO BALs in our sample. The class of each UFO BAL is indicated by the color as shown in the legend and the size of the circle scales with the maximum ∆W W observed. The red dashed lines show the median along the respective axis and the fraction of sources with max( ∆W W ) > 0.67 is indicated in green boxes for the corresponding regions of the plane.
Figure 18 .
18This figure shows the ∆W W vs time for UFOs in Class 1 (top panel), Class 2 (middle panel), and Class 3 (bottom panel) sources. The panels in the right side show the histogram distribution of ∆W W . Horizontal dashed lines mark ∆W W = 0 and ±0.67. It is evident that Class-1 and Class-2 UFO BALs show larger scatter compared to that of Class-3.
Figure 19 .Figure 20 .
1920Top panel: Comparison of ∆W W of low-velocity non-UFO BAL troughs to that of UFO BALs in Class-2 sources. The points contributed by J1322+0524 (cyan) are shown separately. The blue line is the best-fitted linear regression line. The equality line is shown by the orange dotted line. Bottom panel: The ∆W W distribution of non-UFO (red) and UFO BALs (blue) in Class-2 sources as a function of time. Horizontal dashed lines mark ∆W Top panel: ∆W W of low-velocity vs. ∆W W of high-velocity regions in quasars belonging to Class-3. The blue line shows the best-fitted linear regression line. The equality line is indicated by the orange dotted line. Bottom panel: ∆W W of low-and highvelocity regions of BAL troughs belonging to Class-3 as a function of time. Horizontal dashed lines mark ∆W W = 0 and ±0.67.
For this time bin, we looked at the ∆W W distribution as a function of BSBEL as shown in Fig 21. It is clear from the figure that the sources with less blueshift tend to show larger scatter in the ∆W W distribution. The Spearman rank correlation test suggests a possible anti-correlation with a coefficient of −0.25 and a p-value of <0.001. This may be explained as the result of a moderate correlation of Wmax with blueshift from
Figure 21 .Figure 22 .
2122∆W W is shown for each UFO BAL source as a function of blueshift for the time bin 2.0< ∆t <3.5 yrs. The vertical dashed line indicates the median value of the quantity in the abscissa and the horizontal line indicates ∆W W = 0.68. In this figure, we compare the fractional variations in total g-and r-band flux to that of the BAL EW in the top and bottom panels respectively.
Figure 23 .
23We plot the mean and standard deviation of the average ∆W W distribution at the four different time-scales, i.e. short (0-0.5 yr), intermediate (0.5-2 yr) and two long ( 2-3.5 yr and >3.5 yr) time-scales, for the two sub-samples having estimated VAS of the r-band light curve below and above its median.
Figure A1 .
A1In this figure, we show the distribution of differences in the estimated values of M BH (∆M BH ) and L bol (∆L Bol ) betweenShen et al. (2011) and this study for 38 UFO BAL sources and 20 non-BAL sources from the control sample.
few sources 1 https://github.com/legolason/PyQSOFit0
10000
20000
30000
20
40
60
80 CLASS 1
0
20
40
60
80
100
120 CLASS 2
1300 1350 1400 1450 1500 1550 1600 1650
0
20
40
60
80
100 CLASS 3
Velocity (kms 1 )
Rest wavelength(Å)
Flux (10 17
erg/cm 2
/s/Å)
Rest wavelength(Å)
Flux (10 17
erg/cm 2
/s/Å)
Rest wavelength(Å)
Flux (10 17
erg/cm 2
/s/Å)
Table 1 .
1Details of quasars with UFOs in our sampleQSO
zem
z abs
v min
vmax
BI
Class log(M BH ) log(L Bol ) log(λ Edd ) Number
∆t (yr)
(km s −1 ) (km s −1 ) (km s −1 )
(M )
(erg s −1 )
of
(min,max)
epochs
(1)
(2)
(3)
(4)
(5)
(6)
(7)
(8)
(9)
(10)
(11)
(12)
J0028-0539
2.5584 2.36
10378
25096
4138
1
9.50
46.98
-0.62
3
0.53, 2.44
J0034+0309 2.3329 2.14
10799
26307
4442
1
9.46
46.88
-0.68
4
0.29, 3.18
J0046+0104 2.1492 2.01
10202
19450
4649
2
9.21
47.30
-0.01
13
0.00, 6.64
J0052+0909 2.6625 2.43
10672
30000
8866
1
9.41
47.05
-0.46
3
0.29, 2.67
J0054+0027 2.5142 2.42
4426
11048
1249
2
9.48
47.16
-0.42
8
0.03, 5.96
2.31
12220
25899
1246
2
9.48
47.16
-0.42
8
0.03, 5.96
J0138+0124 2.5441 2.42
3000
28896
15374
3
9.21
47.19
-0.12
4
0.24, 3.04
J0152+0929 2.1794 2.04
6840
26925
5516
3
9.31
46.93
-0.48
3
0.32, 3.33
J0200-0037
2.1422 1.99
6846
29777
12261
3
9.29
47.11
-0.27
7
0.70, 6.64
J0216+0115 2.2310 2.19
3191
6227
1337
999
8.98
47.02
-0.06
6
0.31, 6.79
2.02
7538
29943
6501
999
8.98
47.02
-0.06
6
0.31, 6.79
J0220-0812
2.0095 1.86
9509
29709
6129
1
9.70
47.11
-0.69
5
0.35, 7.00
J0224-0528
2.0845 1.91
14234
20623
1473
1
9.85
47.47
-0.48
3
0.34, 3.19
1.85
22059
27995
455
1
9.85
47.47
-0.48
3
0.34, 3.19
J0229-0034
2.1382 2.03
3083
28332
13129
3
9.39
47.02
-0.48
14
0.00, 2.55
J0242+0049 2.0573 1.88
10573
26859
3059
2
9.91
47.09
-0.93
9
0.02, 6.85
J0244-0108
3.9856 3.68
10283
27536
5335
1
9.96
47.47
-0.59
6
0.11, 4.37
J0814-0004
2.5936 2.37
10636
29162
2830
1
9.62
47.32
-0.40
4
0.28, 4.39
J0817+0717 2.4428 2.33
4641
18133
6032
3
9.30
47.33
-0.06
2
2.63, 2.63
J0831+0354 2.0761 1.95
8823
27670
6475
1
9.41
46.94
-0.57
3
2.67, 6.18
J0837+0521 2.3624 2.31
3000
7547
1647
2
9.60
47.12
-0.58
3
2.62, 5.34
2.10
22073
28994
1502
2
9.60
47.12
-0.58
3
2.62, 5.34
J0845+0812 2.3545 2.13
15183
27269
3778
1
9.38
47.08
-0.40
3
0.04, 2.44
J0911+0550 2.7933 2.58
12300
21914
1641
1
9.66
47.52
-0.24
3
1.14, 3.48
J0924-0128
2.4461 2.38
4427
7118
1764
2
8.91
47.16
0.14
2
3.15, 3.15
2.22
9527
28598
4602
2
8.91
47.16
0.14
2
3.15, 3.15
J0932+0237 2.1679 1.96
13042
30000
7982
1
9.50
47.09
-0.51
4
0.27, 6.59
J0951-0157
3.2553 2.97
14909
27271
3953
1
9.73
47.24
-0.59
3
0.26, 2.62
J1006+0119 2.3030 2.16
6674
24194
8743
3
9.39
47.22
-0.27
2
2.97, 2.97
J1007+0304 2.1245 1.92
15770
24885
1986
1
9.44
47.05
-0.49
6
0.30, 6.13
J1054+0150 2.2366 2.08
13434
17152
1978
1
9.43
46.85
-0.68
4
0.30, 6.14
2.01
19284
23542
784
1
9.43
46.85
-0.68
4
0.30, 6.14
J1110-0140
2.8192 2.63
8252
24514
6785
1
9.80
47.29
-0.61
3
0.00, 2.88
J1116+0808 3.2429 2.97
16170
24210
663
1
9.29
47.40
0.01
2
2.04, 2.04
J1135-0240
2.4611 2.23
15141
25096
1281
1
9.51
47.05
-0.55
3
0.28, 3.47
J1143+0912 2.3253 2.20
5665
16556
3775
3
9.32
46.92
-0.49
2
2.69, 2.69
J1156+0856 2.1077 1.94
6919
30000
12496
3
8.92
46.75
-0.27
3
2.84, 6.13
J1205+0134 2.1523 1.96
12909
29458
6180
1
9.57
47.09
-0.59
5
0.32, 6.28
J1208+0355 2.0210 1.95
4995
9738
1286
2
9.40
46.88
-0.62
3
2.95, 6.23
1.82
12972
26326
2808
2
9.40
46.88
-0.62
3
2.95, 6.23
J1215-0034
2.6987 2.54
4151
22790
6568
3
9.94
47.64
-0.40
3
2.46, 5.63
J1301+0314 2.1115 1.91
16175
25380
1656
1
9.58
47.05
-0.63
3
3.22, 6.75
J1317+0100 2.6961 2.59
3816
18015
5987
3
9.17
47.32
0.05
6
0.01, 6.01
2.45
19596
25568
427
3
9.17
47.32
0.05
6
0.01, 6.01
J1331+0042 2.4341 2.35
6555
9453
498
2
9.37
47.09
-0.38
4
0.01, 3.16
2.27
11383
16893
468
2
9.37
47.09
-0.38
4
0.01, 3.16
J1341-0115
2.7682 2.55
8502
25835
7132
1
9.72
47.24
-0.59
3
2.33, 5.02
J1343+0351 2.8686 2.68
12032
19532
2192
1
9.92
47.23
-0.79
3
0.25, 2.81
J1350+0843 2.6157 2.44
13629
17632
953
1
9.72
47.30
-0.52
3
0.25, 2.77
J1357+0055 2.0081 1.80
11851
30000
2757
1
9.27
47.32
-0.05
4
0.27, 7.26
J1359+0310 2.6778 2.52
8662
19833
4439
1
9.49
47.28
-0.31
2
2.70, 2.70
J1400+0507 2.3015 2.16
7038
25561
7335
3
9.34
47.01
-0.43
3
0.29, 3.04
J1405+0229 2.8266 2.67
6487
24861
7057
3
9.43
47.25
-0.28
4
0.25, 5.52
J1424+0441 2.2762 2.05
11006
28320
2923
1
9.44
47.12
-0.42
4
0.29, 3.39
J1445-0023
2.2296 2.05
7822
27489
8866
3
9.68
47.40
-0.39
4
0.02, 4.04
J1452+0932 2.4607 2.29
5694
28047
8540
3
9.40
47.24
-0.26
4
0.31, 4.65
J1500+0033 2.4394 2.23
9914
27649
4574
1
9.88
47.27
-0.70
6
0.30, 6.22
J1547+0606 2.0188 1.95
5432
9467
2587
2
9.64
47.27
-0.48
3
2.31, 5.61
1.87
9984
24909
7467
2
9.64
47.27
-0.48
3
2.31, 5.61
J1606+0718 2.0766 1.99
3835
22135
6819
3
9.46
46.90
-0.66
4
0.34, 5.24
J1606+0746 2.3687 2.20
8414
23723
2005
1
9.72
47.08
-0.74
3
0.29, 2.74
J1609+0526 2.3802 2.27
4115
26115
9608
3
9.29
47.10
-0.29
4
0.28, 4.07
In the three bins, the percentage of variable sources are 89 +16 −14 %, 100 +21 −17 % and 94 +14 −12 % and the percentage of variable BALs are 84 +14 −12 %, 97 +18 −15 % and 94 +12 −10 %. In the case of the Hemler et al. (2019) observations, we find that the variable fraction of sources are 0.96 and 0.94 and the variable fraction of BALs are 0.92 and 0.90 for the short and intermediate time bins respectively. Thus it appears that, with a suffi-0.2
0.4
0.6
0.8
1.0
1.2
1.4
| ∆W
W |
0.0
0.2
0.4
0.6
0.8
1.0
Fraction of sources
Table 2 .
2Dependence of BAL properties on properties of C iv broad emission lineSample
Spearmann coefficient, p value
Wmax
vcent,max
vmax
d BAL,max
∆vmax
BS BEL
0.340, 0.021
0.093,0.538
0.351, 0.016
0.336, 0.022
0.272, 0.068
max( ∆W BEL
W BEL
) -0.447, 0.005 0.041, 0.805 -0.151, 0.364 -0.392, 0.015 -0.417, 0.009
0
1
2
3
4
5
6
7
∆t(yr)
0.1
0.2
0.3
0.4
| ∆W
W
|
VAS r < median
VAS r > median
Table B1 .
B1Log of sources, observations, details of spectra obtained at different epochsQSO
Telescope
Date
MJD
Exposure Wavelength
Spectral
S/N b
(M/D/Y)
time (s)
range (Å)
res. (km s −1 )
J002809.45-053941.6
SDSS
10-06-2013 56571
4500
3562-10341
150
16.52
SALT
07-21-2020 59051
2400
4461-7524
304
33.54
SALT
06-20-2021 59385
2400
3321-6430
304
5.01
SALT
06-08-2022 59738
2400
3312-6426
304
22.42
J003419.19+030913.1
SDSS
12-01-2010 55531
3603
3589-10363
150
19.46
SDSS
11-20-2017 58077
4500
3596-10329
150
20.87
SALT
07-16-2020 59046
2400
4178-7252
304
35.7
SALT
07-06-2021 59401
2450
3189-6296
304
36.43
J004613.54+010425.7
SDSS
09-07-2000 51794
3600
3802-9193
150
19.26
SDSS
10-17-2001 52199
3602
3801-9215
150
23.67
SDSS
12-21-2009 55186
5402
3578-10353
150
27.49
SDSS
01-05-2010 55201
7201
3587-10360
150
33.76
SDSS
09-12-2010 55451
5404
3590-10396
150
31.61
SALT
06-20-2015 57193
1800
3467-6566
304
19.34
SALT
07-27-2015 57230
1800
3462-6564
304
38.89
SDSS
09-13-2015 57278
4500
3576-10334
150
28.48
SALT
09-16-2015 57281
1800
3471-6566
304
55.92
SDSS
12-09-2017 58096
4500
3587-10360
150
17.12
SALT
08-18-2018 58348
1300
3464-6567
304
41.93
SALT
07-04-2019 58668
1350
3466-6569
304
34.61
SALT
08-01-2021 59427
1913
3482-6568
304
24.91
J005251.81+090945.2
SDSS
10-21-2011 55855
2702
3557-10346
150
14.08
SALT
07-16-2020 59046
2300
4601-7655
304
28.96
SALT
08-03-2021 59429
2300
3468-6566
304
12.12
J005419.98+002727.9
SDSS
09-25-2000 51812
4500
3787-9191
150
14.97
SDSS
11-28-2000 51876
5100
3808-9215
150
14.02
SDSS
01-04-2001 51913
13500
3806-9212
150
25.89
SDSS
11-30-2008 54800
10203
3800-9202
150
23.17
SDSS
10-12-2010 55481
4504
3591-10401
150
24.01
SDSS
09-14-2015 57279
4500
3620-10392
150
25.74
SDSS
12-09-2017 58096
4500
3598-10332
150
18.12
SALT
08-30-2021 59456
2230
3475-6571
304
23.8
J013802.07+012424.4
SDSS
11-01-2010 55501
5405
3566-10327
150
23.48
SDSS
01-07-2018 58125
3600
3583-10363
150
20.37
SALT
09-20-2020 59112
2150
4464-7526
304
42.59
SALT
08-02-2021 59428
2400
3484-6568
304
40.98
J015244.15+092950.1
SDSS
01-03-2011 55564
6305
3552-10303
150
31.54
SALT
07-28-2020 59058
2300
3896-6979
304
42.53
SALT
08-03-2021 59429
2300
3339-6433
304
28.74
SALT
09-16-2022 59838
2300
3356-6479
304
11.3
J020006.31-003709.7
SDSS
11-23-2000 51871
2700
3805-9204
150
16.96
SDSS
11-04-2005 53678
6900
3804-9200
150
26.52
SDSS
01-13-2008 54478
12005
3801-9176
150
30.83
SDSS
09-10-2010 55449
5404
3566-10322
150
25.73
SDSS
11-26-2014 56987
6300
3565-10337
150
26.06
SDSS
11-23-2017 58080
3600
3597-10327
150
24.53
SALT
09-29-2021 59486
2400
3895-6979
304
29.81
J021606.40+011509.5
SDSS
09-29-2000 51816
2700
3795-9183
150
11.44
SDSS
10-15-2010 55484
3603
3567-10329
150
22.52
SDSS
01-21-2015 57043
5400
3566-10337
150
20.75
SDSS
12-12-2017 58099
5400
3598-10334
150
19.67
SALT
08-30-2021 59456
2400
4038-7114
304
26.49
SALT
08-27-2022 59818
2450
4037-7111
304
27.37
J022036.27-081242.9
SDSS
09-10-2001 52162
4501
3806-9204
150
18.3
SDSS
12-21-2008 54821
10801
3805-9215
150
31.86
SDSS
09-29-2011 55833
5405
3591-10387
150
25.65
QSO
Telescope
Date
MJD
Exposure Wavelength
Spectral
S/N b
(M/D/Y)
time (s)
range (Å)
res. (km s −1 )
SALT
09-12-2021 59469
2400
3613-6706
304
37.2
SALT
09-27-2022 59849
2400
3603-6703
304
48.38
J022431.59-052818.9
SDSS
10-17-2012 56217
4504
3561-10332
150
38.56
SALT
08-07-2021 59433
2200
3755-6843
304
73.21
SALT
08-21-2022 59812
2200
3754-6841
304
62.41
J022943.90-003458.0
SDSS
12-14-2009 55179
9903
3553-10308
150
29.72
SDSS
12-16-2009 55181
7204
3561-10322
150
23.62
SDSS
01-12-2010 55208
14406
3558-10325
150
25.84
SDSS
02-14-2010 55241
7203
3565-10322
150
26.23
SDSS
09-06-2010 55445
5404
3566-10334
150
31.12
SDSS
10-07-2010 55476
7206
3568-10327
150
22.84
SDSS
09-23-2011 55827
6305
3555-10344
150
25.25
SDSS
10-22-2011 55856
3603
3553-10346
150
27.5
SDSS
01-19-2012 55945
6305
3554-10344
150
24.52
SDSS
10-19-2012 56219
5404
3565-10344
150
27.09
SDSS
10-19-2012 56219
4504
3566-10344
150
23.27
SDSS
09-09-2013 56544
7200
3566-10351
150
33.63
SDSS
10-03-2013 56568
2700
3566-10351
150
19.72
SDSS
10-31-2013 56596
6300
3563-10349
150
22.46
SDSS
12-09-2017 58096
4500
3597-10329
150
21.66
J024221.87+004912.7
SDSS
10-04-2000 51821
3600
3797-9183
150
18.13
SDSS
09-25-2001 52177
3601
3823-9215
150
17.67
SDSS
10-17-2001 52199
2701
3820-9202
150
16.83
SDSS
02-20-2010 55247
6301
3580-10346
150
30.25
SDSS
09-16-2010 55455
4504
3594-10380
150
27.16
SDSS
09-14-2015 57279
2700
3576-10334
150
21.93
SDSS
01-05-2017 57758
2700
3572-10334
150
10.89
SDSS
11-16-2017 58073
5400
3598-10325
150
27.23
SALT
09-11-2021 59468
2400
3757-6844
304
50.11
J024457.18-010809.8
SDSS
11-23-2000 51871
2700
3818-9189
150
10.82
SDSS
02-20-2010 55247
6301
3564-10318
150
19.6
SDSS
09-16-2010 55455
4504
3566-10337
150
10.53
SDSS
01-19-2015 57041
7200
3568-10337
150
19.15
SDSS
12-21-2016 57743
5400
3604-10334
150
20.96
SALT
09-07-2022 59829
2400
5863-8855
304
19.21
J081417.55-000455.0
SDSS
03-22-2006 53816
6900
3801-9191
150
19.04
SDSS
11-24-2011 55889
3603
3558-10346
150
24.15
SALT
12-25-2020 59208
2500
3474-6572
304
59.11
SALT
12-31-2021 59579
2500
3465-6568
304
48.11
J081737.08+071723.9
SDSS
11-30-2011 55895
5405
3582-10356
150
26.86
SALT
12-23-2020 59206
2400
3311-6421
304
83.16
J083126.15+035408.1
SDSS
01-07-2003 52646
4200
3817-9183
150
15.31
SDSS
03-26-2011 55646
4504
3551-10337
150
23.78
SALT
01-07-2022 59586
2400
3754-6842
304
29.06
J083745.74+052109.4
SDSS
03-10-2003 52708
2400
3796-9189
150
15.22
SDSS
01-01-2012 55927
2702
3560-10346
150
19.78
SALT
02-19-2021 59264
2400
4180-7252
304
58.73
J084502.73+081214.2
SDSS
11-21-2003 52964
2240
3808-9227
150
12.19
SDSS
11-28-2011 55893
7206
3581-10384
150
28.26
SDSS
01-20-2012 55946
3603
3604-10365
150
22.3
J091127.61+055054.1
SDSS
01-13-2003 52652
3150
3804-9215
150
25.41
SDSS
12-01-2011 55896
3603
3557-10332
150
22.52
SALT
03-29-2016 57476
1900
4554-6602
304
28.18
J092437.70-012844.2
SDSS
02-16-2010 55243
9001
3583-10358
150
32.02
SALT
12-25-2020 59208
2400
3285-6428
304
43.77
QSO
Telescope
Date
MJD
Exposure Wavelength
Spectral
S/N b
(M/D/Y)
time (s)
range (Å)
res. (km s −1 )
J133138.49+004221.1
SDSS
04-28-2000 51662
2700
3813-9234
150
9.03
SDSS
02-15-2001 51955
6300
3811-9215
150
12.53
SDSS
02-13-2011 55605
3603
3595-10363
150
12.1
SDSS
03-02-2011 55622
4504
3557-10320
150
10.73
J134112.37-011545.6
SDSS
06-02-2002 52427
3003
3816-9212
150
18.34
SDSS
03-10-2011 55630
3603
3558-10327
150
18.49
SALT
04-23-2021 59327
2307
3612-6705
304
33.64
J134320.72+035148.5
SDSS
04-08-2011 55659
3603
3558-10332
150
17.71
SALT
03-07-2021 59280
2400
3898-6980
304
50.5
SALT
02-23-2022 59633
2400
3897-6979
304
26.76
J135031.73+084354.7
SDSS
02-21-2012 55978
3603
3605-10392
150
22.74
SALT
04-02-2021 59306
2300
3617-6708
304
6.51
SALT
02-24-2022 59634
2350
3613-6705
304
19.63
J135721.77+005501.1
SDSS
04-07-2000 51641
3600
3802-9215
150
17.21
SDSS
02-02-2001 51942
3600
3812-9234
150
19.8
SDSS
03-11-2011 55631
4504
3594-10394
150
29.29
SALT
02-07-2022 59617
2400
3610-6705
304
34.08
J135955.65+031025.9
SDSS
04-26-2011 55677
7206
3556-10329
150
19.09
SALT
03-25-2021 59298
2400
3612-6707
304
48.81
J140053.02+050708.8
SDSS
04-26-2011 55677
7206
3593-10392
150
19.41
SALT
05-14-2020 58983
2400
4178-7253
304
23.91
SALT
05-02-2021 59336
2400
3036-6159
304
24.8
J140532.90+022957.3
SDSS
03-25-2001 51993
2702
3801-9219
150
16.06
SDSS
05-16-2010 55332
3603
3590-10389
150
21.12
SALT
05-19-2021 59353
2400
3751-6842
304
42.15
SALT
05-05-2022 59704
2450
3757-6847
304
55.47
J142405.57+044105.5
SDSS
04-02-2011 55653
2702
3597-10392
150
23.39
SALT
05-23-2020 58992
2400
4037-7117
304
56.26
SALT
05-03-2021 59337
2400
3046-6162
304
52.1
SALT
05-05-2022 59704
2450
3038-6152
304
54.02
J144514.86-002358.1
SDSS
06-09-2008 54626
14504
3809-9179
150
48.75
SDSS
07-06-2008 54653
13506
3808-9176
150
31.09
SDSS
06-05-2010 55352
5404
3574-10313
150
35.13
SALT
06-19-2021 59384
2400
4039-7115
304
58.95
J145229.08+093204.9
SDSS
04-02-2006 53827
4900
3808-9204
150
15.0
SDSS
02-29-2012 55986
3603
3599-10389
150
19.78
SALT
04-04-2021 59308
2300
3315-6426
304
39.04
SALT
05-04-2022 59703
2320
3304-6425
304
40.49
J150033.53+003353.6
SDSS
03-13-2000 51616
1800
3805-9212
150
12.86
SDSS
03-22-2001 51990
2701
3811-9219
150
12.92
SDSS
03-12-2011 55632
5405
3595-10353
150
26.79
SALT
05-13-2019 58616
1300
4323-7385
304
30.06
SALT
07-22-2020 59052
2300
4324-7385
304
47.61
SALT
08-03-2021 59429
2300
4324-7386
304
40.08
J154757.71+060626.6
SDSS
06-11-2004 53167
3300
3799-9204
150
26.15
SDSS
05-26-2011 55707
3603
3595-10387
150
34.12
SALT
05-14-2021 59348
2200
3748-6842
304
83.34
J160622.00+071849.9
SDSS
05-08-2005 53498
3000
3817-9189
150
14.54
SDSS
04-01-2012 56018
2702
3564-10341
150
20.25
SALT
05-28-2020 58997
2400
3754-6842
304
28.99
SALT
06-19-2021 59384
2400
3315-6430
304
19.91
QSO
Telescope
Date
MJD
Exposure Wavelength
Spectral
S/N b
(M/D/Y)
time (s)
range (Å)
res. (km s −1 )
J231027.23+074658.2
SDSS
09-17-2012 56187
4504
3611-10392
150
23.26
SALT
07-25-2020 59055
2300
4178-7252
304
28.78
SALT
07-18-2021 59413
2300
3175-6290
304
37.9
SALT
06-27-2022 59757
2300
3160-6283
304
29.01
J235238.08+010552.3
SDSS
09-01-2000 51788
1800
3794-9196
150
19.94
SDSS
10-02-2010 55471
7206
3595-10387
150
35.06
SDSS
10-29-2014 56959
12601
3609-10392
150
65.06
SDSS
10-21-2016 57682
7200
3571-10332
150
37.48
SDSS
11-21-2016 57713
6300
3609-10334
150
42.64
SDSS
11-24-2017 58081
4500
3598-10332
150
34.49
SALT
06-19-2021 59384
2400
3613-6706
304
62.33
SALT
06-26-2022 59756
2400
3615-6705
304
54.81
J235507.36-035709.6
SDSS
09-30-2013 56565
3600
3611-10387
150
17.51
SALT
07-29-2020 59059
2400
4463-7523
304
37.82
SALT
07-07-2021 59402
2400
3305-6432
304
59.01
J132216.25+052446.3
SDSS
04-12-2002 52376
2402
3822-9202
150
13.34
SDSS
03-13-2011 55633
7206
3595-10394
150
26.43
SDSS
05-22-2011 55703
4504
3559-10329
150
26.11
SALT
06-21-2015 57194
2400
3210-6290
304
29.47
SALT
04-15-2016 57493
2400
3136-6282
304
22.87
SALT
05-07-2018 58245
2000
3166-6284
304
15.62
SALT
06-09-2019 58643
2400
3172-6284
304
19.94
SALT
03-23-2020 58931
1200
3177-6290
304
35.68
SALT
02-17-2021 59262
2500
3171-6291
304
42.37
SALT
05-10-2021 59344
2500
3177-6290
304
53.46
SALT
06-02-2022 59732
2500
3184-6294
304
47.61
SALT
06-04-2022 59734
2500
3191-6292
304
39.67
a
b
QSO
MNRAS 000, 1-22(2015)
iraf is distributed by the National Optical Astronomy Observatories, which are operated by the Association of Universities for Research in Astronomy, Inc., under co-operative agreement with the National Science Foundation.MNRAS 000, 1-22(2015)
The errors are calculated using Poisson statistics as discussed inGehrels (1986).
a b MNRAS 000, 1-22 (2015)
ACKNOWLEDGEMENTSWe thank Aseem Paranjape and K. Subramanian for useful discussions. PA thanks Labanya K Guha for helpful discussions on several python programming techniques used in this paper. PPJ thanks Camille Noûs (Laboratoire Cogitamus) for inappreciable and often unnoticed discussions, advice and support. PPJ is partly supported by the Agence Nationale de la Recherche under contract ???.DATA AVAILABILITYData used in this work are obtained using SALT. Raw data will become available for public use 1.5 years after the observing date at https://ssda.saao.ac.za/.
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| {'fraction_non_alphanumeric': 0.05877865566496711, 'fraction_numerical': 0.11744833223056864, 'mean_word_length': 3.6056716738966768, 'pattern_counts': {'":': 0, '<': 30, '<?xml version=': 0, '>': 36, 'https://': 2, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, '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': 'We study the time variability (over 7.3 yrs) of ultra fast outflows (UFOs) detected in a sample of 64 C iv broad absorption line (BAL) quasars (with 80 distinct BAL components) monitored using the Southern African Large Telescope. By comparing the properties of the quasar in our sample with those of a control sample of non-BAL quasars we show that the distributions of black hole mass are different and the bolometric luminosities and optical photometric variations of UFO BAL quasars are slightly smaller compared to that of non-BAL quasars. The detection fraction of C iv equivalent width (W) variability (∼95%), the fractional variability amplitude ( ∆W W ) and the fraction of "highly variable" BAL (i.e., ∆W W >0.67) components (∼ 33%) are higher in our sample compared to the general BAL population. The scatter in ∆W W and the fraction of "highly variable" BALs increase with the time-scale probed. The ∆W W distribution is asymmetric at large time scales. We attribute this to the BAL strengthening time scales being shorter than the weakening time scales. The BAL variability amplitude correlates strongly with the BAL properties compared to the quasar properties. BALs with low W, high-velocity, shallow profiles, and low-velocity width tend to show more variability. When multiple BAL components are present a correlated variability is seen between low-and high-velocity components with the latter showing a larger amplitude variations. We find an anti-correlation between the fractional variations in the continuum flux and W. While this suggests photoionizationinduced variability, the scatter in continuum flux is much smaller than that of W.', 'arxivid': '2305.02352', 'author': ['P Aromal \nIUCAA\nPostbag 4411007Ganeshkind, PuneIndia\n', '† ', 'R Srianand \nIUCAA\nPostbag 4411007Ganeshkind, PuneIndia\n', "P Petitjean \nInstitut d'Astrophysique de Paris\nSorbonne Université and CNRS\n98bis boulevard Arago75014ParisFrance\n"], 'authoraffiliation': ['IUCAA\nPostbag 4411007Ganeshkind, PuneIndia', 'IUCAA\nPostbag 4411007Ganeshkind, PuneIndia', "Institut d'Astrophysique de Paris\nSorbonne Université and CNRS\n98bis boulevard Arago75014ParisFrance"], 'corpusid': 258479901, 'doi': '10.1093/mnras/stad1370', 'github_urls': ['https://github.com/legolason/PyQSOFit0'], 'n_tokens_mistral': 51683, 'n_tokens_neox': 39907, 'n_words': 21039, 'pdfsha': '9ea038b1ddfecdc856c786875c4c7f8a750544bc', 'pdfurls': ['https://export.arxiv.org/pdf/2305.02352v1.pdf'], 'title': ['Time variability of ultra fast BAL outflows using SALT: C iv equivalent width analysis', 'Time variability of ultra fast BAL outflows using SALT: C iv equivalent width analysis'], 'venue': ['MNRAS']} |
arxiv |
PNet: A Python Library for Petri Net Modeling and Simulation
Zhu En Chay [email protected]
Colossus Technologies LLP
Republic of Singapore
Bing Feng Goh
Colossus Technologies LLP, Republic of Singapore School of BioSciences
Singapore Institute of Technology
Republic of Singapore
Maurice Ht Ling [email protected]
The University of Melbourne
Australia
PNet: A Python Library for Petri Net Modeling and Simulation
Network modelingTime-step simulationPetri NetOrdinary Differential EquationPython
Petri Net is a formalism to describe changes between 2 or more states across discrete time and has been used to model many systems. We present PNeta pure Python library for Petri Net modeling and simulation in Python programming language. The design of PNet focuses on reducing the learning curve needed to define a Petri Net by using a text-based language rather than programming constructs to define transition rules. Complex transition rules can be refined as regular Python functions. To demonstrate the simplicity of PNet, we present 2 examplesbread baking, and epidemiological models.
Introduction
Petri Nets are tools designed by C. A. Petri to model concurrent systems, as graphical representations and mathematical modeling tool for system of events [1][2][3]. The use of Petri Nets allows formal analysis of the model or process that is being depicted [1][2][3]. Petri Nets are populated by three types of objectsplaces, transitions and arcs [1][2][3]. A place is an input position that is connected to another transition via an arc. Place is usually depicted by circles and transitions as bars [1][2][3].
Petri Nets are often used for software engineering [4], system modeling [5] and even in biochemistry; such as biochemical reactions [6], signal transduction networks [7] and gene regulation networks [8]. An example of the use of Petri Nets is by Liu and Heiner [9] where they investigate biochemical reaction networks with the use of unifying Petri Net framework to model and analyze such networks [4]. This is because the properties of the processes can be studied: Terminating, Reachability, Safeness, Boundedness, Liveness, Reversibilty and Home State, Coverability, Persistence and Fairness [1]. Such properties can be studied [1] using Coverability tree method, Reachability Graphs and Incidence Matrix and State Equation.
There are libraries developed for modeling Petri Nets, such as SimForge GUI [10] incorporated within OpenModelica, and Petri Net Simulink Block (PNSB [11]) for MATLAB. SNAKES had been developed by Pommereau [12] as a library for implementing Petri Nets in Python programming language. SNAKES [12] adopts a high-level of object-oriented programming as tokens and all transition rules are implemented as Python objects. Although this provides flexibility, it may present as a steeper learning curve. It may be considerably more difficult to translate a text-based Petri Net specification into a model in SNAKES. At the same time, SNAKES [12] does not cater for complex transition rules that can only be implemented as a function. However, a strong advantage of SNAKES [12] is the incorporation of plugins, Petri Net analysis tools, and the ability to convert implemented Petri Nets into C language.
In this work, we present PNet an alternative pure Python library for Petri Net modeling by reducing its objectoriented programming overheads to its minimum, and adding Python functions as an alternative type of transition rule. Hence, PNet is likely to reduce the learning curve needed for a beginner to start experiencing Petri Nets before transitioning to more extensive library, such as SNAKES [12]. PNet has been incorporated into COPADS (https://github.com/mauriceling/copads), a library of algorithms and data structures, developed entirely in Python programming language and has no third-party dependencies. Future work aims to implement a GUI for improved usability and tools to analyze Petri Nets.
Description of PNet
In this section, we will describe PNet by using the steps required to write a simulation. There are 5 steps to writing a Petri Net simulation using PNetestablishing a Petri Net, adding places or states, adding transition rules, simulating the Petri Net, and generating the results file.
A Petri Net is established by importing PNet as a module and instantiating the PNet class within the imported module. This is followed by adding of places or states into the Petri Net using add_places method, which takes 2 parametersthe name of the place and a dictionary representing the initial tokens, where the dictionary key and value is the name of the token and the number of named tokens respectively. This allows for a place/state to have more than 1 type of tokens. For example, a mixed bowl of 100 red and green beans each can be stated as However, there are situations whereby an infinite source or sink is needed; for example, the number of people to be born may be virtually infinite. In electronics, Earth is considered an infinite source of positive and negative charges. To cater for this need, a special place known as ouroboros (ouroboros is the name for the "infinity" symbol in mathematics) is defined with an infinite number of "U" tokens.
The third step is the addition of transition rule(s), using add_rules method. Each transition rules is named.
Transitions are channels where the tokens move from a place to another, whereas the rules determine how the move occurs. Generally, a transition rule consists of a source place and destination place to direct the tokens, source token type for identification purpose and destination token type for getting a precise result. The processes of transitions rules will then be checked against the find value of tokens, using logical operators. The logical operators of the checks are determined by criterions, which are also known as the intended result after the going through the transitions.
There are 5 types of rules: step, ratio, delay, incubate, and function rules. The execution/firing of transition rules is time-step dependent. Although in the strictest sense, each rule should define only one transition; in practice, a single rule can trigger 1 or more transitions as it is possible to specify more than 1 transition in a rule. This can be seen as a syntactic shorthand provided by PNet.
Step rule based on a step-wise execution where the rule will be triggered at every time-step. The origin place and the token at the origin place have to be indicated; at the same time, the destination place and the affected token at the destination place must be specified. This represents a single transition. For example, given a bowl of red and green beans each, the following step rule defines the swapping of a bean at each time step,
net.add_rules('swap_bean, 'step', ['B1.red_bean -> B2.red_bean; 1', 'B2.green_bean -> B1.green_bean; 1'])
Ratio rule is also a step-wise execution. Both step and ratio rules have similar parameters, with the difference of using the ratio of tokens to trigger the execution. The ratio that is intended for the transition is indicated, which will be check by a logical operator against the limit indicated. Function(s) to be used in function rule(s) takes only one parameter, places, which is the dictionary of states/places in PNet. Each state/place can be accessed using the name of the state/place as key to the places dictionary. Tokens linked to a particular state/place are implemented as an attributes dictionary and be accessed using the name of token.
After rules definition, the fourth step is to simulate the Petri Net using simulate or simulate_yield method. The rules mentioned above will then be executed, by setting the wall time of current simulation and the interval. The wall time will be check against the rules to ensure that it fulfills the time interval indicated.
The simulate method will stores the simulation results in the memory space; thus, increased intervals of reporting causes more reports to be generated, which causes memory to run out at a faster rate. On the other hand, simulate_yield method is a generator function, which does not pre-store all the simulation results in memory. The parameters of simulate method are length of time to simulate, time step, and reporting frequency. However, simulate_yield method only requires length of time to simulate, and time step.
Finally, PNet provides a method to process the simulation results into a format suitable for CSV file output. The reports will be generated with each step count of current simulation, in the memory of each token status. The status of the tokens can be also report by generating a list representing the status from one step or the entire simulation.
Simulation and reporting are usually related to each other. For example, the following code snippet demonstrates the simulation and report generation using both simulate or simulate_yield method:
Examples
Two examples are presented to illustrate the use of PNet. The first example is a light-hearted example of bread baking while the second example is a more serious but simple model of epidemiology.
Example 1: Bread Baking. In this example, a bread baking recipe was modeled (see Appendix A for implementation) and simulated for 90 time steps. It is worth noting that this recipe utilized all features of PNet except the use of infinite tokens from Ouroboros. This recipe calls for 1000 g of flour, 500 g of water, 20 g of sugar, and 1 g of yeast in the following steps:
1. Turn on the mixer and add 100 g of flour, 50 g of water, and 2 g of sugar at each time step. Add 0.5 g of yeast into mixer with 5 time step interval in between each addition. 2. In each time step, the mixer will turn 80 g of flour, 40 g of water, 1.5 g of sugar, and 1 g of yeast into dough. 3. After mixing is completed, leave dough to rise in mixer for 10 time steps. 4. Transfer dough into pan, and leave dough to rise in pan for another 10 time steps. 5. Bake at 400 o C. In each time step, 30% of the remaining dough will be baked into bread. Baking is completed when there is less than 1 g of dough remaining. 6. Transfer the bread to the table for cooling. In each time step, 10% of the heat will dissipate until room temperature of 30 o C is reached. 7. Enjoy your bread. Steps 1 and 2 are the addition and mixing the ingredients into bread dough. The rate of dough formation is slower than the rate of ingredients addition; for example, 100 g of flour and 50 g of water are added into the mixer per time step but only 80 g of flour and 40 g of water are converted to dough. This results in the accumulation of 20 g of flour and 10 g of water in the mixer until all flour and water are added (as seen in the mixing stage in Figure 1A); after which, the accumulated flour in the mixer is converted into bread dough. Once all ingredients are mixed into bread dough, it undergoes 2 stages of rising (Steps 3 and 4). After which, the dough is being baked at 400 o C (Step 5).
The baking process is represented as transferal of 30% of the dough tokens into bread tokens at each time step. Once the bread is baked as represented by negligible remains of dough (less than 1 g), the bread is transferred to a table and cooled (Step 6). The cooling from 400 o C to room temperature of 30 o C is represented by another function.
Example 2: Epidemiological Models. Epidemiological
Models are frameworks of ecological and epidemiological phenomena that are often used to study interactions between the host and the pathogen [13]. Epidemiological models have proven useful for the study of evolutionary dynamics of evolutionary dynamics and predicting properties of the spread of pathogen like prevalence and duration [13]. Alphabet models are frameworks of a population whereby susceptible individuals are considered to be invaded by an infectious agent [13]. The population is divided into three epidemiological subclasses: S denotes susceptible to diseases, I denotes number of infected individuals and R denotes number of individuals who at time no longer contribute to spread of diseases [13].
The Susceptible-Infectious-Susceptible (SIS) model is predicated on the pathogen infects susceptible humans, resulting in an infection and recovers from infection and returns back to the susceptible class again [14]. Infectious hosts recover at a constant per capita rate, γ and β is the rate of infection of the susceptible class [14]. SIS model is for fast evolving virus and infections that do not provide immunity [14]. The Susceptible-Infectious-Recovered (SIR) model is similar to SIS model except that the pathogen leads to lifelong immunity [15]. The individuals who were infected and recovered from infection are immune to reinfection, possessing lifelong immunity [15]. This model is used for viral diseases such as measles, mumps and rubella [14]. The Susceptible-Infectious-Recovered-Susceptible (SIRS) model is similar to that of SIR model, except that the immunity that are acquired is temporary [16]. The individuals who were infected and recovered from infection are not immune to reinfection [14]. Example of such infection modeled by SIRS is tuberculosis [16].
However, it is common for most epidemiological models to be implemented as a system of ordinary differential equations (ODEs) [17][18][19][20]. ODEs and Petri Net are two of the most common mathematical constructs for mathematical modeling [21]. Hence, a method to represent an ODE in the form of Petri Net notation is needed and the correspondence between ODE and state-transition network is provided by Soliman and Heiner [22]. Briefly, an ODE models the change of a state over time while Petri Net models the movement which results in the change of state over time (Figure 2). In the context of states (nodes) and transitions (arcs), this suggests that ODEs models the nodes while Petri Net models the transitions, which gives rise to an easy conversion between ODE representations and Petri Net representations, assuming that the unit for time is the same under both representations. Hence, the standard system of ODEs for SIRS [17,18] can be readily converted into Petri Net representations ( Figures 3 and 4). The implementation of SIRS model using PNet is given as Appendix B. Our simulation results show that the proportion of infected population and susceptible population reaches equilibrium over time ( Figure 5a). As there is also no immunity conferred after recovery, it is expected to have a constant infected population [23], also known as endemic population. This is under the assumption that there are no birth and death for the entire duration, and the disease is not death causing. When there is immunity as a result of recovery, SIS model becomes SIR model where the population gradually becomes fully immune if there is no additional birth (Figure 5b). This is similar to the case of chickenpox [24], which confers lifelong immunity to most recovered patients, leading to children being most susceptible to chicken pox and most adults immune. However, if the conferred immunity is temporary, reinfection is possible and this leads to SIRS model from SIR model (Figure 5b). Our results show that SIRS model behaves in similar manner compared to SIS model where there is a constant pool of infected (endemic) individuals [25]. In spite of this, there is also a constant pool of immune individuals whom had recently recovered from the disease. This is expected when the infection agent can re-infect a recovered person.
net.add_places('bowl',{'red_beans': 100, 'green_beans': 100})
Fig. 1
1Token Values in Bread Baking Simulation. Graph A shows the mixing of ingredients into bread dough and the rising of the dough. Graph B re-illustrate the rising of the dough and continue through the baking and cooling process.
Fig. 2
2Fig. 2 Correspondence between Ordinary Differential Equation and Petri Net Transition Rule.
Fig. 4
4SIR/SIRS Epidemiological Model.
Fig. 5
5Schematics of a Data Frame.
This can be used to define increasing or decreasing number of tokens moved. For example, given that there are 2 bowl (B1 and B2) where B1 contains all the red beans and B2 is an empty bowl, moving 10% of the remaining red beans in B1 to B2 can be defined as follow, net.add_rules('swap_ratio', 'ratio', ['B1.red_beans -> B2.red_beans; 0.1; \ B1.red_beans < 1; 0']) Delay rule is a step rule with time interval between each token movement. In effect, delay rule can be used to produce a regular spiking movement. For example, moving 10 beans from bowl B1 to B2 once every 5 th time step, can be defined as net.add_rules('interval_transfer', 'delay', ['B1.beans -> B2.beans; 10; 5'])Incubation rule can be seen as a "do nothing for a period of time before a specific action". It requires a value and a timer, which has a logical check within to make sure that the conditions are met before sending to the destination place. For example, soaking a bowl of beans for 60 time steps (such as 60 minutes) once water is added, and transfer the soaked beans into a pot after soaking for 60 time steps, can be defined asnet.add_rules('soak', 'incubate',
['60; bowl.beans -> pot.beans; \
bowl.water > 0'])
Function rule is a user-defined condition. Usually, function
rules are used when the previous 4 rules do not fit the
user's requirement. However, all forms of transition rules
can be written as function rule; thus, function rule is the
superset. Another common application of function rule is
to change the type of tokens from one type to another. The
variations between the transition rules are the conditions
that trigger of the transition rules. The origin and
destination place together with the initial token and the
final token are required for the computation. For example,
the above ratio rule
net.add_rules('swap_ratio', 'ratio',
['B1.red_beans -> B2.red_beans; 0.1; \
B1.red_beans < 1; 0'])
can be written as the following function rule,
def bean_swap(places):
place = places['B1']
n = place.attributes['red_beans']
if n > 0.0:
return 0.0
else:
return 0.1 * n
net.add_rules('swap_ratio', 'function',
['B1.red_beans -> B2.red_beans',
bean_swap, 'B1.red_beans > 0'])
AcknowledgementThe authors wish to thank HJ Wang (Nanyang Technological University) for valuable discussions during the developmental and testing phases.
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The Global Stability of an SIRS Model with Infection Age. Y Chen, J Yang, F Zhang, Mathematical Biosciences and Engineering. 11Y. Chen, J. Yang and F. Zhang, "The Global Stability of an SIRS Model with Infection Age", Mathematical Biosciences and Engineering 11, 2014, 449-469.
# The steps net.add_rules('add_flour', 'step', ['flour.flour -> mixer.flour. 100# The steps net.add_rules('add_flour', 'step', ['flour.flour -> mixer.flour; 100'])
step', ['mixer.flour -> mixer.dough; 80', 'mixer.water -> mixer.dough; 40', 'mixer.sugar -> mixer.dough; 1.5', 'mixer.yeast -> mixer. dough; 1'net.add_rules('blend', 'step', ['mixer.flour -> mixer.dough; 80', 'mixer.water -> mixer.dough; 40', 'mixer.sugar -> mixer.dough; 1.5', 'mixer.yeast -> mixer.dough; 1'])
10; mixer.dough -> pan.dough; \ mixer.flour == 0; mixer.water == 0; \ mixer.sugar == 0; mixer. net.add_rules('rise', 'incubate. net.add_rules('rise', 'incubate', ['10; mixer.dough -> pan.dough; \ mixer.flour == 0; mixer.water == 0; \ mixer.sugar == 0; mixer.yeast == 0'])
bread -> table.bread; \ oven.dough == 0']) def cooling(places): place = places. net.add_rules('transfer', 'incubate', ['1; oven.. 'table'] temp = place.attributes['temperature'] if temp <= 30.0: return 0.0 else: return 0.1 * temp net.add_rules('cool', 'function', ['table.temperature -> air.heat', cooling, 'table.bread > 0'net.add_rules('transfer', 'incubate', ['1; oven.bread -> table.bread; \ oven.dough == 0']) def cooling(places): place = places['table'] temp = place.attributes['temperature'] if temp <= 30.0: return 0.0 else: return 0.1 * temp net.add_rules('cool', 'function', ['table.temperature -> air.heat', cooling, 'table.bread > 0'])
w') f.write(','.join(headers) + '\n') for tdata in data: tdata =. tdata[0]] + \ [str(x) for x in tdata[2f = open('bread.csv', 'w') f.write(','.join(headers) + '\n') for tdata in data: tdata = [tdata[0]] + \ [str(x) for x in tdata[2]]
| {'fraction_non_alphanumeric': 0.05571836628704974, 'fraction_numerical': 0.023370553103090108, 'mean_word_length': 4.62680025046963, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 28, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, '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': 'Petri Net is a formalism to describe changes between 2 or more states across discrete time and has been used to model many systems. We present PNeta pure Python library for Petri Net modeling and simulation in Python programming language. The design of PNet focuses on reducing the learning curve needed to define a Petri Net by using a text-based language rather than programming constructs to define transition rules. Complex transition rules can be refined as regular Python functions. To demonstrate the simplicity of PNet, we present 2 examplesbread baking, and epidemiological models.', 'arxivid': '2302.12054', 'author': ['Zhu En Chay [email protected] \nColossus Technologies LLP\nRepublic of Singapore\n', 'Bing Feng Goh \nColossus Technologies LLP, Republic of Singapore School of BioSciences\nSingapore Institute of Technology\nRepublic of Singapore\n', 'Maurice Ht Ling [email protected] \nThe University of Melbourne\nAustralia\n', 'Zhu En Chay [email protected] \nColossus Technologies LLP\nRepublic of Singapore\n', 'Bing Feng Goh \nColossus Technologies LLP, Republic of Singapore School of BioSciences\nSingapore Institute of Technology\nRepublic of Singapore\n', 'Maurice Ht Ling [email protected] \nThe University of Melbourne\nAustralia\n'], 'authoraffiliation': ['Colossus Technologies LLP\nRepublic of Singapore', 'Colossus Technologies LLP, Republic of Singapore School of BioSciences\nSingapore Institute of Technology\nRepublic of Singapore', 'The University of Melbourne\nAustralia', 'Colossus Technologies LLP\nRepublic of Singapore', 'Colossus Technologies LLP, Republic of Singapore School of BioSciences\nSingapore Institute of Technology\nRepublic of Singapore', 'The University of Melbourne\nAustralia'], 'corpusid': 16614835, 'doi': '10.48550/arxiv.2302.12054', 'github_urls': ['https://github.com/mauriceling/copads),'], 'n_tokens_mistral': 7667, 'n_tokens_neox': 6664, 'n_words': 4096, 'pdfsha': '315aeee23d4894c57a1b3e6f05f6e549f02a9b88', 'pdfurls': ['https://export.arxiv.org/pdf/2302.12054v1.pdf'], 'title': ['PNet: A Python Library for Petri Net Modeling and Simulation', 'PNet: A Python Library for Petri Net Modeling and Simulation', 'PNet: A Python Library for Petri Net Modeling and Simulation', 'PNet: A Python Library for Petri Net Modeling and Simulation'], 'venue': []} |
arxiv |
Nonlinear large scale flow in a precessing cylinder and its ability to drive dynamo action
(Dated: August 1, 2018) 18 Dec 2017
André Giesecke
Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany
Tobias Vogt
Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany
Thomas Gundrum
Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany
Frank Stefani
Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany
Nonlinear large scale flow in a precessing cylinder and its ability to drive dynamo action
(Dated: August 1, 2018) 18 Dec 2017numbers: 9125Cw4765-d4780Cb4720Ky5265Kj Keywords: dynamoprecessionrotating fluidsexperimentsdirect numerical simulations
We have conducted experimental measurements and numerical simulations of a precession driven flow in a cylindrical cavity. The study is dedicated to the precession dynamo experiment currently under construction at Helmholtz-Zentrum Dresden-Rossendorf (HZDR) and aims at the evaluation of the hydrodynamic flow with respect to its ability to drive a dynamo. We focus on the strongly nonlinear regime in which the flow is essentially composed of the directly forced primary Kelvin mode and higher modes in terms of standing inertial waves arising from nonlinear self-interactions. We obtain an excellent agreement between experiment and simulation with regard to both, flow amplitudes and flow geometry. A peculiarity is the resonance-like emergence of an axisymmetric mode that represents a double roll structure in the meridional plane. Kinematic simulations of the magnetic field evolution induced by the time-averaged flow yield dynamo action at critical magnetic Reynolds numbers around Rm c ≈ 430 which is well within the range of the planned liquid sodium experiment.
Magnetic fields of celestial bodies like planets, moons or asteroids are ubiquitous in the solar system with a wide diversity of manifestations [1]. While it is undisputed that these fields are generated by conversion of mechanical energy from the flow of an electrically conductive fluid, there are various possibilities to drive the underlying fluid motions. Usually, it is assumed that the flow in liquid planetary cores is driven by thermo-compositional convection [2], yet alternative approaches invoke mechanical stirring by libration [3], tidal forcing [4] or precession [5]. In particular precession has been repeatedly proposed as source for dynamo action of the ancient lunar magnetic field [6] or the geodynamo [7]. Indeed, simulations and experiments revealed that precession may excite vigorous flows [8] which are supposed to drive a dynamo [9]. Precessional forcing has become of great interest from the experimental point of view, because it represents a natural mechanism which allows an efficient driving of conducting fluid flows on the laboratory scale without making use of propellers or pumps [10]. At HZDR a precession dynamo experiment is under development [11] which will provide a flow of liquid sodium in a cylindrical cavity with a magnetic Reynolds number of up to Rm = Ω c R 2 /η ≈ 700 (defined with the achievable angular velocity of the cylinder Ω c = 63 s −1 , the radius R = 1 m, and the magnetic diffusivity for liquid sodium η = 0.09 m 2 /s). The project is further motivated by previous precession experiments conducted by Gans [12] who achieved an amplification of an applied magnetic field by a factor of 3 with a device smaller by a factor of 8 and by numerical studies yielding precession driven dynamos in different geometries with a critical magnetic Reynolds number of O(10 3 ) [5,13]. However, so far numerical models of the planned experiment have not shown conclusively that the achievable magnetic Reynolds number will be sufficient to allow for dynamo action [14,15].
In the present study we address preparatory simulations and flow measurements at a water experiment that represents a down-scaled model of the planned sodium dynamo. The re-sults provide flow patterns and amplitudes in dependence on Reynolds number Re = Ω c R 2 /ν and on the relation of precession frequency Ω p to rotation frequency Ω c , the Poincaré number Po = Ω p /Ω c . Finally, the three-dimensional velocity fields from the simulations are used in kinematic dynamo models in order to estimate parameter regimes that will be appropriate for dynamo action.
We conduct direct numerical simulations in the precession reference frame using the code SEMTEX [17]. In this frame the observer resides on the turntable following the rotation around the precession axis, thereby watching at the spinning cylinder (Fig. 1a). The flow is described by the Navier Stokes equation including a time-independent term for the Coriolis force due to precession [18]:
∂ ∂t u + u∇u = −∇P − 2Ω p × u + ν∇ 2 u.(1)
Here, u is the incompressible velocity field, P the reduced pressure, ν the viscosity, and Ω p the angular velocity of the precessional motion. The flow obeys no-slip boundary conditions for the poloidal components, u r = u z = 0, whereas the azimuthal flow at the boundaries is prescribed by u ϕ = rΩ c . Fluid velocities are measured using Ultrasonic Doppler Velocimetry (UDV) which provides instantaneous profiles of the velocity component in direction of an ultrasonic beam [19,20] oriented parallel to the cylinder axis. Four ultrasound transducers are fixed at one end cap of the cylinder (Fig. 1a) and co-rotate with the container thus providing measurements in the cylinder frame. This reference frame is well suited for flow characterization in terms of eigenmodes of rotating flows which are the solutions of the linearized inviscid version of Eq. (1). In a cylinder these solutions are inertial waves, or Kelvin modes, U j (r, z, ϕ, t) =ũ j (r)e i(ω j t+mϕ+kz) [21,22] sion relation for an inertial wave
ω j λ j J m−1 (λ j ) + m 2−ω j J m (λ j ) = 0 with ω j = ±2 1+ λ j 2kπ 2 (2)
where J m denotes the Bessel function of order m, and λ j plays the role of a radial wave number. Precession causes a steady volume forcing with an odd symmetry with respect to the equatorial plane. Therefore the primary response of the fluid is a flow with an azimuthal wave number m = 1 and an odd axial wave number k that is stationary in the precession reference frame. If the frequency of the corresponding eigenmode (ω j ) exactly matches the frequency of the forcing (Ω c ), the mode becomes resonant, and the linear inviscid approach for the computation of the amplitude fails [23]. The resonance condition delicately depends on the aspect ratio, and the primary forced mode with the simplest possible structure, i.e. m = 1, k = 1 and n = 1 becomes resonant at H/R = 1.98982 which is close to the geometry envisaged for our planned experiment (H/R = 2). In the present study the corresponding cylinder utilized in the water experiment has radius R = 163 mm and height H = 326 mm, and the angle between rotation axis and precession axis is fixed at α = 90 • . Typical measurements of a single UDV probe are shown in Fig.1b (top) in terms of the axial velocity versus time and depth. The alternation of the sign of u z with the periodicity of Ω c and the asymmetry with respect to the equatorial plane illustrate the dominance of the m = 1 component superposed by higher azimuthal modes (essentially m = 2). We find a very good agreement between experiments and simulations (Fig. 1b, central and bottom panel). For sufficiently large Po, the flow is concentrated in the vicinity of the cylinder walls (Fig. 1c) and can be decomposed into few large scale modes. These modes represent standing inertial waves in the preces-sion reference frame, and time-dependent contributions only appear as weak small-scale fluctuations (see movie in supplementals [16]).
A quantitative analysis of the flow is done by decomposing axial profiles of u z in k-modes ∝ sin(πzk/H) which is the characteristic z-dependence of the axial component of an inertial wave in a cylinder with height H [23,24]. In a second step we take the individual k-modes from this decomposition and apply a 2D Fourier transformation in azimuthal direction and in time which finally yields spectra that allow the identification of individual modes labeled by (m, k). Typical spectra from simulations at Re = 10 4 and Po = 0.1 are shown in Fig. 2a which represents the signature of the primary forced mode (m, k) = (1, 1) and its first multiple (m, k) = (2, 2) resulting from nonlinear self-interaction. For sufficiently strong precession the spectra of all (m, k)-modes qualitatively look similar with one single peak at ω = 0 that corresponds to a standing inertial wave in the precession reference frame. The amplitudes of individual modes, estimated from spectral peaks at ω = 0, show that, independent of Po, the flow is always dominated by the primary forced mode (m, k) = (1, 1) (Fig. 2b, blue curve). A characteristic feature is the concise maximum of the amplitude at Po c ≈ 0.09. Immediately following this maximum we find three phenomena that are intimately connected: a strong and abrupt reduction of the amplitude of the directly forced flow with (m, k) = (1, 1), a gradual increase of higher modes that originate from nonlinear self-interaction according to (m, k) → (2m, 2k) (Fig. 2b, red curve) and a sudden appearance of a non-geostrophic axisymmetric flow with k even (Fig. 2b, green curve). The axisymmetric mode only exists with noteworthy amplitude within a rather narrow band with a width ∆Po∼0.006 (Fig. 2c). This axisymmetric mode is of interest with regard to the dynamo problem because its geometric pattern corresponds to a double roll structure (Fig. 3a similar to the mean poloidal flow in the von-Kármán sodium dynamo in which the flow was driven by two opposite counterrotating impellers [25]. It is well known that this flow can drive a dynamo at comparatively low Rm [26] when the relation between toroidal and poloidal components is of order unity. However, there are further contributions to the axisymmetric flow in terms of a geostrophic azimuthal circulation (Fig. 3b) directed opposite to the solid body rotation which worsen this relation in our model.
✵ ✺ ✵ ✶ ✵ ✵ ✶ ✺ ✵ (c) (a) ✲ ✺ ✵ ✲ ✵ ✲ ✁ ✵ ✲ ✂ ✵ ✲ ✶ ✵ ✵ ✉ ϕ ♠ ✄ ❬ ☎ ☎ ✆ ✝ ✞ (b) ✵ ✺ ✵ ✶ ✵ ✵ ✶ ✺ ✵ ✂ ✵ ✵ ✂ ✺ ✵ ✁ ✵ ✵ ✲ ✶ ✵ ✲ ✺ ✵ ✺ ✶ ✵ ③ ✟ ✠ ✠ ✡ ✉ ☛ ♠ ✄ ❬ ☎ ☎ ✆ ✝ ✞ ☞ ✌ ✍ ☞ ✎ ✍ ☞ ✏ ✍ ☞ ✑ ✍ ✍ ✒ ϕ ✓ ✔ ✔ ✕ ✖ ✗ ✘ ✙ ✘ ✚ ✘ ✘ ✚ ✙ ✘ ✘ ✷ ✹ ✻ ✽ ϕ ✛ ✜ ✢ ✸ ✵ ✺ ✵ ✶ ✵ ✵ ✶ ✺ ✵ ✂ ✵ ✵ ✂ ✺ ✵ ✁ ✵ ✵ s ✣ ✤ ✥ ✦ ✧ ✣ ✦ ★ ✩ ✣ ✪ ✫ ✪ ✥ ✣ ✬ ✣ ✭ ✮ ✯ ✮ ✰ ✮ ✣ ✭ ✮ ✯ ✮ ✱ ✮ ✣ ✭ ✮ ✯ ✮ ✳ ✮ ✣ ✭ ✮ ✯ ✮ ✴ ✳ ✣ ✭ ✮ ✯ ✮ ✼ ✴ ✳ ✣ ✭ ✮ ✯ ✰ ✮ ✮ ✣ ✭ ✮ ✯ ✰ ✾ ✳
The experiments show that the basic flow properties remain unchanged up to Re = 10 5 except the decrease of the critical value Po c at which the previously discussed phenomena emerge. The occurrence of the non-geostrophic axisymmetric resonance is a robust feature which does not disappear when increasing Re (Fig. 2c). This mode can be excited by interacting inertial waves according to (m, k, ω) → (0, 2k, 0) [24]. However, this is unlikely without the presence of singularities [27] so that these interactions must happen within no-slip boundary layers [28] or internal shear layers [29]. A more descriptive explanation rests upon the modification of the basic azimuthal circulation, which for sufficiently large Po compensates the bulk fluid's solid body rotation. The azimuthal fluid motion opposite to the cylinder rotation can even become so strong that eventually the Rayleigh criterion for stability of rotating fluids may be violated by developing a negative radial derivative of the angular momentum, i.e., d dr (u ϕ r) < 0 (Fig. 3c), immediately leading to the formation of Taylor vortices. Finally, the further increase of Po leads to the breakdown of the large scale structures into smaller scales which, at Re ≈ O(10 6 ), corresponds to a transition into a fully turbulent flow without significant large scale contributions [32].
In the following we use the velocity fields obtained from the hydrodynamic simulations, validated by UDV flow measurements, as basis for kinematic dynamo models. We concentrate on the strongly precessing regime around Po ≈ 0.1 so that the flow is determined by standing inertial waves which makes the time-averaged velocity field appropriate for the application in kinematic simulations. The flow field is further decomposed into separate azimuthal modes m = 0, 1, 2 in order to carve out
∂ ∂t B = ∇ × (ū × B − η∇ × B) .(3)
With the Ansatz B(r, t) = B 0 (r)e σt the solution of Eq. (3) is a linear problem with the real part of the eigenvalue σ representing the magnetic field growth rate γ. We solve Eq. (3) numerically with pseudo-vacuum boundary conditions for the magnetic field, and the growth rates are computed from the timeevolution of the magnetic field. Except for the velocity field at Po = 0.1, the kinematic models either show no dynamo or do so at best for magnetic Reynolds numbers far above the values that will be attainable in the planned dynamo experiment (e.g. Rm c ≈ 5000 for Po = 0.0875). Taking the time-averaged flow field from hydrodynamic simulations at Re = 10 4 and Po = 0.1, we find dynamos at much reduced Rm. The kinematic growth rates for this particular case are shown in Fig. 4 where we distinguish five different set-ups. We find that neither the axisymmetric flow (m = 0, orange curve) nor the directly forced flow (m = 1, green curve) alone are capable of driving a dynamo. The latter was expected because the structure of the primary flow is too simplistic for dynamo action [14]. The failure of the pure axisymmetric flow to drive a dynamo confirms our previous assumption of the inappropriate relation of axisymmetric poloidal and toroidal flow components. However, when summing up both contributions we obtain dynamo action at a critical magnetic Reynolds number Rm c ≈ 560 (blue curve). This value decreases to Rm c ≈ 430 when further including the m = 2 modes (red curve), most probably because this contribution, which is dominated by the (m, k) = (2, 2) mode, increases the breaking of the equatorial symmetry, which is beneficial for precession driven dynamos [5]. Other contributions with higher wave numbers are less important and no significant further reduction of Rm c is obtained when using the total time-averaged flow which yields Rm c ≈ 428 (black curve).
The water experiments indicate that the flow structure does not change much when increasing Re [31], albeit the corresponding decrease of Po c does not follow a simple scaling law (Fig. 2c). However, it is known from measurements of the internal pressure that the sudden drop of the m = 1 mode, which constitutes the second criteria for Po c , only weakly depends on Re if Re 5 × 10 5 [32]. This is already indicated in our experiments when increasing Re from 4 × 10 4 to 10 5 . The width within which we observe the axisymmetric mode (∆Po ≈ 0.006) corresponds nearly exactly to the width of the hysteresis found in [32] around Re ∼ O(10 6 ) at a precession ratio comparable with Po c in our experiments at Re = 10 5 . It seems likely that both phenomena are closely connected, with the (m, k) = (0, 2) mode being a precursor for the transition to the turbulent state observed in [32]. In the limit of large Re as they will occur in the liquid sodium experiment (up to Re ≈ 10 8 ), we thus expect dynamo action to arise in connection with the non-geostrophic axisymmetric mode within a width of ∆Po ≈ 0.006 around Po not much smaller than Po c in our experiments at Re = 10 5 .
Our results reveal a first promising -though narrowregime, defined by the presence of the axisymmetric mode, within which we expect dynamo action in the planned dynamo experiment. This is not a turbulent dynamo since there is no significant amount of turbulence as it would result, for example, from the resonant collapse reported in experimental studies of precessing flows with small nutation angles [33]. Our model rather constitutes a laminar dynamo driven by few large scale velocity modes, and our simulations and measurements indicate that time-dependent contributions remain weak even at the largest Re with the spectra always being determined by standing inertial waves. This is in contrast to the flow in the VKS dynamo where the fluctuations are of the same order as the mean flow. Instead, a comparison with the Riga Dynamo is more appropriate, in which a fully developed turbulence arises on top of a mean flow [34], and calculations based on the time-averaged flow field still provided good agreement with the experiment [35] proving that the turbulent β-effect remains negligible for such flows.
So far, we did not consider more realistic magnetic boundary conditions, like an insulating outer domain or the finite conductivity of the container made of stainless steel which will be focus of a future study. Preliminary results from models including a thin outer layer with the electrical conductivity reduced by a factor of 8 show an increment of Rm c by roughly 10% which is still well within the capabilities of the planned facility.
This study has been conducted in the framework of the project DRESDYN (DREsden Sodium facility for DYNamo and thermohydraulic studies) which provides the platform for the precession dynamo experiment at HZDR. The au-
Figure 1 .
1labeled by j that abbreviates a triple index comprising the azimuthal wave number m, the axial wave number k, and a radial wave number index n. The last index counts the roots of the disper-arXiv:1708.06314v2 [physics.flu-dyn] (a) Sketch of the experimental set-up. The red dots denote the locations of the UDV probes in the water experiment and the arrows illustrate the propagation of the ultrasound signal. (b) Temporal evolution of the axial velocity u z at r = 150 mm (top: UDV measurements, center: simulations, bottom: comparison of simulations and experiments in the equatorial plane). (c) Isosurfaces showing a snapshot of u z from simulations at Re = 10 4 and Po = 0.1. Blue (red) colors indicate flow in negative (positive) direction (see movie at [16] for temporal evolution of u z ).
Figure 2 .
2(a) Fourier spectra for the (m, k) = (1, 1) mode and for the strongest secondary mode (m, k) = (2, 2) from simulations at r = 150 mm, Re = 10 4 and Po = 0.1. (b) Amplitude of the time-independent part of directly forced mode (m, k) = (1, 1), its multiple (m, k) = (2, 2) and the non-geostrophic axisymmetric mode (m, k) = (0, 2) (Re = 10 4 , r = 150 mm). (c) Relative amplitude of the non-geostrophic axisymmetric mode (m, k) = (0, 2) with respect to (m, k) = (1, 1). The solid curves in (b) and (c) denote results from the water experiment, and the diamonds denote results from simulations.
Figure 3 .
3(a) Time-averaged axisymmetric velocity field at Re = 10 4 and Po = 0.1. Colors denote u ϕ (without solid body rotation) and arrows represent u r and u z . (b) Axial profile of u ϕ and u z at r = 150 mm. Grey curves represent temporal variations of instantaneous profiles from simulations and red curves show the time average. The black curve in the bottom panel shows the time-averaged profile obtained in the water experiment. (c) Radial profiles of the time-averaged angular momentum including solid body rotation from simulations at Re = 10 4 .
Figure 4 .
4Growth rates for combinations of various azimuthal modes from the velocity field obtained at simulations at Re = 10 4 and Po = 0.1. The insert drawing depicts isosurfaces of the magnetic energy density mapped with B ϕ . In the precession frame the field structure propagates around the cylinder axis (see movie at[30]). the impact of the individual contributions on the dynamo. The temporal evolution of the magnetic flux density B induced by a given time-averaged flowū of a conducting liquid is determined by the induction equation
)Amplitude [mm/s]
0.00
0.05
0.10
0.15
0.20
0
5
10
15
20
25
30
m=1,k=1
m=2,k=2
m=0,k=2
Precession ratio
Amplitude [mm/s]
0.04
0.06
0.08
0.10
0.12
0.00
0.05
0.10
0.15
0.20
Re=1x10 5
Re=4x10 4
Re=2x10 4
Re=1x10 4
Re=1x10 4
(Simulations)
Precession ratio
u
z (m=0,k=2)/u
z (m=1,k=1)
further acknowledge support by the Helmholtz Allianz LIMTECH and Bernd Wustmann for the mechanical design of the experiment. * [email protected] acknowledge support by the Helmholtz Allianz LIMTECH and Bernd Wustmann for the mechanical design of the experiment. * [email protected]
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| {'fraction_non_alphanumeric': 0.07550312754963286, 'fraction_numerical': 0.07325945063910796, 'mean_word_length': 3.898751040799334, 'pattern_counts': {'":': 0, '<': 1, '<?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': 32, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We have conducted experimental measurements and numerical simulations of a precession driven flow in a cylindrical cavity. The study is dedicated to the precession dynamo experiment currently under construction at Helmholtz-Zentrum Dresden-Rossendorf (HZDR) and aims at the evaluation of the hydrodynamic flow with respect to its ability to drive a dynamo. We focus on the strongly nonlinear regime in which the flow is essentially composed of the directly forced primary Kelvin mode and higher modes in terms of standing inertial waves arising from nonlinear self-interactions. We obtain an excellent agreement between experiment and simulation with regard to both, flow amplitudes and flow geometry. A peculiarity is the resonance-like emergence of an axisymmetric mode that represents a double roll structure in the meridional plane. Kinematic simulations of the magnetic field evolution induced by the time-averaged flow yield dynamo action at critical magnetic Reynolds numbers around Rm c ≈ 430 which is well within the range of the planned liquid sodium experiment.', 'arxivid': '1708.06314', 'author': ['André Giesecke \nHelmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany\n', 'Tobias Vogt \nHelmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany\n', 'Thomas Gundrum \nHelmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany\n', 'Frank Stefani \nHelmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany\n'], 'authoraffiliation': ['Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany', 'Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany', 'Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany', 'Helmholtz-Zentrum Dresden-Rossendorf Bautzner Landstrasse 400D-01328DresdenGermany'], 'corpusid': 6927776, 'doi': '10.1103/physrevlett.120.024502', 'github_urls': [], 'n_tokens_mistral': 11490, 'n_tokens_neox': 9202, 'n_words': 4779, 'pdfsha': 'edf88c49dc51c10ee3af4c0fd636ea83a35309b0', 'pdfurls': ['https://arxiv.org/pdf/1708.06314v2.pdf'], 'title': ['Nonlinear large scale flow in a precessing cylinder and its ability to drive dynamo action', 'Nonlinear large scale flow in a precessing cylinder and its ability to drive dynamo action'], 'venue': []} |
arxiv |
Orientation relationship of FeNiC and FeNiCSi from variant detection in EBSD data
Mattis Seehaus [email protected]
Institut für Metallkunde und Materialphysik
RWTH Aachen University
52074, 2022Aachen, Germany
Risheng Pei
Institut für Metallkunde und Materialphysik
RWTH Aachen University
52074, 2022Aachen, Germany
Sandra Korte-Kerzel
Institut für Metallkunde und Materialphysik
RWTH Aachen University
52074, 2022Aachen, Germany
Stefanie Sandlöbes-Haut
Institut für Metallkunde und Materialphysik
RWTH Aachen University
52074, 2022Aachen, Germany
Orientation relationship of FeNiC and FeNiCSi from variant detection in EBSD data
1orientation relationshipEBSDmartensitic steel 2
The determination of orientation relationships in dual or multi-phase materials is very important in the field of interface engineering for the design of materials with tailored properties. In this work, a code is developed for the automated and statistical analysis of the orientation relationship of electron backscatter diffraction data. On the example of Fe-Ni-(Si)-C alloys containing lenticular martensite and retained austenite, the code is applied and it is shown that the orientation relationship (OR) corresponds to the Greninger-Troiano OR and that a statistically reliable investigation of the OR between the retained austenite and the related martensite variants is feasible using the code developed in this study.
Introduction
In the field of interface engineering, it is important to determine and identify the present orientation relationships (OR) between the different phases in dual or multi-phase materials. This helps to understand and tailor the material properties such as fracture toughness, strength, ductility and the mechanisms of (co-)deformation [1,2]. An example for materials with distinct ORs are steels containing martensite (bct) and austenite (fcc), whereby the character of the interphase boundary of the constituent phases affects the mechanical properties of the material.
In these steels, the OR between neighbouring austenite and martensite grains is mostly investigated by transmission electron microscopy (TEM) due to the usually small grain sizes [3][4][5][6][7]. However, the investigation by means of electron backscatter diffraction (EBSD) has been established in the last decade as a sufficiently performant method to investigate large, representative areas with respect to crystallographic information such as crystal structure (phase), grain orientation or misorientation across grain boundaries in martensitic, low-carbon or duplex steels [8][9][10][11][12][13].
The EBSD orientation maps, revealing the spatial orientation distribution, are recorded using a scanning electron microscope (SEM) and have the advantage of a higher statistical significance of the discovered ORs compared to the acquisition of individual orientation measurements in the TEM. The orientation maps obtained in this way contain a large amount of crystallographic data, which then leads to a statistically reliable analysis. Prior studies on the determination of ORs between martensite and austenite have revealed some advantages of EBSD over TEM, but have been carried out on individual laths only [14][15][16]. By reducing the acceleration voltage and / or using transmission Kikuchi diffraction (TKD) on thin (~100 nm) lamellae, the interaction volume between the electron beam and the sample can be reduced, enabling the measurement of small grains similar to TEM [17]. In contrast to the statistically low relevance of the TEM data, the large amount of orientation data obtained by EBSD provides the possibility to determine the differences between experimental pole figures and theoretical pole figures calculated from the respective OR models [18].
A basic prerequisite for such investigations is an OR model. Typically, the OR models define lattice directions and lattice planes. The statistically relevant identification of ORs using austenitic-martensitic steels as an example is shown in this work. Examples for well-established OR models for these steels are the Kurdjumov-Sachs (KS) and Nishiyama-Wassermann (NW) orientation relationships, that were discovered in low-carbon steels on a single austenite grain after the martensitic transformation and in iron-nickel alloys (30 % Ni), respectively [19][20][21]. These models appear in many studies to classify the OR of martensitic steels [10,22]. Plate martensite forms at temperatures down to those of liquid nitrogen, unlike lath martensite, resulting in a morphology of accommodated twins exhibiting the Greninger-Troiano (GT) OR [15,[23][24][25][26][27]. However, the KS and NW OR models are considered untenable based on the high indexed martensite habit planes and, moreover, numerous measurements show significant deviations from the ideal model conditions, which could still be referred to as near KS or near NW ORs. [7,28]. In Table 1 the main ORs proposed or observed are summarized [14,15,29]. Table 1 Observed orientation relationships (ORs) between austenite, γ, and martensite, α', in steels.
OR
Plane Direction
Bain {100}γ//{100}α <100>γ //<110> α NW {111} γ//{110} α <112> γ//<110> α KS {111} γ//{110} α <110> γ//<111> α GT {111} γ//{110} α <123> γ//<133> α Pitsch {100} γ//{110} α <110> γ//<111> α
In the present work, the OR between austenite and martensite is determined using TKD and EBSD in Fe-Ni-C-(Si) alloys, where lenticular martensite and retained austenite co-exist at room temperature. In steels, Si increases the strength and also the amount of retained austenite.
Further, Si addition also increases the temperature at which stable C clusters can form, which prevents carbide precipitation like cementite during tempering [30][31][32]. A code was developed to automatically determine the ORs from the EBSD and TKD data. As a result of the large data sets resulting from these EBSD and TKD mappings, a statistically reliable determination with regard to the martensite/austenite ORs is feasible.
Code development
For the identification of ORs, a code was developed that uses various methods for analysing two-phase or multi-phase materials in terms of orientation relationships and is accessible as supplementary material and at [34]. In this work, the code is used taking martensite and austenite in Fe-Ni-C(-Si) steels as an example.
In order to determine the orientation relationship between austenite and martensite as well as to compare them, the austenite grains were detected from the EBSD maps of the partially Moreover, it is feasible to select individual Phase 2 grains and either automatically determine the respective fraction of (converted) Phase 1 grains originating from these grains, or to select certain other phase 1 grains and display them correlatively in the pole figures. In Fig. 4a) such a combined EBSD/phase map is depicted with the three individual modes for partial intergranular orientation relationship plots. In Fig. 4b) the intergranular orientations of a single martensite (green box in the extracted region of Fig. 4c)) are shown. Based on the adaptive selection method, the orientations of the individual martensite grains can be compared and plotted as a predefined patterns (Fig. 4d)), as a manually created rectangular area or as individual EBSD data points with the orientations of the parent phase, depending on the specific use case. Using this mode, it is feasible to analyse the OR in specific areas close to interfaces, as the ORs at individual interfaces may vary.
Results
The procedures available in the code are demonstrated by using FeNiCSi, however, the analysis results of FeNiC obtained using the same procedures will be also presented in the final sections. (Table 1), enabling the direct comparison between experimentally measured and simulated orientation relationships, here using the (001), (011) and (111) (Table 2), as exemplified in Fig. 6d).
Discussion
An approach to compare the similarity of experimental data with potential orientation relationships is the analysis of the misorientation angles. The distributions of the misorientation angles of the martensite/austenite-phases are depicted in Fig. 7 within a range of ± 5 ° for FeNiC and FeNiCSi, respectively. The martensite-austenite misorientation profiles show clear peaks at a misorientation angle of about 44.05 °, indicating that the Greninger-Troiano orientation relationship is matching best to the experimental data.
Fig. 7 Misorientation angle distribution for FeNiC and FeNiCSi.
Specifically, the mean misorientation angle of FeNiC and FeNiCSi amounts to 44.43°, very close to the misorientation postulated by Greniger and Troiano, which lies between the KS and the NW orientation relationships (Table 3, Fig. 7) [35]. The analysis of the misorientation angle is not always sufficient to accurately determine an OR.
To confirm this finding, a subsequent pole figure analysis can be performed. As an approach to validate the obtained OR, normalised cross correlation (NCC), an inverse Fourier transform of the convolution of the Fourier transform of, e.g., two images, can be applied [36,37]. The normalisation is then done using the local cumulative sums and standard deviations. This type of correlation helps to ascertain the agreement between two data sets and has been applied to compare the simulated and experimental pole figures of the automatic selection mode visually in order to obtain quantitative correlation coefficients. In Fig. 8a) Table 1 are shown in Fig. 8d), indicating that the dominant OR in both alloys corresponds to the GT-OR with the highest correlation factors.
The effects of Si on the initial austenite grain sizes and hence on the thermally transformed martensites have no major impact on the determination of the OR as GT for both, FeNiC and FeNiCSi, in this work. In addition, the reference systems can be modified. In principle, the code was designed as a workflow for several use cases with some defined pre-settings, but it is also intended to be kept as user-friendly as possible, which means that functional components can be adjusted in the respective source codes. It is mainly optimised for two-phase systems, but can also visualise at least individual orientation relationships in multi-phase systems through the manual selection of grains. If the EBSD / TKD map quality is insufficient for a statistical evaluation, manual grain selection may be an option to determine the orientation relationship of individual areas.
Since the local misorientations within individual grains can diverge significantly, the manual mode enables the visualisation of the orientation relationship of single, acquired orientation spots or areas within a grain. Although this method could also be useful for e.g. phase shape memory alloys (Ti-Ni [38,39]) or intermetallic phases (Mg-Ag-Al [40]), it has so far only been investigated for the material systems described in this work.
The aforementioned concept is limited to use cases in which grains of both phases and their boundaries remain for OR analysis. Thus, it is not applicable after a complete phase transformation. In an Fe-24Ni-0.3C alloy, by comparing ultrafine with coarse grained austenitic microstructure, it was observed that the martensite/austenite interface exhibited the GT OR on both sides in coarse grains and in addition the K-S OR on the outer side of the martensite in ultrafine grains [41]. Micromechanical effects, such as strain fields or distortions between neighbouring martensite variants, could have an influence on the crystal orientation of the martensite variants, which complicates a precise differentiation of individual ORs [42]. In an Fe-33Ni alloy, it was found that the OR in lenticular martensite varied from the midrib (GT) to the austenite/martensite interface (KS) [15]. However, in an Fe-31%Ni-0.01C alloy, where partially transformed lenticular martensite in conjunction with austenite was present, it was reported that although there is a scatter of orientations, near both, the interface and the midrib, the misorientations are closer to the GT orientation, resulting in the OR being the same between the midrib and the interfacial region [43]. Due to the potential existence of several ORs in one material and to enable a more representative demonstration of the ORs, further work is required.
All the codes discussed in this work are maintained in an online repository available at the following reference: [34].
Conclusions
A Matlab® code based on the MTEX toolbar for the quantitative identification of orientation relationships with statistical relevance has been developed. As an example, the OR between retained austenite and martensite was determined and statistically evaluated. From our study the following conclusions have been derived:
-The code contains different modes that can be used for an in-depth analysis of obtained ORs, specifically, automatic, semi-automatic and manual selection of grains or microstructure areas. -For quantitative analysis of the observed OR, the minimum rotation angle deviations between the experimental and the theoretically calculated orientation relationships were determined for different ORs. This method was successfully applied to confirm the previously identified Greninger-Troiano OR for the present example case.
-Although the capabilities of the developed code on the example of martensite and austenite in steels have been demonstrated, it can be used for many other applications where an orientation relationship between different phases is relevant. An adaptation of the present method for the investigation of ORs in other material systems such as intermetallic phases, shape memory alloys, thin films or other composite materials could provide insights into the existing ORs.
referred to as FeNiC and FeNiCSi, respectively) have been produced by casting, hot rolling at 1050 °C, subsequent cold rolling and recrystallization treatment at 950 °C. At room temperature, these two materials are completely austenitic. The martensitic transformation takes place at temperatures below 0 °C, which facilitates the investigation of the austenitic microstructure at RT before the martensitic transformation. Quenching of only half of the samples in liquid nitrogen preserves a high proportion of austenite, which further enables the determination of the orientation relationship between α' and γ. Samples for microstructure characterisation were cut from the rolled sheet by electrical discharge machining, mechanically ground, polished to 0.25 μm with diamond suspension and finally electrolytically polished with A2 electrolyte at 30 V for 30 s. SEM and BSE images were taken to investigate the microstructure of the two alloys. Different substructures of the lenticular martensite are clearly visible caused by the presence and absence of silicon as shown in Fig. 1. Characteristic for lenticular martensite is the midrib, which consists of a structure of fine transformation twins from which growth to the lenticular form is initiated.
Fig. 1
1Microstructure of the lenticular martensite in a) FeNiC and b) FeNiCSi by means of backscatter electron imaging, respectively.EBSD was applied to perform microstructure analysis in FEG-SEM (Helios NanoLab 600i, FEI Co) at 15 kV and 1.4 nA (Hikari, EDAX Inc.). The EBSD maps cover an area of 40 x 30 μm², were recorded with a step size of 150 nm, and were analyzed using the free Matlab® toolbox MTEX 5.7.0[33].
transformed samples and their orientation matrices were rotated according to the standard projection along (001). In conjunction with this orientation transformation, the transformed martensite grains were selected and rotated with the corresponding rotation matrices with reference to the sample symmetry. Several different workflows are enabled and visualised inFig. 2.
Fig. 2
2Workflow of the OR analysis code. Orange segments show preceding preparations and measurements, blue shows modes 1, 2 and 3 whereas green illustrates the three submodes of mode 2 and purple depicts potential analysis after the code has been executed.The developed Matlab code enables the fully automated characterisation of an entire EBSD map and plotting of the result as pole figures. If only specific grains or individual interfaces of an EBSD are of interest, the manual selection of grains is also possible. Furthermore, the different modes can be used to correlate the orientations within a Phase 1 (parent phase) grain or individually selected EBSD data points with the Phase 2 (transformed) grain orientations.In the automated mode, the EBSD data set is first arranged in descending order of the size of the phase 2 grains and the orientations of a predefined number of the largest grains are stored.This selection is expanded to all austenite grains that exhibit the same orientations within a deviation angle of φ=5° to improve statistics. Furthermore, the adjacent Phase 1 grains sharing a grain boundary with the filtered Phase 2 grains are assigned to them and their orientations are stored. Consequently, the Phase 2 grains are rotated to the standard orientation and the corresponding transformation matrices are stored in order to rotate the corresponding Phase 1 grains accordingly. InFig. 3an example of an EBSD map of martensite and austenite, the corresponding phase map and the combined EBSD/phase map after completion of the code analysis are depicted.
Fig. 3 a
3) EBSD map of FeNiC revealing the orientation data b) Phase map showing martensite/austenite grains, respectively and c) combined EBSD/Phase map revealing both the analysed selected austenite grains with the corresponding martensite grains in terms of their OR and the remaining unexamined grains according to their orientation.
Fig. 4 a
4) Combined EBSD and phase map of FeNiC representing the mean austenite orientations as indicated by the inverse pole figure colour code with reference to the norm direction. The martensite is represented in red colour only. b) intergranular orientations of martensite and the corresponding austenite grain in c); d) distinct predefined range, individually selected rectangular area and individually predefined data points of the EBSD data within the martensite and the corresponding austenite grain. The white spots indicate non-indexed points.
Fig. 5
5shows the measured orientations of martensite and austenite of FeNiCSi and also simulated pole figures of the possible ORs
pole figures. The blue orientations in the pole figures of FeNiCSi correspond to the austenite grains after rotation into the reference position.
Fig. 5
5Pole figures calculated for different theoretical orientation relationships, namely Kurdjumov-Sachs (KS), Nishiyama-Wasserman (NW), Greninger-Troiano (GT), Pitsch and experimentally obtained data exemplary shown for FeNiCSi showing the reference austenite orientation in blue. An example for the semi-automatic selection of grains is depicted in Fig. 6. The blue orientations in the pole figures of FeNiCSi correspond to the austenite grains after rotation into the reference position. Likewise, the rotation angle between the martensite and austenite orientations are utilised to identify the variants. The lowest rotation angle deviation and thus the variant is determined by comparison with martensite variants previously calculated based on the closest matching OR
Fig. 6 a
6) Selection of grains to be analysed (white) with respect to their orientation relationship or variants. b) Extraction and highlighting of the selected grains visualised in c) for martensite with respect to the rotated austenite orientations shown in blue in the pole figure. d) Histogram of the variants based on the smallest deviation of the rotation angle of the exemplary martensite variants from the extracted grains with respect to the possible simulated variants (here GT).
-c) the subtracted simulated and experimental (001), (011) and (111) pole figures of FeNiCSi are depicted. In addition, the correlation factors of the possible ORs given in
Fig. 8
8Subtraction of simulated (here GT) from experimental a) (001) b) (011) and c) (111) pole figures for FeNiCSi; d) Correlation coefficients of FeNiC and FeNiCSi, respectively, with the simulated pole figures to enable quantitative analysis. Another approach to statistically analyse the experimentally measured with the simulated pole figures, the orientation data were compared in terms of the minimum deviation of the rotation angles between the measured orientation and all martensite variants of the respective ORs according to a certain threshold value. Fig. 9 a) displays the experimental raw data of the (011) pole figure, while b) and c) present the filtered experimental data for a minimum rotation angle deviation with a threshold of <10 ° and <5 °, respectively. For a threshold value of <5°, about ~86 % of all orientations of the original EBSD map are preserved, allowing a statistical analysis of the large majority of the experimental data. As a result of multiple phases or grain boundaries in material systems, the EBSD measurement is partially unable to assign the correct diffraction patterns unambiguously. In comparison of all ORs at the different threshold values and the average values of both alloys, the smallest rotational deviations are obtained for the GT OR, demonstrating that the misorientation distribution and combined pole figure comparison are suitable to determine orientation relationships.
Fig. 9
9Pole figures of the raw and filtered orientation data measured for FeNiCSi and the statistical evaluation of the orientation data exemplary for the (011) pole figure. a) raw pole figure, b) pole figure filtered by rotation angle deviation <10° and c) pole figure filtered by rotation angle deviation <5° and d) histograms showing the mean values for each set of minimum rotation angle deviation for each OR revealing that the lowest rotational deviations are present for the GT orientation relationship.
-
The automatic mode allows a statistically relevant and fully automatic evaluation of the orientation relationship of a large range of grains and thereby allows the determination of the predominant OR. This is done by comparing the pole figures of the corresponding phases of interest, e.g., retained parent austenite grains and the adjacent martensite grains, with a calculated theoretical orientation relationship. -Additionally, the manual evaluation mode provides local insights into the precise OR for chosen grains or individual interfaces, in the present case a determination of the occurring martensite variants was possible. The selection of individual measurement points instead of whole grains is also possible and could be used to evaluate the behavioural impact of intragranular misorientations on the OR or can be applied if the indexing rate of the EBSD data is low. -Exemplarily, the code was applied to an austenitic-martensitic steel where the successful identification and evaluation of the OR between austenite and martensite was demonstrated. The comparison of the misorientation angle distribution with the misorientation angles of specific ORs supports the identification of the Greninger-Troiano OR for both FeNiC and FeNiCSi. This was further confirmed by an image correlation algorithm, comparing calculated theoretical variants with the experimental pole figures.
Table 2
2Calculated variants from the GT ORVariant No.
Mis. Angle from V1
Rotation axis from V1
φ1
Φ
φ2
1
0
-
324.79
170.10
191.32
2
55.26
[-0.71 0.00 0.71]
98.12
84.31
136.13
3
60
[-0.71 0.00 0.71]
351.88
95.70
316.13
4
4.74
[-0.71 0.00 0.71]
125.21
9.90
11.32
5
60
[-0.71 0.00 0.71]
275.75
98.08
46.94
6
60.25
[-0.53 0.53 0.66]
174.25
81.92
226.94
7
50.67
[-0.64 0.42 0.64]
261.88
95.70
316.13
8
16.61
[-0.69 0.23 0.69]
35.21
9.90
11.32
9
55.24
[-0.67 0.23 0.71]
185.75
98.08
46.94
10
50.14
[-0.47 0.57 0.68]
84.25
81.92
226.94
11
13.99
[-0.55 0.06 0.83]
234.79
170.10
191.32
12
50.65
[-0.67 0.26 0.70]
8.12
84.31
136.13
13
13.99
[-0.06 0.55 0.83]
54.79
170.10
191.32
14
50.14
[-0.57 0.47 0.68]
188.11
84.31
136.13
15
52.21
[-0.21 0.65 0.73]
81.88
95.70
316.13
16
11.60
[-0.69 0.19 0.69]
215.21
9.90
11.32
17
49.64
[-0.60 0.52 0.60]
5.75
98.08
46.94
18
56.85
[-0.19 0.66 0.72]
264.25
81.92
226.94
19
55.24
[-0.23 0.67 0.71]
354.25
81.92
226.94
20
50.65
[-0.26 0.67 0.70]
95.75
98.08
46.94
21
19.71
[-0.14 0.00 0.99]
305.21
9.90
11.32
22
56.84
[-0.66 0.19 0.72]
171.88
95.69
316.13
23
52.21
[-0.65 0.21 0.73]
278.11
84.31
136.13
24
19.80
[-0.20 0.00 0.98]
144.79
170.10
191.32
Table 3
3Overview of the different ORs between fcc and bcc crystals and the respective misorientation axis/angle.OR
<uvw>
ωmin
Bain
<100>
45°
NW
<0.98 0.08 0.20>
45.98°
KS
<0.97 0.18 0.18>
42.85°
GT
<0.97 0.19 0.13>
44.23°
Pitsch
<0.08 0.20 0.98>
45.98°
Table 4
4presents the correlation coefficients and rotational deviations of FeNiC and FeNiCSi
when considering the possible orientation relationships given in Table 1. The higher the
correlation coefficient of the experimental data with one of the possible ORs, the higher the
similarity between them. Similarly, a lower rotational deviation indicates higher similarity. In
the present examples, all methods to determine the OR show the highest match with the GT-
OR.
Table 4
4Mean correlation coefficients and mean orientation angle deviation calculated forcomponent is the automatically generated statistical evaluation of all grain orientations within an entire EBSD / TKD measurement and relating them to a predefined orientation relationship.FeNiC and FeNiCSi.
OR
NW
KS
GT
Pitsch
Mean Ccorr (FeNiC)
28.50
24.05
30.51
23.33
Mean Ccorr (FeNiCSi)
31.05
25.41
32.84
23.66
Mean σ (FeNiC)
3.48
4.39
2.97
6.09
Mean σ (FeNiCSi)
3.70
4.54
3.12
6.04
The experimental data of the martensite variants of both, FeNiC and FeNiCSi, presented in
austenite reference pole figures in this work (Fig. 5) provide a statistically close correspondence
with the GT OR in comparison with other established ORs like KS, NW or Pitsch. It has been
shown that the code developed for identifying orientation relationships gives quite
unambiguous results for reasonably accurate EBSD and / or TKD datasets. The main
AcknowledgementThe authors gratefully acknowledge funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through project 406912286 (C-Tram).Data AvailabilityThe raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.
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arxiv |
20 May 2001
R K Bhaduri
Department of Physics and Astronomy
McMaster University
L8S 4M1HamiltonOntarioCanada
Diptiman Sen
Centre for Theoretical Studies
Indian Institute of Science
560012BangaloreIndia
20 May 2001Comment on "Low-dimensional Bose liquids: beyond the Gross-Pitaevskii approximation"PACS numbers: 0530Jp, 0375Fi
This is a comment on the work of Kolomeisky et al., Phys. Rev. Lett. 85, 1146. We point out that they are using the wrong form of the energy functional for one-dimensional fermions.We point out two possible forms of the energy functional, both of which can be derived from first principles but using different methods. One is obtained from the collective field theory method, while the other is derived from the extended Thomas-Fermi method. These two forms of the energy functional do not support the soliton solutions which are obtained by Kolomeisky et al.
The Gross-Pitaevskii (GP) mean-field theory replaces the bosonic field operator by a classical field Φ(r, t) [1]. This approach has been highly successful in describing a dilute gas of trapped bosonic atoms [2]. However, some authors have pointed out that the usual potential energy |Φ| 4 which arises from the zero-range pseudopotential in three dimensions, needs to be modified in lower dimensions. Kolomeisky et al. [3] have proposed replacing the quartic term by a |Φ| 6 term in one dimension. This can be motivated by considering a one-dimensional Bose gas in which particles interact pair-wise via a repulsive δ-function potential. This is the Lieb-Liniger model which is exactly solvable [4]. The model has only one dimensionless parameter, namely, g =h 2 ρ/mu 0 , where ρ = |Φ| 2 is the density, u 0 is the strength of the δ-function interaction, and m is the particle mass. In the limit of g → 0 (called the dilute approximation), this system is equivalent to a gas of noninteracting fermions whose energy density is given by π 2h2 ρ 3 /6m. Kolomeisky et al.
use this energy density to study a dilute Bose gas and to obtain a stationary soliton solution.
Our main criticism of their work is that they are using the wrong form of the second derivative terms in the energy functional for one-dimensional fermions, as indicated below. Since a dilute Bose gas with repulsive interactions in one dimension is equivalent to a system of noninteracting fermions, one way to proceed is to use the collective field theory (CFT) method to derive the Hamiltonian in terms of the density variables. This was found long ago [5]. In the absence of an external potential, the energy density is given by
H =h 2 2m [ ρ( ∂θ ∂x ) 2 + π 2 3 ρ 3 ] ,(1)
where the density and phase fields ρ and θ are related to the order parameter field by Φ = √ ρe iθ . The equations of motion for this system are easily obtained [6], and these do not support any solutions in which ρ is timeindependent and inhomogeneous. This seems to contradict the fact that a system of noninteracting fermions can have a time-independent and inhomogeneous density profile [7], for instance, in the vicinity of a hard 1 wall. The exact density profile in both those cases has damped oscillations whose wavelength is of the order of the interparticle separation. This shows a limitation of the CFT method; the CFT Hamiltonian given in Eq.
(1) gives the correct description of a system of noninteracting fermions only for long-wavelength density fluctuations [6]. A different derivation of the kinetic energy functional for noninteracting fermions placed in an external potential is based on the extended Thomas-Fermi method [8]. In one dimension, to orderh 2 , this gives the inhomogeneous terms in the kinetic energy density to bē Our second comment on the work of Kolomeisky et al. is that the dilute approximation is not obtained in any of the Bose condensates studied so far [2]. In the experimental systems, the equivalent of the parameter g is N a/a HO , where a is the scattering length, and a HO is the harmonic confinement length. This equivalence follows because the particle density ρ in the center of the trap is of the order of N/a HO , while the scattering length a is proportional to the strength of the pseudopotential in three dimensions and is therefore analogous to u 0 in the one-dimensional problem. Ref. [2] states that N a/a HO typically goes from a number of order 1 to several thousands, because a/a HO is usually of the order of 10 −3 while N typically goes from 10 3 to 10 6 . Hence the experimental systems cannot be considered to be dilute in the Lieb-Liniger sense, and the mapping from interacting bosons to noninteracting fermions in one dimension is not valid for such systems.
h 2 /2m(−ρ ′2 /12ρ + ρ ′′ /3
This research was partially supported by NSERC of Canada.
). This is different, in both sign and magnitude, from the term used by Kolomeisky et al., and has no solitonic solutions. In contrast to this, the Hamiltonian of Kolomeisky et al. does support stationary solutions with inhomogeneous densities such as solitons. However, this does not by itself justify the addition of the termh 2 ρ ′2 /8mρ to the Hamiltonian (1) which describes noninteracting fermions.
. L P Pitaevskii, Sov. Phys. JETP. 13451L. P. Pitaevskii, Sov. Phys. JETP 13, 451 (1961);
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. M D Girardeau, E M Wright, Phys. Rev. Lett. 845691M. D. Girardeau and E. M. Wright, Phys. Rev. Lett. 84, 5691 (2000).
M Brack, R K Bhaduri, Semiclassical Physics. ReadingAddison-WesleyM. Brack and R. K. Bhaduri, Semiclassical Physics (Addison-Wesley, Reading, 1997).
| {'fraction_non_alphanumeric': 0.056181533646322376, 'fraction_numerical': 0.03630672926447574, 'mean_word_length': 3.9351351351351354, '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': 'This is a comment on the work of Kolomeisky et al., Phys. Rev. Lett. 85, 1146. We point out that they are using the wrong form of the energy functional for one-dimensional fermions.We point out two possible forms of the energy functional, both of which can be derived from first principles but using different methods. One is obtained from the collective field theory method, while the other is derived from the extended Thomas-Fermi method. These two forms of the energy functional do not support the soliton solutions which are obtained by Kolomeisky et al.', 'arxivid': 'cond-mat/0105385', 'author': ['R K Bhaduri \nDepartment of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada\n', 'Diptiman Sen \nCentre for Theoretical Studies\nIndian Institute of Science\n560012BangaloreIndia\n'], 'authoraffiliation': ['Department of Physics and Astronomy\nMcMaster University\nL8S 4M1HamiltonOntarioCanada', 'Centre for Theoretical Studies\nIndian Institute of Science\n560012BangaloreIndia'], 'corpusid': 7459048, 'doi': '10.1103/physrevlett.86.4708', 'github_urls': [], 'n_tokens_mistral': 2027, 'n_tokens_neox': 1732, 'n_words': 1073, 'pdfsha': '803a9c55c19c46890d023858162f210ae928c951', 'pdfurls': ['https://arxiv.org/pdf/cond-mat/0105385v1.pdf'], 'title': [], 'venue': []} |
arxiv |
A Comprehensive Performance Evaluation of a DF-Based Multi-Hop System Over α − κ − µ and α − κ − µ-Extreme Fading Channels
22 Mar 2019
Tau Raphael Rasethuntsa
are with the Central Research Laboratory
Bharat Electronics Limited
Uttar PradeshGhaziadbadIndia
Sandeep Kumar
are with the Central Research Laboratory
Bharat Electronics Limited
Uttar PradeshGhaziadbadIndia
Manpreet Kaur [email protected]
are with the Central Research Laboratory
Bharat Electronics Limited
Uttar PradeshGhaziadbadIndia
Moreneng, Maseru, LesothoAbia / Mahlabatheng
are with the Central Research Laboratory
Bharat Electronics Limited
Uttar PradeshGhaziadbadIndia
Sandeep Kumar
are with the Central Research Laboratory
Bharat Electronics Limited
Uttar PradeshGhaziadbadIndia
Manpreet Kaur
are with the Central Research Laboratory
Bharat Electronics Limited
Uttar PradeshGhaziadbadIndia
A Comprehensive Performance Evaluation of a DF-Based Multi-Hop System Over α − κ − µ and α − κ − µ-Extreme Fading Channels
22 Mar 20191 2
In this work, an integrated performance evaluation of a decode-and-forward (DF) multi-hop wireless communication system is undertaken over the non-linear generalized α − κ − µ and α − κ − µ-Extreme fading models. Analytical formulas for the probability density function (PDF) and the cumulative distribution function (CDF) of the received signal-to-noise ratio (SNR) as well as its generalized moments and moment generating function (MGF) are derived. Based on the derived PDFs, novel closed-form expressions for traditional performance metrics such as amount of fading (AF), outage probability (OP), bit error rate (BER) under coherent and non-coherent modulation schemes as well as channel capacity under various adaptive transmission techniques are derived. Additionally, asymptotic analyses of BER based on Poincare series expansions of SNR PDFs are carried out and results show good approximations for low SNR regimes. The correctness of the proposed solutions has been corroborated by comparing them with Monte Carlo simulation results. Index Terms Multi-hop system, decode-and-forward relay, bivariate Fox H-function, non-linear generalized fading, Marcum Q-function, Nuttall Q-function Tau Raphael Rasethuntsa is based in, Lenasia Extension 9, Gauteng Province, Johannesburg, South Africa and Ha
I. INTRODUCTION
The use of multi-hop communication systems as a way of increasing network coverage area and energy-efficiency with high data rates is gaining importance in next-generation wireless networks. In a multi-hop wireless communication system, information is transmitted from a source device to a destination device through several intermediate nodes, referred to as relays.
Based on the nature and complexity of the relaying system, multi-hop communication systems can be classified into two categories, namely (i) decode-and-forward (DF) and (ii) amplify-andforward [1]. For a DF-based multi-hop system, the relaying node receives the encoded signal, decodes it to regenerate the original symbol and forwards the newly encoded signal to the next node [2]. DF-based multi-hop systems have several applications in ad-hoc networks, sensor networks and microwave links among many others [3].
The performance analysis of multi-hop communication systems over numerous fading distributions has been well documented in the literature [3], [4], [5], [6], [7]. The ergodic capacity of a multi-hop relaying system over Rayleigh fading channels has been studied in [4]. The end-toend performance of a DF relaying system based on Ricean and Nakagami-m fading channels was explored in [5]. Yang et al. [8] studied the unified performance analysis of a DF-based multi-hop system based on second order statistics in average outage duration (AOD) and level crossing rate (LCR) over Nakagami, Rayleigh and Rician fading channels. The performance study of a multi-hop communication system with regenerative relays has been investigated over generalized η − µ, κ − µ and α − µ fading channels in [6], [7] and [9], respectively. Using traditional performance metrics as well as LCR and amount of fade duration (AFD), Cao et al. [3] presented the performance analysis of end-to-end wireless links over the Generalized-K fading channel.
The modeling of mobile radio channel is a challenging task as the wireless signal is subjected to various hurdles/corruptions in its propagation path from the transmitter to the receiver [10].
Accurate characterization of the wireless channel is therefore of paramount importance for the realistic assessment of wireless systems performances. Several statistical distributions were developed in the literature to model small-scale fluctuations of the fading channel envelope [11].
For instance, motivated by the pioneering works in [12] and [13], the non-linear generalized α − κ − µ and α − κ − µ-Extreme fading models were proposed in [14], [15] and [16]. The α − κ − µ distribution is a very flexible model that does not assume homogeneous scattering and can account well for non-linearity of multi-path fading [17]. The model includes as special cases several other well-known distributions. In particular, when α = 2, the α − κ − µ distribution reduces to the κ − µ distribution [12] and for κ → 0, the α − µ is obtained [7]. The α − κ − µ-Extreme fading model is derived from the α − κ − µ distribution by allowing µ → 0 and κ → ∞ in order to better characterize small-scale variations of mobile radio propagation under non-linear severe multi-path fading effects usually encountered in enclosed environments [15]. For α = 2, the α − κ − µ-Extreme distribution reduces to a linear severe fading model, the κ − µ-Extreme distribution introduced by Rabelo et al. [13]. Although there are many works related to the study of DF-based multi-hop relaying systems over different fading channels, none of the previous works have investigated the end-to-end performance over α−κ−µ and α−κ−µ-Extreme fading channels. Motivated by this void, we have derived the closed-form mathematical expressions for amount of fading (AF), outage probability (OP), bit error rate (BER) (coherent and non-coherent) and channel capacity for ene-to-end DF relaying system under various adaptive transmissions over α−κ−µ and α−κ−µ-Extreme fading channels. The expressions derived here are presented in a simple manner to optimize clarity and readability. Furthermore, the results produced here are generalized expressions and valid for other fading distributions as special cases. The rest of the paper is organized as follows : The system and channel model are described in Section II. In Section III, we provide the closed-form expressions for the probability density function (PDF), cumulative distribution function (CDF) generalized moments and moment generating function of the end-to-end signal-to-noise ratio (SNR). Different performance metrics for the DF-based multi-hop wireless system over α − κ − µ and α − κ − µ-Extreme fading channels are presented in Section IV. Asymptotic analysis of BER based on Poincare expansions of fading model (PDFs) is carried out in Section V. Performance evaluation, simulation results and discussions are provided in Section VI. Finally, we round up the paper in Section VII with conclusions and possible future considerations.
II. MODEL FORMULATION AND DESCRIPTION
A. System and Channel Models
Consider an n-hop system shown in Figure 1 transmitting information from the source S to the destination D via n − 1 intermediate relays R 1 , R 2 , . . . , R n−1 in time-sharing principles. If the SNR of the i-th hop is denoted by γ i , with PDF f γ i , i = 1, . . . n, the statistical properties
1 2 1 n n S D R1 R2
Rn-1 Figure 1. A n-link DF-based multi-hop system.
Relays
of the end-to-end system SNR are then determined by the SNR of the last hop γ n . According to [18], the PDF of the end-to-end system SNR γ can be written as
f γ (γ) = Aδ(γ) +Āf γn (γ),(1)
where A denotes the probability of the first outage occurring for the first n − 1 hops,Ā = 1 − A and δ(·) is the Dirac delta function. The probability A is given by [18]
A = 1 − P r (γ 1 > γ th , γ 2 > γ th , · · · , γ m−1 > γ th ) = 1 − m−1 k=1 P r (γ n > γ th ) = 1 − m−1 k=1 [1 − F γn (γ th )] ,(2)
where F γn is the CDF of γ n .
1) The α − κ − µ and α − κ − µ-Extreme fading models: The PDF of the instantaneous SNR of the n-th hop γ n > 0 following the α n − κ n − µ n distribution is given by
f γn (γ n ) = K n γ ( αn 2 )ωn−1 n e −σnγ αn 2 n I µn−1 θ n γ αn 2 n ,(3)
where, in order to improve clarity in presentation, we have set
K n = α n σ ωn n e −µnκn 2(µ n κ n ) ωn−1 , ω n = µ n + 1 2 , σ n = µ n (κ n + 1) γ αn 2 n , θ n = 4µ n κ n σ n(4)
with α n , µ n , κ n > 0. The average SNR per symbolγ n is given bȳ
γ n = e µnκn [µ n (κ n + 1)] 2 αn Γ 2 αn + µ n Φ 2 αn + µ n ; µ n ; µ n κ n E b N 0 ,
where E b denotes the energy per symbol and N 0 is the single-sided power spectral density. Using the method of [19], one can easily show that the CDF of γ n is written as
F γn (γ n ) = 1 − Q µn 2κ n µ n , 2σ n γ αn 2 n , γ n > 0,(5)
where Q(·, ·) is the generalized Marcum Q-function. The PDF of the instantaneous SNR of the n-th hop γ n > 0 following the α n − κ n − µ n -Extreme distribution is given by g γn (γ) = a n γ αn/4−1 n e bnγ
αn 2 n I 1 c n γ αn 2 n + e −2mn 2 √ γ nγn δ γ n γ ,(6)
where a n = α n m n e −2mn
γ αn/4 n , b = 2m n γ αn 2 n , c n = (16m 2 n ) γ αn 2 n ,(7)
with α n , µ n , κ n > 0 and m n = µ n (κ n + 1) 2 /(2κ n + 1) as the Nakagami parameter which is inversely proportional to the fading severity. The average SNR per symbol is given bȳ
γ n = e 2mn (2m n ) 2 αn −1 Γ 2 αn + µ n Φ 2 αn + µ n ; µ n ; µ n κ n E b N 0 .
The CDF of the α − κ − µ-Extreme fading model can also be derived using the method of [19] resulting in
G γ (γ) = 1 − Q 0 2 √ m n , 2 m n (γ/γ) α 2 .(8)
The symbols in (4) and (7) will be employed throughout the paper unless the original notation produces shorter expressions.
Remark 1. It is also worth noting that the proposed PDFs representations renders, purely on a symbolic level, the α − κ − µ-Extreme fading model PDF without the extra δ(·) term as a special case of the PDF of the α − κ − µ model in a sense that K n = a n , ω n = 1/2, σ n = b n , θ n = c n and µ n = 2. This observation will be particularly useful for reducing redundancy in the forthcoming derivations.
III. STATISTICAL PROPERTIES
A. PDF and CDF of End-to-End SNR Case I : Assume that the SNRs of the n hops are independent and not necessarily identically distributed (INID) random variables following the α − κ − µ distribution. Then we have from (3) and (5) that the PDF and CDF of the end-to-end SNR γ must be given respectively by
p γ (γ) = Aδ(γ) + K nĀ γ ( αn 2 )ωn−1 e σnγ αn 2 I µn−1 θ n γ αn 2 (9) where A = 1 − n−1 k=1 Q µ k √ 2κ k µ k , 2σ k γ α k 2 th and P γ (γ) = 1 −ĀQ µn 2κ n µ n , 2σ n γ αn 2 .(10)
Case II : On the other hand if the SNRs of the n hops are INID random variables following the α − κ − µ-Extreme distribution, then we have from (6) and (8) that the PDF of the end-to-end SNR γ must be given by p e γ (γ) = A e δ(γ) +Ā e a n γ
I 1 c n γ αn 2 + δ * (γ)(11)whereδ * (γ) = e −2mn 2 √ γγ n δ γ n γ and A e = 1 − n−1 k=1 Q 0 2 √ m k , 2 m k (γ k /γ k ) α k 2 .
The CDF of γ can be written as
P e γ (γ) = 1 −Ā e Q 0 2 √ m n , 2 m n (γ/γ) αn 2 .(12)
B. Moments of End-to-End SNR
Generalized moments of the end-to-end SNR are useful for obtaining r-th-order performance metrics such as the mean, variance as well as kurtosis and skewness of the distribution of γ.
The r-th moment of a positive random variable X with density f X is defined by
E X r ∞ 0 x r f X (x)dx(13)
where E · denotes the expectation operation. For Case I, substituting (9) into (13) and setting y = γ α 2 /2 , the r-th moment of γ can be written with the aid of [20, Eq. 6.643 (2)] and [20, Eq.
6.220 (2)] as E γ r =Āγ r n Γ 2r αn + µ n Φ 2r αn + µ n ; µ n ; µ n κ n e µnκn [µ n (κ n + 1)] 2r αn (14) whereΦ is the regularized confluent hypergeometric function [20, Eq. 9.210(1)]. Similarly, for
Case II, we have the r-th moment of γ as E e γ r =Ā eγr n Γ 2r αn + 1 Φ 2r αn + 1; 2; 2m n e 2mn (2m n ) 2r αn −1 (15) It is also worth noting that as expected, upon substituting r = 1 in (14) and (15), one obtains γ =Āγ n andγ =Ā eγ n for each case, respectively.
C. Moment Generating Function
Another useful statistical characteristic of random variables is the moment generating function (MGF) used to obtain n-th order moments or BER under non-coherent modulation, see Section
IV. The MGF of γ, denoted by M γ , is defined as M γ (s) ∞ 0 e −sγ f γ (γ)dγ.(16)
For Case I, we substitute (9) into (16) to get the integral
M γ (s) = A + K nĀ I 1 ∞ 0 γ ( αn 2 )ωn−1 e sγ e σnγ αn 2 I µn−1 θ n γ αn 2 dγ(17)
In order to evaluate I 1 , we set y = γ αn/2 and employ [21, Eq. and reversing the order of integration, we have that
I 1 =π ∞ 0 y ωn−1 e −σny H 1,0 0,1 s αn 2 y − 0, αn 2 × H 1,0 1,3 θ n 4 y µn 2 , 1 (ω n − 1, 1), (1 − ω n , 1), µn 2 , 1 dy(18)I 1 =π 1 (2πi) 2 Lr Lt G ∞ 0 y ωn+r+t−1 e −σny dy × Γ − αn 2 r Γ(ω n − 1 − t) s αn 2 r θn 4 t drdt Γ(ω n + t)Γ 1 − µn 2 + t Γ µn 2 − t(19)
where L r and L t are two suitable contours in the complex r and t planes respectively. The integral G can be evaluated with the aid of [20, Eq. 3.326(2)] to yield
I 1 = π σ ωn n (2πi) 2 Lr Lt Γ − αn 2 r Γ(ω n − 1 − t) Γ(ω n + t) × Γ (ω n + r + t) s αn 2 /σ n r θn 4σn t drdt Γ 1 − µn 2 + t Γ µn 2 − t(20)M γ (s) = A +ĀK n π σ ωn n H 0,1:1,0;1,0 1,0:1,0;1,3 s αn 2 σ n θ n 4σ n ∆ (α n , µ n , ω n ) ∇ (α n , µ n , ω n ) (21)
M e γ (s) = A e +Ā e a n π √ b n H 0,1:1,0;1,0
1,0:1,0;1,3 s αn 2 b n c n 4b n ∆ α n , 2, 1 2 ∇ α n , 2, 1 2 . (22) ∆ ≡ (1 − ω n ; 1, 1) : − ; µ n 2 , 1 ∇ ≡ − : 0, α n 2 ; (ω n − 1, 1), (1 − ω n , 1), µ n 2 , 1
where (ω n +r +t) > 0. Now comparing (20) (22) of Table I.
IV. PERFORMANCE ANALYSIS OF MULTI-HOP DF-BASED RELAY SYSTEMS
A. Amount of Fading
The second-order AF is a useful performance metric that can be used to account for the severity of fading. AF is defined as the ratio of the variance to the square average SNR per symbol, AF
(2) γ Var(γ)/γ 2 . That is, AF (2) γ E γ 2 − E γ 2 E γ 2 .(23)
This can be generalized to higher-order fading as AF (14) into (23) yields after some basic simplifications the second-order AF of γ under Case I as
(n) γ E γ n /(E γ n ) − 1. Substitution of E γ and E γ 2 fromAF γ = e µnκn Γ 4 αn + µ n Φ µ n + 4 αn ; µ n ; κ n µ n Ā Γ 2 αn + µ n 2Φ µ n + 2 αn ; µ n ; κ n µ n 2 − 1(24)
Likewise, for Case II, the second-order amount of fading can be derived using (15) in (23) resulting in
AF e γ = e 2mn Γ 4 αn + 1 Φ 4 αn + 1; 2; 2m n Āe 2m n Γ 2 αn + 1 2 Φ 2 αn + 1; 2; 2m n 2 − 1(25)
B. Outage probability
The OP is defined as the probability that the SNR per symbol falls below a certain threshold γ th . By employing (10) and (12), this probability can be written for Case I as
P out = P (γ ≤ γ th ) = 1 −ĀQ µn 2κ n µ n , 2σ n γ αn/2 th .(26)
Similarly, for Case II, we have that OP can be written as
P e out = 1 −Ā e Q 0 2 √ m n , 2 m n (γ th /γ) αn/2 .(27)P b (e|γ) (φ/2)erfc ρ γ/2(28)
where erfc(·) is the complementary error function [20, Eq. 8.250 (4) and ρ = 6/(M 2 − 1). Thus, for Case I we average (9) over (28) to obtain the integral
P b = φ 2 ∞ 0 p γ (γ)erfc ρ γ/2 dγ.(29)
Substituting (9) into (29), one has that
P b = φA 2 ∞ 0 δ(γ)erfc ρ γ/2 dγ + φK nĀ 2 I 2(30)P b = φ 2 A +ĀK n φ √ π 2 ( √ 2) αn ρ αn ωn × H 0,2:1,0;1,0 2,1:0,1;1,3 σ n ( √ 2) αn ρ αn θ n ( √ 2) αn 4ρ αn Φ(α n , µ n , ω n ) Ψ(α n , µ n , ω n ) (32) P e b = φ 2 A e + a nĀ e φ √ π 2 ( √ 2) αn ρ αn 1 2 × H 0,2:1,0;1,0 2,1:0,1;1,3 b n ( √ 2) αn ρ αn c n ( √ 2) αn 4ρ αn Φ α n , 2, 1 2 Ψ α n , 2, 1 2 . (33) Φ ≡ 1 − α n 2 ω n ; α n 2 , α n 2 , 1 2 − α n 2 ω n ; α n 2 , α n 2 : − ; µ n 2 , 1 Ψ ≡ − α n 2 ω n ; α n 2 , α n 2 : (0, 1) ; (ω n − 1, 1), (1 − ω n , 1), µ n 2 , 1 where I 2 = ∞ 0 γ (αn/2)ωn−1 e σnγ αn/2 I µn−1 θ n γ αn 2 erfc ρ γ 2 dγ.(31)
Following the same method as in Section III, using [
P bn φ ∞ 0 e −ργ f γ (γ)dγ.(34)
The values of φ and ρ also vary depending on the modulation scheme. Specifically, for BFSK, φ = 1 2 and ρ = 1 2 . For DBPSK, φ = 1 2 and ρ = 1 and for M -FSK φ = (M − 1)/2 and ρ = 1 2 . Upon substituting (9) into (34), it is clear that the resulting integral is of the form of the MGF integral given in (17). It is therefore straightforward to show that
P bn = φM γ (ρ).(35)
Similarly, the BER for non-coherent modulation schemes under Case II, can be derived using (22) resulting in P e bn = P bn = φM e γ (ρ).
C O = B n ln 2 ∞ 0 ln(1 + γ)f γ (γ)dγ.(37)
For Case I, we substitute (9)
I 3 = 1 (2πi) 2 Lr Lt Γ ω n + 2 αn r + t Γ(ω n − 1 − t) Γ 1 − µn 2 + t Γ µn 2 − t × πΓ(1 − r)Γ(r)Γ(r) 1 σn 2r αn θn 4σn t drdt σ ωn n Γ(1 + r)Γ(ω n + t)Γ µn 2 − t(39)
where Re (ω n + 2r/α n + t) > 0.
1 σ n 2 αn θ n 4σ n Ω(α n , µ n , ω n ) Π(α n , µ n , ω n ) (40) C e O =
BĀ e πα n a n 2n √ b n ln 2 H 0,1:1,2;1,0
1,0:2,2;1,3 1 b n 2 αn c n 4b n Ω α n , 2, 1 2 Π α n , 2, 1 2 (41) Ω ≡ 1 − ω n ; 2 α n , 1 : (1, 1), (1, 1); µ n 2 , 1 Π ≡ − : (1, 1), (0, 1); (ω n − 1, 1), (1 − ω n , 1), µ n 2 , 1
inspection from (40) the ORA capacity for Case II as in (41) of Table III. Using integration by parts, [25] showed that (37) has the alternative representation
C O = B n ln 2 ∞ 0 [1 − F γ (γ)] 1 + γ dγ.(42)
Therefore, substituting (10) and (12)
C O = BĀ n ln 2 ∞ 0 Q µn 2κ n µ n , 2σ n γ αn 2 /(1 + γ)dγ,(43)C e O = BĀ e n ln 2 ∞ 0 Q 0 2 √ m n , 2 m n (γ/γ) αn 2 /(1 + γ)dγ.(44)
2) Optimal Power and Rate Adaptation: Under OPRA, the channel capacity of the end-to-end SNR γ of a n-link DF-based multi-hop system with density f γ is defined as [25, Eq. 10]
C P B n ∞ γo log 2 γ γ o f γ (γ)dγ = γ o B n ln 2 ∞ 1
ln(y)f γ (γ o y)dy, upon setting γ = γ o y.
(45) 13 The optimum cutoff γ o must satisfy the following relation
∞ γo (1/γ o − 1/γ) f γ (γ)dγ = 1.(46)
In order to evaluate (45) for Case I, we express the PDF (9) in terms of the Fox H-function as was done in Section III to obtain after some simple manipulations the integral
C P =K * 1 (2πi) 2 Lr Lt H ∞ 1 ln y 2 αn y 1−ωn−r−t dy × Γ(−r)Γ(ω n − 1 − t) σ n γ αn 2 o r θn 4 γ αn 2 o t drdt Γ(ω n + t)Γ 1 − µn 2 + t Γ µn 2 − t(47)
where K * = 2γ αn 2 ωn o πBK n /(α n n ln 2Ā −1 ). The integral H can be evaluated using integration by parts as H = 2/ [α n (ω n + r + t) 2 ] resulting in the double Mellin-Barnes contour integral
C P = 2K * α n 1 (2πi) 2 Lr Lt Γ(−r)Γ(ω n + r + t) 2 Γ(1 + ω n + r + t) 2 × Γ(ω n − 1 − t) σ n γ αn 2 o r θn 4 γ αn 2 o t drdt Γ(ω n + t)Γ 1 − µn 2 + t Γ µn 2 − t(48)
where (ω n + r + t) < 0.
[1 − F γ (γ o )] /γ o − ∞ γo f γ (γ)/γdγ = 1 =⇒ γ o = 1 − F γ (γ o ) 1 + ∞ γo f γ (γ)/γdγ = ξ(γ o ).C P = B n ln 2 ∞ γo [1 − F γ (γ)] /γdγ.(52)
Therefore, we can use (10) and (12) in (52) to obtain the OPRA channel capacities for Case I and Case II respectively as
C P = BĀ n ln 2 ∞ γo Q µn 2κ n µ n , 2σ n γ αn 2 )/γdγ,(53)C e P = BĀ e n ln 2 ∞ γo Q 0 2 √ m n , 2 m n (γ/γ) αn 2 /γdγ(54)
Numerical integration routines from standard packages can be employed to evaluate (53) and
(54).
3) Channel Inversion With Fixed Rate:
Under this technique, the channel capacity of the end-to-end SNR γ of a n-link DF-based multi-hop system with density f γ is given by
C c B n log 2 1 + 1 ∞ 0 (f γ (γ)/γ) dγ(55)
Since the expression obtained using (14) and (15) are highly restrictive, we can obtain more general ones by employing a similar procedure to the one used in deriving the MGF function of Section III. We only express the bessel function in each SNR PDF in terms of the Meijer G-function and then reverse the order of integration to obtain after some simplifications the
single Mellin-Barnes integral ∞ 0 p γ (γ) γ dγ = 2πK nĀ α n σ ωn− 2 αn n 1 2πi Lr Γ ω n − 2 αn + r Γ(ω n + r) × Γ(ω n − 1 − r) θn 4σn r dr Γ 1 − µn 2 + r Γ µn 2 − r(56)
where Re ω n − 2 αn + r > 0. Comparing the contour integral in (56)
nC c B = log 2 1 + 2πK nĀ α n σ ωn− 2 αn n G 1,1 2,3 θ n 4σ n −1 ,(57)
where ≡ 1 + 2 αn − ω n , µn 2 and ≡ ω n − 1, 1 − ω n , µn 2 . In an analogous fashion, the CIFR channel capacity for Case II can be derived with the contour condition Re 1/2 − 2 αn + r > 0 as
C T B n log 2 1 + 1 ∞ γo f γ (γ)/γdγ P r (γ > γ o ) .(59)
Note that C T → C c from above as γ o → 0. That is, the gap between C T and C c depends on P r (γ > γ o ) in (59). Making the substitution y = γ/γ o , the integral in (59) can be re-written as
σ n γ αn 2 o θ n 4 γ αn 2 o ♠ * (µ n , ω n ) ♣ * (µ n , ω n ) (61) J = −2πĀ e γ αn 4
o a n α n G 0,1:1,0;1,0
1,1:0,1;1,3 b n γ αn 2 o c n 4 γ αn 2 o ♠ * 2, 1 2 ♣ * 2, 1 2 (62) ♠ * ≡ 1 + 2 α n − ω n ; 1, 1 : −; µ n 2 , 1 ♣ * ≡ 2 α n − ω n ; 1, 1 , 2 α n − ω n ; 1, 1 : (0, 1); (ω n − 1, 1), (1 − ω n , 1), µ n 2 , 1 J = ∞ γo f γ (γ)/γdγ = ∞ 1 1 y f y (γ o y)dy(60)
Substituting (9) into (60) and following the simple procedure undertaken in deriving OPRA channel capacity yields The double Mellin-Barnes contour integral must satisfy Re ω n − 2 αn + r + t < 0. Therefore, putting (61) and (62) into (59) gives the TIRF channel capacity for Case I and Case II, respectively. Alternatively, for Case I, another closed-form expression for the integral in (59) can be obtained by making use of the consecutive substitutions x = γ αn/2 followed by N Q N +1 (a, b). The latter is clearly evident from (63) in the case α = 2.
√ x = u/ √ 2σ n which yield J =Ā (2σ n ) 2 αn 2(2µ n κ n ) ωn−1 Q µ * ,
V. BER AND CAPACITY APPROXIMATIONS FOR LOW SNR REGIMES
In this section, we explore the asymptotic behaviour of the BER by employing asymptotic
Poincare series expansions for both the α − κ − µ and α − κ − µ-Extreme fading models in (9) and (11). Following the idea from [27] of using [20,Eq. 8.445] and [20, Eq. 0.316], we have that the PDFs of the SNRs for the n-th hop of Case I and Case II fading channels have
Poincare asymptotic series expansion near the origin given respectively by
p γ (γ)∼ Aδ(γ) +ĀK n θ n 4 ωn−1 N −1 k=0 d k γ t k + O γ t+N −1(64)
where O denotes the order of a term in the asymptotic series and
d k = k j=0 (−1) j σ j n θn 4 k−j j!(k − j)!Γ(k + µ n − j) , t k = α n (k + µ n )/2 − 1.(65)
p e γ (γ)∼ A e δ(γ) +Ā e a n √ c n 2
N −1 k=0 d e k γ t e k + O γ t+N −1(66)
where
d e k = k j=0 (−1) j b j n cn 4 k−j j!(k − j)!Γ(k + 2 − j)
, t e k = α n (k + 1)/2 − 1.
A. Asymptotic BER at low SNR levels
We will now attempt to derive asymptotic BER expressions based on the series expansions in (64) and (66).
1) Coherent Modulation
P b∼ φ 2 A +ĀK n θ n 4 ωn−1 φ √ π N −1 k=0 d k 2 t k Γ 3 2 + t k (t k + 1)ρ 2(t k +1) ,(68)
P e b∼ φ 2 A e + φĀ e a n √ c n
2 √ π N −1 k=0 d e k 2 t e k Γ 3 2 + t e k
(t e k + 1)ρ 2(t e k +1) .
2) Non-coherent Modulation Schemes: Similarly, by substituting (64) and (66)
P bn∼ φA + φĀK n θ n 4 ωn−1 N −1 k=0 d k Γ(t k + 1) ρ t k +1(70)
P e bn∼ φA e + φĀ e a n √ c n 2 Figure 4 illustrates the OP vs average SNR curves for Case I under moderate fading conditions. From the figure, we can deduce that increasing any of the fading parameters values results in notable system performance enhancement with steeper increases noted for increments in µ, the multi-cluster parameter. Increasing the number of hops also improves system performance, especially at lower SNR levels which is a particularly desirable trait. The increase in performance is albeit of a diminishing type. This implies that there is an optimal number/range of hops that maximizes system performance. That is, relaying for the sole purpose of improving outage is only fruitful up to a certain number of relays upon which any additional relaying is futile. Figure 5 depicts the behaviour of BER under non- Figure 7 in moderate fading. However, the improvements are very thin and even larger increments in parameter values do not cause a bigger gap between the curves. The gap between OPRA and ORA capacities gets smaller as SNR increases. Based on the analyses for AF, OP and BER, a similar behavior is expected for the other three transmission protocols. However, there appears to be an a diminishing reduction in channel capacity as more relays are added to the DFbased system. According to [7] this is due to the half-duplex capability of the relays which employs time-sharing multiple-access mechanism. Consequently, increasing the number of hops will require more time to deliver information from S to D, which hampers system capacity.
N −1 k=0 d e k Γ(t e k + 1) ρ t e k +1 .(71)
Finally, Figure 8 shows the channel capacity under all the adaptive transmission schemes for moderate fading conditions. The results are in agreement with the findings of [29] and prove that as expected C P ≥ C O ≥ C T ≥ C c . We expect similar results to the trend set in the preceding analyses for Case II and we therefore avoid displaying the figure for the sake of brevity.
VII. CONCLUSIONS
In this paper, various performance measures of a DF-based multi-hop system over generalized α−κ−µ and α−κ−µ-Extreme fading channels have been examined in detail. We have provided the closed-form expressions for the OP, amount of fading, BER and channel capacity under various adaptive schemes for an end-to-end DF-based multi-hop relaying system. The effects of the variation of the fading parameters and the number of hops on the system performance is demonstrated. System performance was shown to improve in a diminishing manner as the number of hop links increased under moderate fading. However, for severe fading conditions, there is little gain or actual performance degradation. From the analyses carried out in the present work, a few points which could also be of interest for further research are : (i) Determining the number of hops at which a negligible gain in system performance kicks in so as to avoid futile additional relaying if the purpose of relaying is to improve system performance. (ii) Determining the exact limit of BER under non-coherent modulation schemes as in [13] for the present fading models.
]. The constants φ and ρ vary depending on the type of modulation scheme. For BFSK, φ = 1 and ρ = 1. For BPSK, φ = 1 and ρ = √ 2. For 4-QAM and QPSK, φ = 2 and ρ = 1. Finally, for M-PAM, φ = 2(1 − 1/M )
section, we investigate the capacity of α−κ−µ and α−κ−µ-Extreme fading channels under various adaptive transmission schemes.1) Optimal Rate Adaptation:The channel capacity of the received ene-to-end SNR γ with density function f γ (γ) under ORA (also known as ergodic capacity) is obtained by averaging the capacity of an additive AWGN C O = B log 2 (1 + γ) over f γ (γ), where B is the bandwidth of the channel. For a n-hop DF-based multi-hop system, we have
clear that the solution to (51) is a fixed point of the non-linear equation γ o = ξ(γ o ). According to [25], γ o will always lie in the interval [0, 1] regardless of the fading model and number of relays used for the wireless system. The explanation for the latter relies on the properties of the CDF of γ and is fully documented in Section 3 of [25]. Thus, choosing a starting point in [0, 1], γ o can be computed via iterative schemes such as the Newton-Raphson method as γ i o = ξ(γ i o )
the i-th iterate of γ o . The iterative procedure will be stopped when |γ i+1 o − γ i o | issufficiently small. We also note again that using integration by parts on (45) yields[25, Eq. 11]
, 1
1The Meijer G-function has been implemented in MATHEMATICA and MATLAB .4) Truncated Channel Inversion With Fixed Rate: The channel inversion technique with fixed rate may suffer large capacity penalties as compared to other techniques. To circumvent this shortcoming, a modified inversion which inverts the channel fading above a predetermined truncated fade γ o is often used and is defined for n-link DF-based multi-hop systems as
* = µ n − 4 αn , α n µ n > 4 and Q M,N (·, ·) is the Nuttall Q-function defined as [26, Eq. 2] with b, M, N ≥ 0 and a > 0. A similar procedure can be followed for Case II. However, the resulting integral does not satisfy the M ≥ 0 condition and hence fails to attain a Nuttall Q-function representation. The Nuttall Q-function is a generalization of the Marcum Q-function. Specifically, when M = N + 1, Q N +1,N (a, b) = a
VI. NUMERICAL EXPERIMENTS AND SIMULATIONSIn this section we have presented the graphical results of the performance metrics derived in the preceding sections. All computations are carried out in MATLAB (version 2014a).Nonetheless, based on linear optimization,[28] recently proposed versatile GPU-enabled MATLAB and C/M EX codes with automated contour computation for the general multivariate Fox H-function of the type in[21] and we use a modified version of it in the present work.Figure 2andFigure 3present the AF vs average SNR under severe and moderate fading conditions for both Case I and Case II, respectively. It is evident that AF decreases with increments in values for any of the fading parameters of both models. This makes sense since the parameter m is inversely proportional to the fading severity. We also note that both models perform almost identically for moderate fading with the extreme distribution showing a slightly lower AF. Furthermore, AF under Case II stabilizes to a constant as SNR decreases indefinitely. While increasing the number of hops appears to lower the AF for moderate fading conditions, for severe fading, there is a steep increase in AF at low SNR with increases in signal power, the number of multi-clusters µ or m having little to no positive effect.
Figure 2 .
2AF curves for the α − κ − µ case showing the effects of variation in model parameter values and hop numbers under moderate and severe fading conditions.
Figure 3 .
3AF curves for the α − κ − µ-Extreme case showing the effects of variation in model parameter values and hop numbers under moderate and severe fading conditions similar to Case I.
Figure 4 .
4OP curves for the α − κ − µ case under moderate fading showing the effects of increments in hop number and fading model parameter values. they showed that as SNR increases, BER under non-coherent DPSK modulation approaches the constant (1/2) exp(−2m).
Figure 5 .Figure 6 .
56BER curves for the α − κ − µ case under non-coherent DBPSK modulation showing the effects of increasing model parameters and the number of hops and N in Poincare approximations for moderate fading conditions Under similar fading conditions of Case I, Figure 6 presents BER curves under non-coherent DBPSK modulation for Case II. We once more observe a consistent behaviour in terms of parameter and hop number variation. Furthermore, BER levels are considerably lower as com-BER curves for the α − κ − µ-Extreme case under non-coherent DBPSK modulation showing the effects of increasing model parameters, the number of hops and N in Poincare approximations for moderate fading conditions pared to Case I even under moderate fading severity. Figure 6 also shows that Poincare seriesbased asymptotic BER approximations perform quite well at low SNR regimes for the choice in parameter values, but that an increase in N does not seem to improve convergence or guarantee improvement in accuracy of approximations. Moreover, an increase in hop number or µ results in a higher non-null BER level under Case I than for increases in m under Case II.
Figure 7
7
Figure 7 .Figure 8 .
78CO curves for the α − κ − µ case showing the effect of increasing number relays and fading model parameter values under moderate fading conditions Channel capacity curves for the α − κ − µ case under various adaptive transmission schemes for a 3-hop DF-based system presents the channel capacity under ORA adaptive transmission for Case I. There is improvement in channel capacity for increments in any of the fading model parameter values in
Now, expanding the two Fox H-functions in terms of their definition as given in [23, Eq. 1.1.1]
Table I
IMOMENT GENERATING FUNCTOINS
Remark 1 suggests that the MGF for Case II can be obtained from (21) by inspection resulting in the MGF for Case II being as shown inwith the definition of the bivariate Fox H-function
given in [21, Eq. 2.57] yields a closed-form expression for the MGF as shown in (21) of Table I.
C. Bit Error Rate 1) Coherent Modulation Schemes: For a given SNR γ, the BER for a variety of coherent modulation schemes can be obtained by averaging the PDF f γ over the conditional additive white Gaussian noise (AWGN) BER given as[24, Eq. 17]
Table II
IIBER UNDER COHERENT MODULATION
22, Eq. 8.4.14(2)] to express the complementary error function in terms of its Mellin-Barnes representation, we obtain after some simple mathematical manipulations the BER under coherent modulation schemes for Case I and Case II as shown in (33) and (33) ofTable II, respectively.2) Non-Coherent Modulation Schemes: Under a range of different non-coherent modulation
schemes, the BER of the end-to-end SNR γ can be written as [24, Eq. 27]
Table III
IIIORA CHANNEL CAPACITYC O =
BĀπα n K n
2nσ ωn
n ln 2
H 0,1:1,2;1,0
1,0:2,2;1,3
into (42) yields expressions for channel capacity with ORA under Case I and Case II respectively as
Table IV
IVOPRA CHANNEL CAPACITYC P =
4γ
αn
2 ωn
o
Table V
VTIFR CHANNEL CAPACITY INTEGRAL JJ =
−2πĀ γ
αn
2 ωn
o
K n
α n γ o
G 0,1:1,0;1,0
1,2:0,1;1,3
coherent DBPSK modulation scheme for Case I in moderate fading conditions. Consistent with observations from AF and OP analyses, there is notable system improvement with increments in fading model parameter values and a diminishing system performance improvement with hop number increase at low SNR levels. An additional system behavior is observed for high SNR level which shows that there is a non-null BER even as signal power increases indefinitely. This characteristic was observed by Rabelo et al.[13] for the linear κ−µ-Extreme fading model where10
15
20
25
30
35
40
45
50
SNR (dB)
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| {'fraction_non_alphanumeric': 0.07385352498288843, 'fraction_numerical': 0.036344969199178644, 'mean_word_length': 3.4611704834605597, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 17, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 71, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this work, an integrated performance evaluation of a decode-and-forward (DF) multi-hop wireless communication system is undertaken over the non-linear generalized α − κ − µ and α − κ − µ-Extreme fading models. Analytical formulas for the probability density function (PDF) and the cumulative distribution function (CDF) of the received signal-to-noise ratio (SNR) as well as its generalized moments and moment generating function (MGF) are derived. Based on the derived PDFs, novel closed-form expressions for traditional performance metrics such as amount of fading (AF), outage probability (OP), bit error rate (BER) under coherent and non-coherent modulation schemes as well as channel capacity under various adaptive transmission techniques are derived. Additionally, asymptotic analyses of BER based on Poincare series expansions of SNR PDFs are carried out and results show good approximations for low SNR regimes. The correctness of the proposed solutions has been corroborated by comparing them with Monte Carlo simulation results. Index Terms Multi-hop system, decode-and-forward relay, bivariate Fox H-function, non-linear generalized fading, Marcum Q-function, Nuttall Q-function Tau Raphael Rasethuntsa is based in, Lenasia Extension 9, Gauteng Province, Johannesburg, South Africa and Ha', 'arxivid': '1903.09353', 'author': ['Tau Raphael Rasethuntsa \nare with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia\n', 'Sandeep Kumar \nare with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia\n', 'Manpreet Kaur [email protected] \nare with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia\n', 'Moreneng, Maseru, LesothoAbia / Mahlabatheng \nare with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia\n', 'Sandeep Kumar \nare with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia\n', 'Manpreet Kaur \nare with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia\n'], 'authoraffiliation': ['are with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia', 'are with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia', 'are with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia', 'are with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia', 'are with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia', 'are with the Central Research Laboratory\nBharat Electronics Limited\nUttar PradeshGhaziadbadIndia'], 'corpusid': 85459420, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 16554, 'n_tokens_neox': 13823, 'n_words': 8299, 'pdfsha': '5afcba2a2a2adb9ab7eb4f6b280826d6080c263f', 'pdfurls': ['https://arxiv.org/pdf/1903.09353v1.pdf'], 'title': ['A Comprehensive Performance Evaluation of a DF-Based Multi-Hop System Over α − κ − µ and α − κ − µ-Extreme Fading Channels', 'A Comprehensive Performance Evaluation of a DF-Based Multi-Hop System Over α − κ − µ and α − κ − µ-Extreme Fading Channels'], 'venue': []} |
arxiv |
Representation-Driven Reinforcement Learning
Ofir Nabati
Guy Tennenholtz
Shie Mannor
Representation-Driven Reinforcement Learning
We present a representation-driven framework for reinforcement learning. By representing policies as estimates of their expected values, we leverage techniques from contextual bandits to guide exploration and exploitation. Particularly, embedding a policy network into a linear feature space allows us to reframe the exploration-exploitation problem as a representation-exploitation problem, where good policy representations enable optimal exploration. We demonstrate the effectiveness of this framework through its application to evolutionary and policy gradient-based approaches, leading to significantly improved performance compared to traditional methods. Our framework provides a new perspective on reinforcement learning, highlighting the importance of policy representation in determining optimal exploration-exploitation strategies.
Introduction
Reinforcement learning (RL) is a field in machine learning in which an agent learns to maximize a reward through interactions with an environment. The agent maps its current state into action and receives a reward signal. Its goal is to maximize the cumulative sum of rewards over some predefined (possibly infinite) horizon (Sutton & Barto, 1998). This setting fits many real-world applications such as recommendation systems (Li et al., 2010), board games , computer games (Mnih et al., 2015), and robotics (Polydoros & Nalpantidis, 2017).
A large amount of contemporary research in RL focuses on gradient-based policy search methods (Sutton et al., 1999;Silver et al., 2014;Schulman et al., 2015;Haarnoja et al., 2018). Nevertheless, these methods optimize the pol-icy locally at specific states and actions. Salimans et al. (2017) have shown that such optimization methods may cause high variance updates in long horizon problems, while have shown possible convergence to suboptimal solutions in continuous regimes. Moreover, policy search methods are commonly sample inefficient, particularly in hard exploration problems, as policy gradient methods usually converge to areas of high reward, without sacrificing exploration resources to achieve a far-reaching sparse reward.
In this work, we present Representation-Driven Reinforcement Learning (RepRL) -a new framework for policysearch methods, which utilizes theoretically optimal exploration strategies in a learned latent space. Particularly, we reduce the policy search problem to a contextual bandit problem, using a mapping from policy space to a linear feature space. Our approach leverages the learned linear space to optimally tradeoff exploration and exploitation using well-established algorithms from the contextual bandit literature (Abbasi-Yadkori et al., 2011;Agrawal & Goyal, 2013). By doing so, we reframe the exploration-exploitation problem to a representation-exploitation problem, for which good policy representations enable optimal exploration.
We demonstrate the effectiveness of our approach through its application to both evolutionary and policy gradientbased approaches -demonstrating significantly improved performance compared to traditional methods. Empirical experiments on the MuJoCo (Todorov et al., 2012) and MinAtar (Young & Tian, 2019) show the benefits of our approach, particularly in sparse reward settings. While our framework does not make the exploration problem necessarily easier, it provides a new perspective on reinforcement learning, shifting the focus to policy representation in the search for optimal exploration-exploitation strategies.
Preliminaries
We consider the infinite-horizon discounted Markov Decision Process (MDP). An MDP is defined by the tuple M = (S, A, r, T, β, γ), where S is the state space, A is the action space, T : S × A → ∆(S) is the transition kernel, r : S × A → [0, 1] is the reward function, β ∈ ∆(S) is the initial state distribution, and γ ∈ [0, 1) is the discount factor. A stationary policy π : S → ∆(A), maps states into a distribution over actions. We denote by Π the set of stationary stochastic policies, and the history of policies and trajectories up to episode k by H k . Finally, we denote S = |S| and A = |A|.
The return of a policy is a random variable defined as the discounted sum of rewards
G(π) = ∞ t=0 γ t r(s t , a t ),(1)
where s 0 ∼ β, a t ∼ π(s t ), s t+1 ∼ T (s t , a t ), and the policy's value is its mean, i.e., v(π) = E[ ∞ t=0 γ t r(s t , a t ) | β, π, T ]. An optimal policy maximizes the value, i.e., π * ∈ arg max π∈Π v(π).
We similarly define the per-state value function, v(π, s)
as v(π, s) = E[ ∞ t=0 γ t r(s t , a t ) | s 0 = s, π, T ], and note that v(π) = E s∼β [v(π, s)].
Finally, we denote the discounted state-action frequency distribution w.r.t. π by ρ π (s, a) = (1 − γ) ∞ t=0 γ t P r s t = s, a t = a|β, π, T , and let K = {ρ π : π ∈ Π}.
Linear Bandits
In this work, we consider the linear bandit framework as defined in Abbasi-Yadkori et al. (2011). At each time t, the learner is given a decision set D t ⊆ R d , which can be adversarially and adaptively chosen. The learner chooses an action x t ∈ D t and receives a reward r t , whose mean is linear w.r.t x t , i.e., E[ r t | x t ] = ⟨x t , w⟩ for some unknown parameter vector w ∈ R d .
A general framework for solving the linear bandit problem is the "Optimism in the Face of Uncertainty Linear bandit algorithm" (OFUL, Abbasi-Yadkori et al. (2011)). There, a linear regression estimator is constructed each round as follows:ŵ where y t , x t are the noisy reward signal and chosen action at time t, respectively, and V 0 = λI for some positive parameter λ > 0.
It can be shown that, under mild assumptions, and with high probability, the self-normalizing norm ∥ŵ t − w∥ Vt can be bounded from above (Abbasi-Yadkori et al., 2011). OFUL then proceeds by taking an optimistic action (x t ,w t ) ∈ arg max x∈Dt,w∈Ct ⟨x,w⟩, where C t is a confidence set induced by the aforementioned bound on ∥ŵ t − w∥ Vt . In Figure 1. RepRL scheme. Composed of 4 stages: representation of the parameters, constructing a decision set, choosing the best arm using an off-the-shelf linear bandit algorithm, collect data with the chosen policy.
Linear Bandits Policy Representation
Construct Decision Set
RepRL
practice, a softer version is used in Chu et al. (2011), where an action is selected optimistically according to
x t ∈ arg max x∈Dt ⟨x,ŵ t ⟩ + α x T V −1 t x,(OFUL)
where α > 0 controls the level of optimism.
Alternatively, linear Thompson sampling (TS, Abeille & Lazaric (2017)) shows it is possible to converge to an optimal solution with sublinear regret, even with a constant probability of optimism. This is achieved through the sampling of a parameter vector from a normal distribution, which is determined by the confidence set C t . Specifically, linear TS selects an action according to
x t ∈ arg max x∈Dt ⟨x,w t ⟩,w t ∼ N ŵ t , σ 2 V −1 t ,(TS)
where σ > 0 controls the level of optimism. We note that for tight regret guarantees, both α and σ need to be chosen to respect the confidence set C t . Nevertheless, it has been shown that tuning these parameters can improve performance in real-world applications (Chu et al., 2011).
RL as a Linear Bandit Problem
Classical methods for solving the RL problem attempted to use bandit formulations (Fox & Rolph, 1973). There, the set of policies Π reflects the set of arms, and the value v(π) is the expected bandit reward. Unfortunately, such a solution is usually intractable due to the exponential number of policies (i.e., bandit actions) in Π.
Alternatively, we consider a linear bandit formulation of the RL problem. Indeed, it is known that the value can be expressed in linear form as v(π) = E (s,a)∼ρ π [r(s, a)] = ⟨ρ π , r⟩.
Here, any ρ π ∈ K represents a possible action in the linear bandit formulation (Abbasi-Yadkori et al., 2011). Notice that |K| = |Π|, as any policy π ∈ Π can be written as π(a|s) = ρ π (s,a) a ′ ρ π (s,a ′ ) , rendering the problem intractable. Nevertheless, this formulation can be relaxed using a lower dimensional embedding of ρ π and r. As such, we make the following assumption. Assumption 3.1 (Linear Embedding). There exist a mapping f : Π → R d such that v(π) = ⟨f (π), w⟩ for all π ∈ Π and some unknown w ∈ R d .
We note that Assumption 3.1 readily holds when d = SA for f (π) ≡ ρ π and w = r. For efficient solutions, we consider environments for which the dimension d is relatively low, i.e., d ≪ SA.
Note that neural bandit approaches also consider linear representations (Riquelme et al., 2018). Nevertheless, these methods use mappings from states S → R d , whereas we consider mapping entire policies Π → R d (i.e., embedding the function π). Learning a mapping f can be viewed as trading the effort of finding good exploration strategies in deep RL problems to finding a good representation. We emphasize that we do not claim it to be an easier task, but rather a different viewpoint of the problem, for which possible new solutions can be derived. Similar to work on neural-bandits (Riquelme et al., 2018), finding such a mapping requires alternating between representation learning and exploration.
RepRL
We formalize a representation-driven framework for RL, inspired by linear bandits (Section 2.1) and Assumption 3.1. We parameterize the policy π and mapping f using neural networks, π θ and f ϕ , respectively. Here, a policy π θ is represented in lower-dimensional space as f ϕ (π θ ). Therefore, searching in policy space is equivalent to searching in the parameter space. With slight abuse of notation, we will denote f ϕ (π θ ) = f ϕ (θ).
Pseudo code for RepRL is presented in Algorithm 1. At every episode k, we map the policy's parameters θ k−1 to a latent space using f ϕ k−1 (θ k−1 ). We then use a construction algorithm, ConstructDecisonSet(θ k−1 , H k−1 ), which takes into account the history H k−1 , to generate a new decision set D k . Then, to update the parameters θ k−1 of the policy, we select an optimistic policy π θ k ∈ D k using a linear bandit method, such as TS or OFUL (see Section 2.1). Finally, we rollout the policy π θ k and update the representation network and the bandit parameters according to the procedure outlined in Equation (2), where x k are the learned representations of f ϕ k . A visual schematic of our framework is depicted in Figure 1.
In the following sections, we present and discuss methods for representation learning, decision set construction, and propose two implementations of RepRL in the context of evolutionary strategies and policy gradient. We note that Algorithm 1 RepRL 1: Init: H 0 ← ∅, π θ0 , f ϕ0 randomly initialized 2: for k = 1, 2, . . . do
3:
Representation Stage: Map the policy network π θ k−1 using representation network f ϕ k−1 (θ k−1 ).
4:
Decision Set Stage: D k ← ConstructDecisonSet(θ k−1 , H k−1 ).
5:
Bandit Stage: Use linear bandit algorithm to choose π θ k out of D k .
6:
Exploitation Stage: Rollout policy π θ k and store the return G k in H k .
7:
Update representation f ϕ k .
8:
Update bandit parametersŵ t , V t (Equation (2)) with the updated representation. 9: end for RepRL is a framework for addressing RL through representation, and as such, any representation learning technique or decision set algorithm can be incorporated as long as the basic structure is maintained.
Learning Representations for RepRL
We learn a linear representation of a policy using tools from variational inference. Specifically, we sample a representation from a posterior distribution z ∼ f ϕ (z|θ), and train the representation by maximizing the Evidence Lower
Bound (ELBO) (Kingma & Welling, 2013) L(ϕ, κ) = −E z∼f ϕ (z|θ) [log p κ (G|z)] + D KL (f ϕ (z|θ)∥p(z)), where f ϕ (z|θ)
acts as the encoder of the embedding, and p κ (G|z) is the return decoder or likelihood term.
The latent representation prior p(z) is typically chosen to be a zero-mean Gaussian distribution. In order to encourage linearity of the value (i.e the return's mean) with respect to the learned representation, we chose the likelihood to be a Gaussian distribution with a mean that is linear in the representation, i.e., p κ (G|z) = N (κ ⊤ z, σ 2 ). When the encoder is also chosen to be a Gaussian distribution, the loss function has a closed form. The choice of the decoder to be linear is crucial, due to the fact that the value is supposed to be linear w.r.t learned embeddings. The parameters ϕ and κ are the learned parameters of the encoder and decoder, respectively. Note that a deterministic mapping occurs when the function f ϕ (z|θ) takes the form of the Dirac delta function. A schematic of the architectural framework is presented in Figure 2.
Constructing a Decision Set
The choice of the decision set algorithm (line 4 of Algorithm 1) may have a great impact on the algorithm in terms of performance and computational complexity. Clearly, choosing D k = Π, ∀k will be unfeasible in terms of computational complexity. Moreover, it may be impractical to learn a linear representation for all policies at once. We present several possible choices of decision sets below.
Policy Space Decision Set. One potential strategy is to sample a set of policies centered around the current policy
D k = {θ k + ϵ i } N i=1 , ϵ i ∼ N (0, ν 2 I),(4)
where ν > 0 controls how local policy search is. This approach is motivated by the assumption that the representation of policies in the vicinity of the current policy will exhibit linear behavior with respect to the value function due to their similarity to policies encountered by the learner thus far.
Latent Space Decision Set. An alternative approach involves sampling policies in their learned latent space, i.e.,
D k = {z k + ϵ i } N i=1 , ϵ i ∼ N (0, ν 2 I),(5)
where z k ∼ f ϕ (z|θ k ). The linearity of the latent space ensures that this decision set will improve the linear bandit target (UCB or the sampled value in TS), which will subsequently lead to an improvement in the actual value. This approach enables optimal exploration w.r.t. linear bandits, as it uniformly samples the eigen directions of the precision matrix V t , rather than only sampling specific directions as may occur when sampling in the parameter space.
Unlike Equation (4) constructing the set in Equation (5) presents several challenges. First, in order to rollout the policy π θ k , one must construct an inverse mapping to extract the chosen policy from the selected latent representation. This can be done by training a decoder for the policy parameters q(θ|z). Alternatively, we propose to use a decoder-free approach. Given a target embedding z * ∈ arg max z∈Dt ⟨z,ŵ⟩, we search for a policy θ * ∈ arg max θ f ϕ (z * |θ). This optimization problem can be solved using gradient descent-based optimization algorithms by varying the inputs to f ϕ . A second challenge for latent-based decision sets involves the realizability of such policies. That is, there may exist representations z ∈ D k , which are not mapped by any policy in Π. Lastly, even for realizable policies, the restored θ may be too far from the learned data manifold, leading to an overestimation of its value and a degradation of the overall optimization process. One way to address these issues is to use a small enough value of ν during the sampling process, reducing the probability of the set members being outside the data distribution. We leave more sophisticated methods of latent-based decision sets for future work.
History-based Decision Set. An additional approach uses the history of policies at time k to design a decision set. Specifically, at time episode k we sample around the set of policies observed so far, i.e.,
D k = ℓ∈[k] {θ ℓ + ϵ ℓ,i } N i=1 , ϵ ℓ,i ∼ N (0, ν 2 I),(6)
resulting in a decision set of size N k. After improving the representation over time, it may be possible to find a better policy near policies that have already been used and were missed due to poor representation or sampling mismatch. This method is quite general, as the history can be truncated only to consider a certain number of past time steps, rather than the complete set of policies observed so far. Truncating the history can help reduce the size of the decision set, making the search more computationally tractable.
In Section 5, we compare the various choices of decision sets. Nevertheless, we found that using policy space decisions is a good first choice, due to their simplicity, which leads to stable implementations. Further exploration of other decision sets is left as a topic for future research.
Inner trajectory sampling
Vanilla RepRL uses the return values of the entire trajectory. As a result, sampling the trajectories at their initial states is the natural solution for both the bandit update and repre-Algorithm 2 Representation Driven Evolution Strategy 1: Input: initial policy π 0 = π θ0 , noise ν, step size α, decision set size N , history H. 2: for t = 1, 2, . . . , T do 3:
Sample an evaluation set and collect their returns.
4:
Update representation f t and bandit parameters (ŵ t , V t ) using history.
5:
Construct a decision set D t .
6:
Use linear bandit algorithm to evaluate each policy in D t .
7:
Update policy using ES scheme (Section 4.1). 8: end for sentation learning. However, the discount factor diminishes learning signals beyond the 1 1−γ effective horizon, preventing the algorithm from utilizing these signals, which may be critical in environments with long-term dependencies. On the other hand, using a discount factor γ = 1 would result in returns with a large variance, leading to poor learning. Instead of sampling from the initial state, we propose to use the discount factor and sample trajectories at various states during learning, enabling the learner to observe data from different locations along the trajectory. Under this sampling scheme, the estimated value would be an estimate of the following quantity:
v(π) = E s∼ρ π [v(π, s)].
In the following proposition we prove that optimizingṽ(π) is equivalent to optimizing the real value.
Proposition 3.2. For a policy π ∈ Π,ṽ(π) = v(π) 1−γ .
The proof can be found in the Appendix C. That is, sampling along the trajectory from ρ π approximates the scaled value, which, like v(π), exhibits linear behavior with respect to the reward function. Thus, instead of sampling the return defined in Equation (1), we sampleG(π) = ∞ t=0 γ t r(s t , a t ), where s 0 ∼ ρ π , a t ∼ π(s t ), s t+1 ∼ T (s t , a t ), both during representation learning and bandit updates. Empirical evidence suggests that uniformly sampling from the stored trajectory produces satisfactory results in practice.
RepRL Algorithms
In this section we describe two possible approaches for applying the RepRL framework; namely, in Evolution Strategy and Policy Gradients (Sutton et al., 1999).
Representation Driven Evolution Strategy
Evolutionary Strategies (ES) are used to train agents by searching through the parameter space of their policy and sampling their return. In contrast to traditional gradient-Algorithm 3 Representation Driven Policy Gradient 1: Input: initial policy π θ , decision set size N , history H. 2: for t = 1, 2, . . . , T do 3:
Collect trajectories using π θ .
4:
Update representation f and bandit parameters (ŵ t , V t ) using history.
5:
Compute Policy Gradient loss L P G (θ).
6:
Sample a decision set and choose the best policyθ.
7:
Compute gradient of the regularized Policy Gradient loss with d(θ,θ) (Equation (7)). 8: end for based methods, ES uses a population of candidates evolving over time through genetic operators to find the optimal parameters for the agent. Such methods have been shown to be effective in training deep RL agents in high-dimensional environments (Salimans et al., 2017;Mania et al., 2018).
At each round, the decision set is chosen over the policy space with Gaussian sampling around the current policy as described in Section 3.3. Algorithm 5 considers an ES implementation of RepRL. To improve the stability of the optimization process, we employ soft-weighted updates across the decision set. This type of update rule is similar to that used in ES algorithms (Salimans et al., 2017;Mania et al., 2018), and allows for an optimal explorationexploitation trade-off, replacing the true sampled returns with the bandit's value. Moreover, instead of sampling the chosen policy, we evaluate it by also sampling around it as done in ES-based algorithms. Each evaluation is used for the bandit parameters update and representation learning process. Sampling the evaluated policies around the chosen policy helps the representation avoid overfitting to a specific policy and generalize better for unseen policies -an important property when selecting the next policy.
Unlike traditional ES, optimizing the UCB in the case of OFUL or sampling using TS can encourage the algorithm to explore unseen policies in the parameter space. This exploration is further stabilized by averaging over the sampled directions, rather than assigning the best policy in the decision set. This is particularly useful when the representation is still noisy, reducing the risk of instability caused by hard assignments. An alternative approach uses a subset of D t with the highest bandit scores, as suggested in Mania et al. (2018), which biases the numerical gradient towards the direction with the highest potential return.
Representation Driven Policy Gradient
RepRL can also be utilized as a regularizer for policy gradient algorithms. Pseudo code for using RepRL in policy gradients is shown in Algorithm 6. At each gradient step, a weighted regularization term d(θ,θ) is added, whereθ are the parameters output by RepRL with respect to the current parameters for a chosen metric (e.g., ℓ 2 ):
L reg (θ) = L PG (θ) + ζd(θ,θ).
After collecting data with the chosen policy and updating the representation and bandit parameters, the regularization term is added to the loss of the policy gradient at each gradient step. The policy gradient algorithm can be either on-policy or off-policy while in our work we experiment with an on-policy algorithm.
Similar to the soft update rule in ES, using RepRL as a regularizer can significantly stabilize the representation process. Applying the regularization term biases the policy toward an optimal exploration strategy in policy space. This can be particularly useful when the representation is still weak and the optimization process is unstable, as it helps guide the update toward more promising areas of the parameter space. In our experiments, we found that using RepRL as a regularizer for policy gradients improved the stability and convergence of the optimization process.
Experiments
In order to evaluate the performance of RepRL, we conducted experiments on various tasks on the MuJoCo (Todorov et al., 2012) and MinAtar (Young & Tian, 2019) domains. We also used a sparse version of the MuJoCo environments, where exploration is crucial. We used linear TS as our linear bandits algorithm as it exhibited good performance during evaluation. The detailed network architecture and hyperparameters utilized in the experiments are provided in Appendix F.
Grid-World Visualization. Before presenting our results, we demonstrate the RepRL framework on a toy example. Specifically, we constructed a GridWorld environment (depicted in Figure 4) which consists of spatially changing, noisy rewards. The agent, initialized at the bottom left state (x, y) = (1, 1), can choose to take one of four actions: up, down, left, or right. To focus on exploration, the rewards were distributed unevenly across the grid. Particularly, the reward for every (x, y) was defined by the Normal random variable r(x, y) ∼ N µ(x, y), σ 2 ,
where σ > 0 and µ(x, y) ∝ R 1 exp − (x−x1) 2 +(y−y1) 2 a1 + R 2 exp − (x−x2) 2 +(y−y2) 2 a2 +R 3 1 {(x,y)=goal} .
That is, the reward consisted of Normally distributed noise, with mean defined by two spatial Gaussians, as shown in Figure 4, with R 1 > R 2 , a 1 < a 2 and a goal state (depicted as a star), with R 3 ≫ R 1 , R 2 . Importantly, the values of R 1 , R 2 , R 3 , a 1 , a 2 were chosen such that an optimal policy would take the upper root in Figure 4.
Comparing the behavior of RepRL and ES on the GridWorld environment, we found that RepRL explored the environment more efficiently, locating the optimal path to the goal. This emphasizes the varying characteristics of state-spacedriven exploration vs. policy-space-driven exploration, which, in our framework, coincides with representationdriven exploration. Figure 3 illustrates a two-dimensional t- SNE plot comparing the learned latent representation of the policy with the direct representation of the policy weights.
Decision Set Comparison. We begin by evaluating the impact of the decision set on the performance of the RepRL. For this, we tested the three decision sets outlined in Section 3.3. The evaluation was conducted using the Representation Driven Evolution Strategy variant on a sparse HalfCheetah environment. A history window of 20 policies was utilized when evaluating the history-based decision set. A gradient descent algorithm was employed to obtain the parameters that correspond to the selected latent code in the latent-based setting As depicted in Figure 8 at Appendix E, RepRL demonstrated similar performance for the varying decision sets on the tested domains. In what follows, we focus on policy space decision sets.
MuJoCo.
We conducted experiments on the MuJoCo suitcase task using RepRL. Our approach followed the setting of Mania et al. (2018), in which a linear policy was used and demonstrated excellent performance on MuJoCo tasks. We utilized the ES variant of our algorithm (Algorithm 5). We incorporated a weighted update between the gradients using the bandit value and the zero-order gradient of the sampled returns, taking advantage of sampled information and ensuring stable updates in areas where the representation is weak.
We first evaluated RepES on the standard MuJoCo baseline (see Figure 5). RepES either significantly outperformed or performed on-par with ES. We also tested a modified, sparse variant of MuJoCo. In the sparse environment, a reward was given for reaching a goal each distance interval, denoted as d, where the reward function was defined as:
r(s, a) = 10 − c(a), |x agent | mod d = 0 −c(a), o.w.
Here, c(a) is the control cost associated with utilizing action a, and x agent denotes the location of the agent along the xaxis. The presence of a control cost function incentivized the agent to maintain its position rather than actively exploring the environment. The results of this experiment, as depicted in Figure 5, indicate that the RepRL algorithm outperformed both the ES and SAC algorithms in terms of achieving distant goals. However, it should be noted that the random search component of the ES algorithm occasionally resulted in successful goal attainment, albeit at a significantly lower rate in comparison to the RepRL algorithm.
MinAtar. We compared the performance of RepRL on MinAtar (Young & Tian, 2019) with the widely used policy gradient algorithm PPO (Schulman et al., 2017). Specifically, we compared PPO against its regularized version with RepRL, as described in Algorithm 6, and refer to it as RepPG. We parametrized the policy by a neural network.
Although PPO collects chunks of rollouts (i.e., uses subtrajectories), RepPG adjusted naturally due to the inner trajectory sampling (see Section 3.4). That is, the critic was used to estimate the value of the rest of the trajectory in cases where the rollouts were truncated by the algorithm. Results are shown in Figure 6. Overall, RepRL outperforms PPO on all tasks, suggesting that RepRL is effective at solving challenging tasks with sparse rewards, such as those found in MinAtar.
Related Work
Policy Optimization: Policy gradient methods (Sutton et al., 1999) have shown great success at various challenging tasks, with numerous improvements over the years; most notable are policy gradient methods for deterministic policies (Silver et al., 2014;Lillicrap et al., 2015), trust region based algorithms (Schulman et al., 2015;, and maximum entropy algorithms (Haarnoja et al., 2018). Despite its popularity, traditional policy gradient methods are limited in continuous action spaces. Therefore, suggest to optimize the policy over the policy distribution space rather than the action space.
In recent years, finite difference gradient methods have been rediscovered by the RL community. This class of algorithms use numerical gradient estimation by sampling random directions (Nesterov & Spokoiny, 2017). A closely related family of optimization methods are Evolution Strategies (ES) a class of black-box optimization algorithms that heuristic search by perturbing and evaluating the set members, choosing only the mutations with the highest scores until convergence. Salimans et al. (2017) used ES for RL as a zero-order gradient estimator for the policy, parameterized as a neural network. ES is robust to the choice of the reward function or the horizon length and it also does not need value function approximation as most state-of-art algorithms. Nevertheless, it suffers from low sample efficiency due to the potentially noisy returns and the usage of the final return value as the sole learning signal. Moreover, it is not effective in hard exploration tasks. Mania et al. (2018) improves ES by using only the most promising directions for gradient estimation.
Policy Search with Bandits. Fox & Rolph (1973) was one of the first works to utilize multi-arm bandits for policy search over a countable stationary policies set -a core approach for follow-up work (Burnetas & Katehakis, 1997;Agrawal et al., 1988). Nevertheless, the concept was left aside due to its difficulty to scale up with large environments.
As an alternative, Neural linear bandits (Riquelme et al., 2018;Xu et al., 2020;Nabati et al., 2021) simultaneously train a neural network policy, while interacting with the environment, using a chosen linear bandit method and are closely related to the neural-bandits literature (Zhou et al., 2020;Kassraie & Krause, 2022). In contrast to this line of work, our work maps entire policy functions into linear space, where linear bandit approaches can take effect. This induces an exploration strategy in policy space, as opposed to locally, in action space.
Representation Learning. Learning a compact and useful representation of states (Laskin et al., 2020;Schwartz et al., 2019;, actions Chandak et al., 2019), rewards (Barreto et al., 2017;Nair et al., 2018;Toro Icarte et al., 2019), and policies (Hausman et al., 2018;Eysenbach et al., 2018), has been at the core of a vast array of research. Such representations can be used to improve the performance of agents by utilizing the structure of an environment more efficiently.
In this work, we viewed the representation problem as an alternative to solving the exploration problem in RL. While this problem is not necessarily easier, it shifts the challenge to a different area, where new methods can be established.
Discussion and Future Work
We presented RepRL, a novel representation-driven framework for reinforcement learning. By optimizing the policy over a learned representation, we leveraged techniques from the contextual bandit literature to guide exploration and exploitation. We demonstrated the effectiveness of this framework through its application to evolutionary and policy gradient-based approaches, leading to significantly improved performance compared to traditional methods.
In this work, we suggested to reframe the explorationexploitation problem as a representation-exploitation problem. By embedding the policy network into a linear feature space, good policy representations enable optimal exploration. This framework provides a new perspective on reinforcement learning, highlighting the importance of policy representation in determining optimal explorationexploitation strategies.
As future work, one can incorporate RepRL into more involved representation methods, including pretrained large Transformers (Devlin et al., 2018;Brown et al., 2020), which have shown great promise recently in various areas of machine learning. Another avenue for future research is the use of RepRL in scenarios where the policy is optimized in latent space using an inverse mapping (i.e., decoder), as well as more involved decision sets. Finally, while this work focused on linear bandit algorithms, future work may explore the use of general contextual bandit algorithms, (e.g., SquareCB Foster & Rakhlin (2020)
Representation-Driven Reinforcement Learning -Appendix
A. Algorithms
Algorithm 4 Random Search / Evolution Strategy 1: Input: initial policy π 0 = π θ0 , noise ν, step size α, set size K. 2: for t = 1, 2, . . . , T do 3:
Sample a decision set D t = {θ t−1 ± δ i } K i=1 , δ i ∼ N (0, ν 2 I). 4: Collect the returns {G(θ t−1 ± δ i )} K i=1 of each policy in D t . 5:
Update policy
θ t = θ t−1 + α σ R K K i=1 G(θ t−1 + δ i ) − G(θ t−1 − δ i ) δ i 6: end for
Algorithm 5 Representation Driven Evolution Strategy 1: Input: initial policy π 0 = π θ0 , noise ν, step size α, set size K, decision set size N , history H. 2: for t = 1, 2, . . . , T do 3:
Sample an evaluation set {θ t−1 ± δ i } K i=1 , δ i ∼ N (0, ν 2 I).
4:
Collect the returns {G(θ t−1 ± δ i )} K i=1 from the environment and store them in replay buffer.
5:
Update representation f t and bandit parameters (ŵ t , V t ) using history.
6:
Construct a decision set D t = {θ t−1 ± δ i } N i=1 , δ i ∼ N (0, ν 2 I).
7:
Use linear bandit algorithm to evaluate each policy in D t : {v(θ t−1 ± δ i )} N i=1 .
8:
Update policy
g t = 1 N N i=1 v(θ t−1 + δ i ) −v(θ t−1 − δ i ) δ i ,
θ t =θ t−1 + αg t 9: end for Algorithm 6 Representation Driven Policy Gradient 1: Input: initial policy π θ , noise ν, step size α, decision set size N , ζ, history H. 2: for t = 1, 2, . . . , T do 3:
for 1, 2, . . . , K do 4:
Collect trajectory data using π θ .
5:
Update representation f and bandit parameters (ŵ, Σ) using history. Sample a decision set D = {θ + δ i } N i=1 , δ i ∼ N (0, ν 2 I).
9:
Use linear bandit algorithm to choose the best parameterθ ∈ arg max θ∈D ⟨z,ŵ⟩ for z ∼ f (z|θ).
10:
Compute g =∇ θ L P G (θ) + ζ∥θ −θ∥ 2 , θ =θ − αg 11: end for 12: end for
B. Variational Interface
We present here proof of the ELBO loss for our variational interface, which was used to train the representation encoder.
Proof.
log p(G; ψ, κ) = log ϕ p κ (G|ϕ)p(ϕ)dϕ = log ϕ p κ (G|ϕ) p(ϕ) q ψ (ϕ|π) q ψ (ϕ|π)dϕ = log E ϕ∼q ψ (ϕ|π) p κ (G|ϕ) p(ϕ) q ψ (ϕ|π) ≥ E ϕ∼q ψ (ϕ|π) log p κ (G|ϕ) + E ϕ∼q ψ (ϕ|π) log p(ϕ) q ψ (ϕ|π) = E ϕ∼q ψ (ϕ|π) log p κ (G|ϕ) − D KL (q ψ (ϕ|π)∥p(ϕ)),
where the inequality is due to Jensen's inequality.
C. Proof for Proposition 3.2
By definition:ṽ
(π) = s ρ π (s)v(π, s) = s ρ π (s) a π(a|s) r(s, a) + γ s ′ T (s ′ |s, a)v(π, s ′ ) = v(π) + γ s ρ π (s) a π(a|s) s ′ T (s ′ |s, a)v(π, s ′ ) = v(π) + γ s ′ ρ π (s ′ )v(π, s ′ ) = v(π) + γṽ(π),
where the second equality is due to the Bellman equation and the third is from the definition. Therefore, v(π) = v(π) + γṽ(π) =⇒ṽ(π) = v(π) 1 − γ
D. Full RepRL Scheme
The diagram presented below illustrates the networks employed in RepRL. The policy's parameters are inputted into the representation network, which serves as a posterior distribution capturing the latent representation of the policy. Subsequently, a sampling procedure is performed from the representation posterior, followed by the utilization of a linear return encoder, acting as the likelihood, to forecast the return distribution with a linear mean (i.e. the policy's value). This framework is employed to maximize the Evidence Lower Bound (ELBO).
Policy Network
Representation Encoder z Linear Decoder Figure 7. The full diagram illustrates the networks in RepRL.
E. Decision Set Experiment
The impact of different decision sets on the performance of RepRL was assessed in our evaluation. We conducted tests using three specific decision sets as described in Section 3.3. The evaluation was carried out on a sparse HalfCheetah environment, utilizing the RepES variant. When evaluating the history-based decision set, we considered a history window consisting of 20 policies. In the latent-based setting, the parameters corresponding to the selected latent code were obtained using a gradient descent algorithm. The results showed that RepRL exhibited similar performance across the various decision sets tested in different domains. In the GridWorld environment, a 8 × 8 grid is utilized with a horizon of 20, where the reward is determined by a stochastic function as outlined in the paper: r(x, y) ∼ N µ(x, y), σ 2 , where σ > 0 and µ(x, y) ∝ R 1 exp − (x−x1) 2 +(y−y1) 2 a1 + R 2 exp − (x−x2) 2 +(y−y2) 2 a2 + R 3 1 {(x,y)=goal} . The parameters of the environment are set as R 1 = 2.5, R 2 = 0.3, R 3 = 13, σ = 3, a 1 = 0.125, a 2 = 8.
The policy employed in this study is a fully-connected network with 3 layers, featuring the use of the tanh non-linearity operator. The hidden layers' dimensions across the network are fixed at 32, followed by a Sof tmax operation. The state is represented as a one-hot vector. please rephrase the next paragraph so it will sounds more professional: The representation encoder is built from Deep Weight-Space (DWS) layers (Navon et al., 2023), which are equivariant to the permutation symmetry of fully connected networks and enable much stronger representation capacity of deep neural networks compared to standard architectures. The DWS model (DWSNet) comprises four layers with a hidden dimension of 16. Batch normalization is applied between these layers, and a subsequent fully connected layer follows. Notably, the encoder is deterministic, meaning it represents a delta function. For more details, we refer the reader to the code provided in Navon et al. (2023), which was used by us.
In the experimental phase, 300 rounds were executed, with 100 trajectories sampled at each round utilizing noisy sampling of the current policy, with a zero-mean Gaussian noise and a standard deviation of 0.1. The ES algorithm utilized a step size of 0.1, while the RepRL algorithm employed a decision set of size 2048 without a discount factor (γ = 1) and λ = 0.1.
F.2. MuJoCo
In the MuJoCo experiments, both ES and RepES employed a linear policy, in accordance with the recommendations outlined in (Mania et al., 2018). For each environment, ES utilized the parameters specified by Mania et al. (2018), while RepES employed the same sampling strategy in order to ensure a fair comparison.
RepES utilized a representation encoder consisting of 4 layers of a fully-connected network, with dimensions of 2048 across all layers, and utilizing the ReLU non-linearity operator. This was followed by a fully-connected layer for the mean and variance. The latent dimension was also chosen to be 2048. After each sampling round, the representation framework (encoder and decoder) were trained for 3 iterations on each example, utilizing an Adam optimizer and a learning rate of 3e − 4. When combining learning signals of the ES with RepES, a mixture gradient approach was employed, with 20% of the gradient taken from the ES gradient and 80% taken from the RepES gradient. Across all experiments, a discount factor of γ = 0.995 and λ = 0.1 were used.
F.3. MinAtar
In the MinAtar experiments, we employed a policy model consisting of a fully-connected neural network similar to the one utilized in the GridWorld experiment, featuring a hidden dimension of 64. The value function was also of a similar structure, with a scalar output. The algorithms collected five rollout chunks of 512 between each training phase.
The regulation coefficient chosen for RepRL was 1, while the discount factor and the mixing factor were set as γ = 0.995 and λ = 0.1. The representation encoder used was similar to the one employed in the GridWorld experiments with two layers, followed by a symmetry invariant layer and two fully connected layers.
Figure 2 .
2The diagram illustrates the structure of the networks in RepRL. The policy's parameters are fed into the representation network, which acts as a posterior distribution for the policy's latent representation. Sampling from this posterior, the latent representation is used by the bandits algorithm to evaluate the value that encapsulates the exploration-exploitation tradeoff.
Figure 3 .
3The two-dimensional t-SNE visualization depicts the policy representation in the GridWorld experiment. On the right, we observe the learned latent representation, while on the left, we see the direct representation of the policy's weights. Each point in the visualization corresponds to a distinct policy, and the color of each point corresponds to a sample of the policy's value.
Figure 4 .
4GridWorld visualization experiment. Trajectories were averaged across 100 seeds at various times during training, where more recent trajectories have greater opacity. Background colors indicate the level of mean reward.
Figure 5 .
5MuJoCo experiments during training. The results are for the MuJoCo suitcase (top) and the modified sparse MuJoCo (bottom).
Figure 6 .
6MinAtar experiments during training.
Figure 8 .
8Plots depict experiments for three decision sets: policy space-based, latent space-based, and history-based. The experiment was conducted on the SparseHalfCheetah environment.F. Hyperparameters and Network Architecture F.1. Grid-World
Department of Electrical-Engineering, Technion Institute of Technology, Israel 2 Technion (currently at Google Research) 3 Nvidia Research. Correspondence to: Ofir Nabati <[email protected]>.
t = V −1 t b t , V t = V t−1 + x t x ⊤ t , b t = b t−1 + x t y t ,(2)
AcknowledgmentsThis work was partially funded by the Israel Science Foundation under Contract 2199/20.
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| {'fraction_non_alphanumeric': 0.051210379199409316, 'fraction_numerical': 0.022502329342685864, 'mean_word_length': 4.581787852026298, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, '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 present a representation-driven framework for reinforcement learning. By representing policies as estimates of their expected values, we leverage techniques from contextual bandits to guide exploration and exploitation. Particularly, embedding a policy network into a linear feature space allows us to reframe the exploration-exploitation problem as a representation-exploitation problem, where good policy representations enable optimal exploration. We demonstrate the effectiveness of this framework through its application to evolutionary and policy gradient-based approaches, leading to significantly improved performance compared to traditional methods. Our framework provides a new perspective on reinforcement learning, highlighting the importance of policy representation in determining optimal exploration-exploitation strategies.', 'arxivid': '2305.19922', 'author': ['Ofir Nabati ', 'Guy Tennenholtz ', 'Shie Mannor '], 'authoraffiliation': [], 'corpusid': 258987852, 'doi': '10.48550/arxiv.2305.19922', 'github_urls': [], 'n_tokens_mistral': 16252, 'n_tokens_neox': 14197, 'n_words': 8766, 'pdfsha': '86a575987a11045541c967843945bdcf8dca2d03', 'pdfurls': ['https://export.arxiv.org/pdf/2305.19922v1.pdf'], 'title': ['Representation-Driven Reinforcement Learning', 'Representation-Driven Reinforcement Learning'], 'venue': []} |
arxiv |
Teamwork Is Not Always Good: An Empirical Study of Classifier Drift in Class-incremental Information Extraction
Minqian Liu [email protected]
Computer Science Department Virginia Tech
Lifu Huang [email protected]
Computer Science Department Virginia Tech
Teamwork Is Not Always Good: An Empirical Study of Classifier Drift in Class-incremental Information Extraction
Class-incremental learning (CIL) aims to develop a learning system that can continually learn new classes from a data stream without forgetting previously learned classes. When learning classes incrementally, the classifier must be constantly updated to incorporate new classes, and the drift in decision boundary may lead to severe forgetting. This fundamental challenge, however, has not yet been studied extensively, especially in the setting where no samples from old classes are stored for rehearsal. In this paper, we take a closer look at how the drift in the classifier leads to forgetting, and accordingly, design four simple yet (super-) effective solutions to alleviate the classifier drift: an Individual Classifiers with Frozen Feature Extractor (ICE) framework where we individually train a classifier for each learning session, and its three variants ICE-PL, ICE-O and ICE-PL&O which further take the logits of previously learned classes from old sessions or a constant logit of an Other class as constraint to the learning of new classifiers. Extensive experiments and analysis on 6 class-incremental information extraction tasks demonstrate that our solutions, especially ICE-O, consistently show significant improvement over the previous state-of-the-art approaches with up to 44.7% absolute F-score gain, providing a strong baseline and insights for future research on class-incremental learning. 1
Introduction
Conventional supervised learning assumes the data are independent and identically distributed (i.i.d.) and usually requires a pre-defined ontology, which may not be realistic in many applications in natural language processing (NLP). For instance, in event detection, the topics of interest may keep shifting over time (e.g., from attack to pandemic), and new event types and annotations could emerge 1 The source code, model checkpoints and data are publicly available at https://github.com/VT-NLP/ICE.
Train on Meet
Train on Arrest
Classifier
Test Input: Some 70 people were arrested Saturday. incessantly. Previous studies (Ring et al., 1994;Kirkpatrick et al., 2017;Lopez-Paz and Ranzato, 2017) therefore proposed continual learning (CL), a.k.a., lifelong learning or incremental learning, a learning paradigm aiming to train a model from a stream of learning sessions that arrive sequentially. In this work, we focus on the class-incremental learning (CIL) setting , where a new session 2 is composed of previously unseen classes and the goal is to learn a unified model that performs well in all seen classes. When new learning sessions arrive sequentially, the classification layer must be constantly updated and/or expanded to accommodate new categories to the model. The change of the classifier between different sessions, i.e., classifier drift, can disturb or overwrite the classifier trained on previous classes, which consequently causes catastrophic forgetting (Biesialska et al., 2020). On the other hand, in many NLP tasks such as information extraction, the model also needs to classify nega-tive instances into the Other type (i.e., none-of-theabove). The Other type adds extra difficulty to classification, and even worse, the meaning of Other varies as the model learns new sessions (Zheng et al., 2022). The CIL problem thus becomes even more challenging when Other is involved. We illustrate the event detection task in CIL and the classifier drift problem in Figure 1.
Despite the progress achieved in CIL (Zhao et al., 2022;Zheng et al., 2022), there are two critical limitations that are still remained: (1) Most previous CIL approaches heavily rely on the rehearsalbased strategy which stores samples from previously learned sessions and keeps re-training the model on these examples in subsequent sessions to mitigate catastrophic forgetting, which requires high computation and storage costs and raises concerns about privacy and data leakage (Shokri and Shmatikov, 2015); (2) Previous approaches have mainly focused on regularizing or expanding the overall model, especially feature extractor, to tackle the forgetting issue (Cao et al., 2020), but they rarely investigate whether the drift of the classifier also leads to forgetting, especially in classification tasks that involve the Other category. In this work, we aim to tackle these limitations by answering the following two research questions: RQ1: how does classifier drift lead to forgetting in the setting where no samples are stored from old sessions for rehearsal?, and RQ2: how to devise an effective strategy to alleviate classifier drift, especially when there is an Other category involved?
In this paper, we aim to answer the two research questions above. First, to study how classifier drift alone affects the model, we build a baseline where we use a pre-trained language model as a fixed feature extractor, such that only the parameters in the classification layer will be updated. Second, to alleviate classifier drift, we propose a simple framework named Individual Classifiers with Frozen Feature Extractor (ICE). Instead of collectively tuning the whole classification layer, we individually train a classifier for the classes in each new session without updating old classifiers and combine all learned classifiers to classify all seen classes during inference. As individually trained classifiers may lack the context of all learned sessions (Zhang et al., 2021), they may not be comparable to each other. We further devise a variant ICE-PL which takes the logits of previous classifiers as constraints to encourage contrastivity among all the classes when learning a new classifier for a new session. Third, both ICE and ICE-PL cannot be applied to detection tasks where an Other class is involved, thus we further design two variants of them: ICE-O and ICE-PL&O, which introduce a constant logit for the Other class and use it to enforce each individual classifier to be bounded by a constraint shared across different learning sessions during training.
We extensively investigate the classifier drift and evaluate our approach on 6 essential information extraction tasks across 4 widely used benchmark datasets under the CIL setting. Our major findings and contributions are: (1) By comparing the drifted baseline and our ICE, we find that the classifier drift alone can be a significant source of forgetting and our approaches effectively mitigate the drift and forgetting. Our results reveal that training the classifier individually can be a superior solution to training the classifier collectively in CIL.
(2) We find that the Other type can effectively improve individually trained classifiers, and it is also helpful when we manually introduce negative instances during training on the tasks that do not have Other.
(3) Experimental results demonstrate that our proposed approaches, especially ICE-O, significantly and consistently mitigate the forgetting problem without rehearsal and outperform the previous stateof-the-art approaches by a large margin. (4) Our study builds a benchmark for 6 class-incremental information extraction tasks and provides a superstrong baseline and insights for the following studies on class-incremental information extraction.
Related Work
Existing approaches for CIL can be roughly categorized into three types . Rehearsal-based approaches (a.k.a. experience replay) (Lopez-Paz and Ranzato, 2017;de Masson d'Autume et al., 2019;Guo et al., 2020;Madotto et al., 2021;Qin and Joty, 2021) select some previous examples (or generate pseudo examples) for rehearsal in subsequent tasks. While such approaches are effective in mitigating forgetting, they require high computation and storage costs and suffer from data leakage risk (Shokri and Shmatikov, 2015;Smith et al., 2021;. Regularization-based approaches (Chuang et al., 2020) aim to regularize the model's update by only updating a subset of parameters. Architecturebased approaches (Lee et al., 2020;Ke et al., 2021a,b,c;Feng et al., 2022; adap-tively expand the model's capacity via parameterefficient techniques (e.g., adapter, prompt) to accommodate more data. While most existing approaches consider alleviating the forgetting of the whole model or transferring previous knowledge to new sessions, few of them thoroughly investigate how the classification layer of the model is affected as it expands to incorporate more classes into the model. Wu et al. (2019) find that the classification layer has a strong bias towards new classes, but they only study this issue in image recognition that doesn't contain the Other class. To fill the blank in current research, we aim to take a closer look at how the drift in the classifier alone affects the model under the CIL setting, especially when Other is involved.
For class-incremental information extraction, several studies tackle the CIL problem in relation learning (Wu et al., 2021), and many of them apply prototype-based approaches equipped with memory buffers to store previous samples Cui et al., 2021;Zhao et al., 2022). Others investigate how to detect named entities (Monaikul et al., 2021;Xia et al., 2022) or event trigger (Cao et al., 2020;Liu et al., 2022) in the CIL setting. For instance, Zheng et al. (2022) propose to distillate causal effects from the Other type in continual named entity recognition. One critical disadvantage of existing approaches for continual IE is they heavily rely on storing previous examples to replay, whereas our method does not require any examplar rehearsal.
Problem Formulation
Class-incremental learning requires a learning system to learn from a sequence of learning sessions D = {D 1 , ..., D T } and each session D k = {(x k , y k )|y k ∈ C k } where x k is an input instance for the session D k and y k ∈ C k denotes its label. The label set C k for session D k is not overlapping with that of other sessions, i.e., ∀k, j and k ̸ = j, C k C j = ∅. Given a test input x and a model that has been trained on up to t sessions, the model needs to predict a labelŷ from a label space that contains all learned classes, i.e., C 1 ... C t and optionally the Other class. Generally, the training instances in old classes are not available in future learning sessions.
We consider a learning system consisting of a feature extractor and a classifier. Specifically, we use a linear layer G 1:t ∈ R c×h as the classification layer, where c is the number of classes that the model has learned up to session t and h is the hidden dimension size of features. We denote the number of classes in a learning session k as n k , i.e., n k = |C k |. The classification layer G 1:t can be viewed as a concatenation of the classifiers in all learned sessions, i.e., G 1:t = [W 1 ; ...; W t ], where each of the classifier W k ∈ R n k ×h is in charge of the classes in C k . The linear layer outputs the logits o 1:t ∈ R c for learned classes, where o k refers to the logits for the classes in C k . The term logit we use in this paper refers to the raw scores before applying the Softmax normalization.
In this work, we focus on studying the classincremental problem in information (entity, relation, and event) extraction tasks.We consider two settings for each task: the detection task that requires the model to identify and classify the candidate mentions or mention pairs into one of the target classes or Other, and the classification task that directly takes the identified mentions or mention pairs as input and classifies them into the target classes without considering Other. We first design a DRIFTED-BERT baseline to investigate how classifier drift alone leads to forgetting, and then provide an insightful analysis of how classifier drift happens, especially in the setting of class-incremental continual learning.
DRIFTED-BERT Baseline
In the current dominant continual learning frameworks, both the feature extractor and classifier are continually updated, which results in drift in both components towards the model's predictions on old classes. To measure how the classifier drift along leads to forgetting, we build a simple baseline that consists of a pretrained BERT (Devlin et al., 2019) as the feature extractor and a linear classification layer (shown in Figure 2 (a)). The model first encodes a given input text x into the contextual representation. For event trigger and entity recognition, the model feeds the representation of a candidate span h into the linear layer to predict the logits for learned classes, i.e., o 1:t = G 1:t (h (Optional) Figure 2: Illustration of the training process in a new learning session for DRIFTED-BERT as well as ICE and its variants. "FE" stands for feature extractor. "O" stands for the Other type. Each circle in the classifier represents a category. The models have learned a classifier (W 1 ) with 3 classes in Session 1 (S1) and they are learning a new classifier (W 2 ) with 2 classes in Session 2 (S2 class which has different meanings from other sessions, we follow to set the logit for Other to a constant value δ, i.e., o 0 = δ. We combine o 0 and o 1:t and pick the label with the maximum logit as the prediction. That is, we predict a sample as Other if and only if max(o 1:t ) < δ.
We freeze the parameters in the feature extractor so that the encoded features of a given sample remain unchanged in different learning sessions. In this way, the updates in the classification layer become the only source of forgetting. Note that we do not apply any continual learning techniques (e.g., experience replay) to DRIFTED-BERT. We denote p(x t ) as the predicted probability to compute the loss in training, where p(x t ) = Softmax(o 0:t ). At the learning session t, the model is trained on D t with the Cross Entropy (CE) loss:
L CE = − (x t ,y t )∈Dt log p(x t ).(1)
A Closer Look at Classifier Drift When the model has learned t sessions and needs to extend to the (t + 1)-th session, the classification layer G 1:t needs to introduce new parameters to accommodate the new classes in C t+1 , i.e.,
G 1:t+1 = [W 1 ; ...; W t ; W t+1 ].
As we assume that all previous training instances in D 1:t are not accessible anymore, solely training the model on D t+1 would lead to an extreme class-imbalance problem (Cao et al., 2020), which consequently causes catastrophic forgetting. However, most existing works rarely discuss how the drift in the classifier alone leads to forgetting, especially when the Other class is involved. We first define the classifier drift between two consecutive learning sessions D t and D t+1 as the change from G 1:t to G 1:t+1 that makes the model lose (part of) its acquired capability on the seen classes in C 1:t . Intuitively, the CE loss aims to maximize the probability of the correct label while minimizing the probabilities of all other labels. Thus, there are two possible causes of classifier drift: (1) new logit explosion: the new classifier W t+1 tends to predict logits o t+1 that are higher than those of all previous classes o 1:t so that the model can trivially discriminate new classes, which causes the old classes being overshadowed by new classes. (2) diminishing old logit: as the old instances are not accessible in future learning sessions, the parameters in previous classifiers will be updated from the previous local optimum to a drifted sub-optimum, such that the classifier outputs low logits for old classes and cannot predict correctly. We empirically analyze the DRIFTED-BERT baseline to investigate the classifier drift in Section 5.2 and discuss the drifting patterns in different classification and detection tasks in Section 5.4.
RQ2: How to Alleviate Classifier Drift?
To alleviate the classifier drift, we introduce two solutions ICE and its variant ICE-PL for the classification tasks without Other, and further design two additional variants ICE-O and ICE-PL&O for detection tasks where Other is involved. We illustrate the training process in a new learning session for ICE and its variants in Figure 2. Note that we only focus on the setting of continual learning without experience replay, i.e., the model does not have access to the data of old sessions.
ICE: Individual Classifiers with Frozen Feature
Extractor We revisit the idea of classifier ensemble (Dietterich, 2000) and separated output layers in multi-task learning (Zhang and Yang, 2018) where task-specific parameters for one task do not affect those for other tasks. Inspired by this, we propose to individually train a classifier for each session without updating or using previously learned classifiers G 1:t (shown in Figure 2 (b)). In this way, previous classifiers can avoid being drifted to the sub-optimum, and the new classifier is less prone to output larger logits to overshadow old classes. Specifically, for an incoming session t + 1, we initialize a set of new weights and train the new classifier W t+1 on D t+1 . We only use the logits for the classes in the new session o t+1 to compute the Cross-Entropy loss in optimization, i.e., p(x t+1 ) = Softmax(o t+1 ). During inference, as we need to classify all seen classes without knowing the session identity of each instance, we combine the logits from all classifiers W 1 , ..., W t+1 together to get the prediction for all learned classes, i.e., o 1:t+1 = [o 1 ; ...; o t+1 ], where each classifier yields the logits via o k = W k ·h given the encoded feature h for each mention.
ICE+Previous Logits (ICE-PL) One limitation of ICE is the classifier individually trained in one session may not be comparable to others. To provide contrastivity to classifiers, we first explore a variant named ICE-PL where we preserve the previous classifiers and only freeze their parameters, such that the new classifier is aware of previous classes during training (shown in Figure 2 (c)). That is, the model uses the logits from all classifiers o 1:t+1 to compute the Cross-Entropy loss, i.e., p(x t+1 ) = Softmax(o 1:t+1 ), while only the parameters in the new classifier are trainable. ICE-PL uses the same inference process as ICE.
ICE+Other (ICE-O) Both ICE-O and ICE-PL
can only be applied to classification tasks and handling the Other category for detection tasks is challenging as each session D t only contains the annotated mentions for the classes C t , while the mentions from all the other classes such as C 1:t−1 are labeled as Other, making the meaning of Other varies in different sessions. To tackle this problem, we purpose the ICE-O variant (shown in Figure 2 (d)) where we assign a constant value δ as the logit of the Other category. Specifically, for each prediction, we combine the logit of Other with the logits from the new session o t+1 to obtain the output probability, i.e., p(x t+1 ) = Softmax([δ; o t+1 ]), and then compute the Cross-Entropy loss to train the classifier to make predictions for both positive classes and Other. During the inference, we combine the Other's logit δ with the logits from all trained classifiers o 1:t+1 , i.e., o 0:t+1 = [δ; o 1 ; ...; o t+1 ] to predict for all learned positive types and Other. We select the label with the highest logit among o 0:t+1 as the prediction, and a candidate will be predicted as Other if and only if max(o 1:t+1 ) < δ.
While the Other class introduces additional difficulties to CIL, we argue that it can also be a good remedy to classifier drift. In particular, in each learning session k, while the classifier W k is independently trained on D k , the output logits o k also need to satisfy the constraint that max(o k ) < δ when the classifier is trained on negative instances. Although the logits from any two distinct classifiers W k and W j (k ̸ = j) do not have explicit contrastivity, both classifiers are trained under the constraint that max(o k ) < δ and max(o j ) < δ, which provides a weak contrastivity between them.
ICE+Previous Logits and Other (ICE-PL&O)
To explore the effect of preserving the previous logits when Other is involved, we devise a ICE-PL&O variant that uses both the Other's logit δ and previous logits o 1:t during training (shown in Figure 2 (e)). That is, ICE-PL&O uses the combined logits o 0:t+1 = [δ; o 1 ; ...; o t+1 ] to compute the loss, i.e., p(x t+1 ) = Softmax(o 0:t+1 ). ICE-PL&O adopts the same inference process as ICE-O.
While ICE-O and ICE-PL&O are naturally applied to detection tasks, for classification tasks without the Other class, we can also manually create negative instances based on the tokens or entity pairs without positive labels. Section 5.1 provides more details regarding how to apply ICE-O and ICE-PL&O to classification tasks.
Experiments and Discussions
Datasets and Experiment Setup
We use Few-NERD (Ding et al., 2021) for classincremental named entity recognition and split all the 66 fine-grained types into 8 learning sessions by following which apply a greedy algorithm to split the types into sessions and ensure each session contains the roughly same number of training instances. We use two benchmark datasets MAVEN and ACE-05 (Doddington et al., 2004) for class-incremental event trigger extraction and following the same setting as to split them into 5 learning sessions, respectively. For class-incremental relation extraction, we use TACRED (Zhang et al., 2017) and follow the same setting as Zhao et al. (2022) to split the 42 relations into 10 learning sessions.
For each dataset, we construct two settings: (1) detection where the model classifies each token (or a candidate entity pair in relation extraction task) in a sentence into a particular class or Other; and (2) classification where the model directly takes in a positive candidate (i.e., an entity, trigger, or a pair of entities) and classify it into one of the classes. For the classification setting, as there are no negative candidates that are labeled as Other, we automatically create negative candidates and introduce the Other category so that we can investigate the effect of Other using ICE-O and ICE-PL&O. Specifically, we assign the Other label to the tokens if they are not labeled with any classes for entity and event trigger classification, and assign the Other label to the pairs of entity mentions if they are not labeled with any relations for relation classification. When we apply ICE-O and ICE-PL&O to classification tasks, during inference, we do not consider the logit of the Other class.
Evaluation We use the same evaluation protocol as previous studies Liu et al., 2022). Every time the model finishes the training on Session t, we evaluate the model on all test samples from Session 1 to Session t for classification tasks. For detection tasks, we evaluate the model on the entire test set where we take the mentions or mention pairs of unlearned classes as Other. Following , we randomly sample 5 permutations of the orders of learning sessions and report the average performance.
Baselines We compare our approaches with the DRIFTED-BERT baseline and several state-ofthe-art methods for class-incremental information extraction, including ER , KCN (Cao et al., 2020), KT , EMP (Liu et al., 2022), CRL (Zhao et al., 2022). All these methods adopt experience replay to alleviate catastrophic forgetting. We also design two approaches to show their performance in the conventional supervised learning setting where the model is trained with the annotated data from all the sessions, as the approximate upperbound of the continual learning approaches: (i) BERT-FFE consists of a pre-trained BERT as the feature extractor and a classifier, where, during training, we fix the feature extraction and only tune the classifier; and (ii) BERT-FT which shares the same architecture as BERT-FFE but both the feature extractor and classifier are tuned during training. More details about the datasets, baselines, and model implementation can be found in Appendix A.
RQ1: How does Classifier Drift Lead to Forgetting?
We conduct an empirical analysis on event detection and classification tasks on MAVEN to answer RQ1 and gain more insight into the classifier drift.
Analysis of Old and New Classes Performance
Our first goal is to analyze the classifier drift during the incremental learning process. In Table 1, we analyze how the performance of previously learned classes changes after the model has been trained on a new session for the DRIFTED-BERT baseline and the variants of ICE. After learning in each session k, we compute the (1) F-score on the new classes (C k ) learned in the current session, (2) accumulated F-score on the old classes (C 1:k−1 ) from all previous sessions, and (3) F-score on the old classes (C k−1 ) from the previous session, respectively. By Table 3: Results (Micro-F1 score, %) on named entity recognition and classification on 8 learning sessions. We highlight the best scores in bold and the second best with underline. † indicates approaches with experience replay.
Upperbound (BERT-FFE) - - - - - - - 72.3 - - - - - - - 73.5 Upperbound (BERT-FT) - - - - - - - 78.8 - - - - - - - 80.0
comparing the performance change on the same set of classes in two continuous sessions, e.g., the F-score on the new classes (C k ) learned in session k and the F-score on the classes (C k ) from the previous session after learning in session k +1, we can quantify how much the classifier is drifted. From Table 1, the performance of DRIFTED-BERT on old classes after learning on a new session is always decreased dramatically, verifying that classifier drift does occur in class-incremental learning and leads to severe forgetting. On the other side, our solutions, especially ICE-O, consistently retain similar performance on the old classes from the previous session after learning on a new session, demonstrating that it effectively alleviates the classifier drift and the forgetting issue. Besides, we find that the ICE-PL variant suffers from a considerable performance drop on both new and old classes, which indicates freezing previous classifiers' parameters while preserving the logits of previously learned classes cannot address the classifier drift and forgetting problems. Note that although we only showed the results on event classification and detection on MAVEN, the conclusions are very consistent for other tasks and datasets as shown in Appendix B.1.
RQ2: How to Alleviate Classifier Drift and Forgetting?
To answer RQ2, we evaluate the effectiveness of our proposed approaches to mitigating classifier drift and catastrophic forgetting.
Quantitative Comparison
We conduct an extensive quantitative comparison of the baselines and our approaches on the 6 class-incremental IE tasks. From Table 2 Comparison with CRL (Zhao et al., 2022) Note that, among all the baselines, CRL consistently outperforms others on the classification tasks. CRL is based on a prototypical network where each class is represented with a prototype computed from an embedding space and performs the classification with the nearest class mean (NCM) classifier. Compared with other Softmax-based classification approaches, CRL can accommodate new classes more flexibly without any change to the architecture. However, it still suffers from the semantic drift problem as the embedding network must be continually updated to learn new classes, and it is non-trivial to adapt it to detection tasks where an Other class is involved under the class-incremental learning setting and the meanings of Other in different learning sessions are also different.
Upperbound (BERT-FFE) - - - - - - - - - 51.2 - - - - - - - - - 73.3 Upperbound (BERT-FT) - - - - - - - - - 61.0 - - - - - - - - - 86.9
Comparison with Trainable Feature Extractor
We also investigate if our proposed approaches can be further improved by tuning the BERT-based feature extractor. However, it naturally leads to forgetting as demonstrated by previous studies Cao et al., 2020;. We hypothesize that this is due to the meaning shift of the Other class when incrementally training it on a sequence of learning sessions. Experience replay may not be enough to constrain the feature extractor to handle the Other class properly.
Analysis of Drifting Patterns
To take a closer look into how the classifier drift leads to forgetting and verify the two hypothetical drifting patterns we discuss in Section 4.1, we analyze the output logits (i.e., the scores before Softmax) from the old and new classifiers for DRIFTED-BERT and our ICE, ICE-PL, and ICE-O. Specifically, we take the test samples whose ground truth labels are learned in Session 1 (denoted as X 1 test ), for analysis. Every time the classifier is trained on a new session, we evaluate the classifier on X 1 test , and then take (1) the logit of the gold class (Gold), and (2) the maximum logit from the new classifier (NCP), i.e., New Classifier's Prediction, for analysis. For each type of logit, we report the average of the logits on all the samples in X 1 test . We have the following findings: (1) By examining the Gold logits and the logits from the new classifier (NCP) of DRIFTED-BERT, we observe that every time a new classifier is added and trained on the new session, the new classifier incrementally outputs higher logits than those in the previous session on X 1 test (blue solid line), whereas the Gold logits first decline a bit and stay at a certain level in the remaining sessions (blue dashed line). This observation confirms that two possible drifting patterns (i.e., new logit explosion and diminishing old logit) exist, and they can happen simultaneously and cause the new classifier overshadows the previously learned classifiers, which consequently leads to forgetting. (2) We find that while the old classifiers are not updated in ICE-PL, the new logit explosion issue gets even more severe (orange solid line), which explains why ICE-PL performs worse than ICE and ICE-O. We hypothesize that the presence of previous logits may encourage the new classifier to predict larger logits. (3) When the classifier in each session is trained individually instead of collectively (i.e., in ICE and ICE-O), the Gold logits from the old classifiers stay at a constant level (red dashed lines), whereas the logits from the new classifier are at a relatively lower level (green and red solid line). As such, the new classifier's logits do not have much impact on those of old classes, which mitigates the drift and forgetting.
The Effect of the Logit for Other Class
Throughout all the experiments, we set the logit for Other class δ as 0 constantly. In this section, we further discuss the effect of the value of δ, and the effect of tuning the Other classifier. We show the results of event detection on MAVEN based on different fixed values or a tunable value of δ in Table 5. We found that the value of Other class's logit does Table 5: Results (Micro-F1 score, %) on the effect of the Other class's logit on the event detection task on MAVEN. We show the performance of the models that have learned all 5 sessions. "Tune" means we used a tunable logit for Other class instead of a fixed value. not affect the model's performance much as long as it is fixed. However, we noticed a significant performance decrease if we continually tuned it with a classifier, demonstrating that it is necessary to fix the Other class's logit during the continual learning process in our approach.
Comparison with Recent LLMs
More recently, very large language models (LLMs) such as ChatGPT (OpenAI, 2022) demonstrate strong in-context learning ability without the need of gradient update. Thus, class-incremental learning may also be tackled as a sequence of in-context learning. However, several recent studies (Gao et al., 2023;Qin et al., 2023) have benchmarked several LLMs with in-context few-shot learning on various IE tasks and show worse performance than our approach. Our approach can efficiently achieve a good performance that is close to the supervised performance by only finetuning the last linear layer using a much smaller frozen BERT backbone. More critically, the knowledge LLMs are often bounded by the training data, whereas the goal of our continual learning approach focuses on incorporating up-to-date information into models.
Conclusion
In this paper, we investigate the answers and the solutions to the research questions that how the classifier drift alone affects a model in the classincremental learning setting, and how to alleviate the drift without retraining the model on previous examples. We, therefore, propose to train a classifier individually for each task and combine them together during inference, such that we can maximally avoid the drift in the classifier. Extensive experiments show that our proposed approaches significantly outperform all the considered baselines on both class-incremental classification and detection benchmarks and provide super-strong baselines. We hope this work can shed light on future research on continual learning in broader research communities.
Limitations
Our approaches mainly leverage a fixed feature extractor together with a set of individually trained classifiers to mitigate catastrophic forgetting whereas a tunable feature extractor may also be helpful and complement the individually trained classifiers, so a future direction is to design advanced strategies to efficiently tune the feature extractor in combination with our proposed ICE based classifiers. In addition, we mainly investigate the classifier drift and demonstrate the effectiveness of our solutions under the class-incremental continual learning setting. Another future direction is to explore similar ideas under other continual learning settings, e.g., task-incremental learning, online learning, or the setting where new sessions also contain annotations for old classes.
A More Details on Experiment Setup
A.1 Details of the Datasets Named Entity We use Few-NERD (Ding et al., 2021), a large-scale named entity recognition (NER) dataset to evaluate class-incremental named entity recognition and classification. Compared with the datasets used in previous continual NER works (Zheng et al., 2022), Few-NERD has a more diverse range of entity types and finer granularity, containing 8 coarse-grained and 66 fine-grained entity types. Thus, it is a better benchmark to study continual NER. We construct two settings for the NER task: (1) a detection task where the model is required to examine every token in the text and classify each of them into a learned positive entity type or Other, and;
(2) a classification task where the positive candidate entity mentions have been provided and the model only needs to assign a learned entity type to the given candidate. Following , we split the dataset into 8 learning sessions with the greedy algorithm such that each session contains the roughly same number of training instances.
Relation We use TACRED (Zhang et al., 2017) to evaluate relation detection and classification tasks. TACRED is a large-scale relation extraction dataset that contains 42 relations. In the previous continual relation classification setting (Cui et al., 2021), they ignore the long-tail distribution and assume each relation contains the same number of instances. We instead use the original train/dev/test split in TACRED where relations are imbalanced. We build two settings for the relation task: (1) a detection task where the model needs to assign an ordered entity mentions with a seen positive relation type or Other, and;
(2) a classification task that assumes the given entity pair must belong to one of learned relation, and the model is only required to predict a label it has learned. We follow the previous setting (Zhao et al., 2022) to split the dataset into 10 learning sessions, where we drop the relation with the fewest instances such that each session contains 4 positive relation types. (2) a classification task where the model only needs to classify a positive trigger mention into a learned event type without considering Other. We did not construct the classification task for the ACE dataset as the majority of instances only contain the Other type and removing such instances will result in a very small dataset.
A.2 Baselines
We use the following baselines for our experiments:
(1) DRIFTED-BERT: we build a baseline with a fixed pre-trained BERT as the feature extractor and only train its classification layer. We do not apply any other continual learning techniques to it. We primarily use this baseline to study the classifier drift discussed in this work.
(2) ER : experience replay is introduced to continual IE by . In this work, we use the same strategy as in (Liu et al., 2022) to select examples to store in the memory and replay them in subsequent sessions. (3)
A.3 Implementation Details
We use the pre-trained BERT-large-cased (Devlin et al., 2019) as the fixed feature extractor. We use AdamW (Loshchilov and Hutter, 2019) as the optimizer with the weight decay set to 1e − 2 and a learning rate of 1e − 4 for detection tasks and 5e − 4 for classification tasks. We apply gradient accumulation and set the step to 8. In each learning session D k , we establish a limit of 15 maximum training epochs. We also adopt the early stopping strategy with a patience of 3, where training will be halted if there is no improvement in performance on the development set for 3 epochs. We set the constant value for the Other class δ to 0. We apply the experience replay strategy with the same setting as in (Liu et al., 2022)
Figure 1 :
1Illustration of class-incremental event detection where the model needs to classify each candidate mention into a label from all learned types or Other. The figure shows two classifiers that are incrementally trained from Session 1 and Session 2 and are evaluated on the same sample. After training on session 2, the classifier mistakenly predicts Other for an Arrest mention due to the classifier drift. The model here uses pre-trained features and only the classifier is trained.
:
Results (Micro-F1 score, %) on event detection and classification on 5 learning sessions. We highlight the best scores in bold and the second best with underline. † indicates approaches with experience replay.
Figure 3 :
3Analysis of output logits on the event trigger classification task on MAVEN. Gold refers to the gold logit and NCP refers to the maximum logit from the new classifier. We keep track of how these two types of logits change throughout 5 learning sessions.
Table 1: Analysis of the performance (Macro-F1 %) on new and old classes on the class-incremental event detection and classification tasks on MAVEN. The best performance of accumulated old classes from all previous sessions (Acc-Old) is highlighted in bold, and the best performance of the old classes in the previous session (Prev-Old) is highlighted with underline.MAVEN (Detection)
Type
S1
S2
S3
S4
S5
DRIFTED-BERT
New
50.9 57.8 52.8 52.7 49.1
Acc-Old
-
0
0
0
0
Prev-Old
-
0
0
0
0
ICE-O (Ours)
New
50.9 56.0 53.2 49.9 49.3
Acc-Old
-
50.6 53.8 53.6 52.4
Prev-Old
-
51.4 56.2 53.1 50.0
ICE-O&PL (Ours)
New
50.9 57.4 53.2 50.2 47.7
Acc-Old
-
50.3 53.0 52.7 50.7
Prev-Old
-
51.0 55.8 52.5 49.6
MAVEN (Classification)
Type
S1
S2
S3
S4
S5
DRIFTED-BERT
New
86.9 63.1 54.7 47.6 34.0
Acc-Old
-
36.9 21.8 15.9 10.0
Prev-Old
-
36.4 33.4 29.4 29.1
ICE (Ours)
New
86.9 79.8 72.8 68.0 59.2
Acc-Old
-
77.2 72.0 66.3 62.5
Prev-Old
-
77.5 72.1 65.7 62.6
ICE-PL (Ours)
New
86.9 67.5 57.2 49.2 34.9
Acc-Old
-
51.3 29.7 16.8 13.1
Prev-Old
-
51.1 49.5 37.2 38.5
ICE-O (Ours)
New
86.5 79.8 76.9 73.3 63.8
Acc-Old
-
80.6 76.5 71.2 68.3
Prev-Old
-
81.0 76.1 69.1 69.4
ICE-PL&O (Ours)
New
86.5 80.3 76.9 71.3 62.0
Acc-Old
-
80.7 76.3 70.2 64.9
Prev-Old
-
81.1 77.0 67.9 66.2
ICE-O+TFE&ER †(Ours) 50.7 42.2 45.0 45.4 46.2 48.7 47.5 47.1 94.2 87.7 86.5 83.9 82.0 81.7 80.2 76.11 17.6
ICE-O (Ours)
56.2 57.8 61.7 64.2 65.6 67.3 68.9 68.9 93.5 86.6 83.8 80.4 78.1 76.5 75.4 71.9
ICE-PL&O (Ours)
56.2 54.9 57.1 58.2 58.9 59.7 60.6 58.7 93.5 84.6 80.3 75.1 71.9 68.7 66.0 60.3
Table 4 :
4Results (Micro-F1 score, %) on relation detection and classification on 10 learning sessions. We highlight the best scores in bold and the second best with underline. † indicates approaches with experience replay.ing negative instances during training can constrain
the updates in the classifier, and consequently miti-
gate classifier drift and forgetting. (3) Persevering
the logits of previous classes without updating the
previous classifiers hurts the performance on most
tasks, by comparing ICE-PL with ICE and com-
paring ICE-PL&O with ICE-O. This observation
is consistent with our findings in Section 5.2. (4)
Previous methods generally perform worse than
our solutions even with experience replay. The
possible reasons include overfitting to the stored
examples in the small memory buffer or the regu-
larization from replay may not be effective enough
to mitigate the forgetting.
BERT-FT upperbound on all the classification tasks. However, ICE-O+TFE&ER performs much worse than ICE-O on all the detection tasks.Thus, following these studies, we adopt experi-
ence replay and design a new variant named ICE-
O with Tunable Feature Extractor and Experience
Replay (abbreviated as ICE-O+TFE&ER), which
tunes the BERT-based feature extractor and adopts
the same replay strategy as ER that preserves 20
samples for each class. From Table 2, 3 and 4, ICE-
O+TFE&ER significantly improves over ICE-O
and achieves comparable performance to the super-
vised
KCN (Cao et al., 2020): the original work proposes a prototype-based method to sample examples to store for replay as well as a hierarchical knowledge distillation (KD) to constrain the model's update. We adapt their hierarchical distillation along with ER as the KCN baseline. (4) KT: a framework that transfers knowledge between new and old event types. (5) EMP(Liu et al., 2022): propose a prompt-based technique to dynamically expand the model architecture to incorporate more classes. (6) CRL(Zhao et al., 2022) proposes consistent representation learning to keep the embeddings of historical relations consistent. Since CRL is designed for the classification tasks without Other, we only evaluate this baseline on the classification tasks we build. (7) Upperbound: we train a model jointly on all classes in the dataset as an upperbound in the conventional supervised learning setting. We devise two different upperbounds: (i) BERT-FFE is the upperbound of our ICE-O model, where we only train the classifier and the feature extractor is fixed. The negative instances are used in the classification tasks without Other; and (ii) BERT-FT is the upperbound that trains both the whole BERT and the classifier.
to ER, KCN, KT, and EMP as an assistant technique to mitigate forgetting. We store 20 examples for each class using the herding algorithm(Welling, 2009) and replay one stored instance in each batch during training to limit the computational cost brought by rehearsal. For CRL, we use the same sample selection and replay strategy as in the original work. For baselines, we adopt a frozen pre-trained BERT-large and a trainable Multi-Layer Perceptron (MLP) as the feature extractor.B More Discussions B.1 More Analysis on Old and New Type Performance
Table 6
6and 7 show the performance of old and new classes for each learning session of the classincremental named entity detection and classification and class-incremental relation detection and classification tasks.Few-NERD (Detection)
Few-NERD (Classification)
Session
Type
1
2
3
4
5
6
7
8
Type
1
2
3
4
5
6
7
8
ER † (Wang et al., 2019)
New
56.9 65.3 75.9 55.8 61.9 56.5 59.35 64.0
New
88.39 74.2 53.8 42.1 33.3 25.0 34.6 26.0
Acc-Old
-
34.4 11.0 18.1 20.3 21.0 23.9 19.1 Acc-Old
-
50.6 52.9 33.7 32.7 34.2 31.3 33.9
Prev-Old
-
32.0 10.3 41.9 39.3 18.8 38.7 36.3 Prev-Old
-
48.9 58.1 48.8 51.0 37.3 42.3 46.1
KCN † (Cao et al., 2020)
New
64.3 58.6 57.9 61.3 56.3 76.0 69.8 56.0
New
88.3 75.1 63.1 46.5 33.4 24.2 31.1 25.9
Acc-Old
-
33.0 35.5 18.6 12.8 6.5
6.5
11.9 Acc-Old
-
57.5 52.4 33.8 29.4 27.2 25.5 21.92
Prev-Old
-
24.44 39.8 18.6 17.9 11.9 25.0 44.6 Prev-Old
-
56.1 61.5 49.1 49.0 46.6 36.8 42.0
KT † (Yu et al., 2021)
New
64.3 60.4 57.4 62.3 56.5 75.7 69.2 58.2
New
88.3 73.1 60.6 45.4 34.7 24.1 34.0 24.8
Acc-Old
-
29.6 34.1 18.9 13.7 6.4
6.4
8.6 Acc-Old
-
51.2 49.6 27.6 30.4 27.6 24.4 26.7
Prev-Old
-
17.3 39.8 21.9 25.0 13.8 22.0 35.0 Prev-Old
-
49.3 58.8 45.8 48.8 34.7 33.1 43.6
EMP † (Liu et al., 2022)
New
61.9 56.6 53.1 58.4 55.1 74.1 64.4 53.8
New
88.1 75.4 65.7 49.1 37.2 31.7 40.4 23.1
Acc-Old
-
37.2 41.0 36.9 32.3 18.7 25.0 26.5 Acc-Old
-
49.6 56.5 46.3 44.6 44.7 39.7 10.1
Prev-Old
-
27.7 40.1 36.22 32.8 22.5 40.2 48.8 Prev-Old
-
46.7 70.1 58.6 56.0 47.9 47.1 15.5
DRIFTED-BERT
New
55.6 67.4 75.5 58.2 60.6 56.4 59.0 59.4
New
88.1 69.2 60.5 44.3 32.0 24.4 36.8 21.2
Acc-Old
-
5.5
3.0
1.2
1.8
2.0
1.4
2.4 Acc-Old
-
9.0
4.4
1.8 10.7 3.6
6.4
6.3
Prev-Old
-
0
0
0
0
0
0
6.34 Prev-Old
-
3.8
3.7
6.2 29.9 14.1 16.5 14.0
ICE (Ours)
New
-
-
-
-
-
-
-
-
New
88.1 85.4 85.5 61.2 63.7 67.4 65.4 59.5
Acc-Old
-
-
-
-
-
-
-
-
Acc-Old
-
79.9 76.4 72.0 65.2 63.7 60.8 59.0
Prev-Old
-
-
-
-
-
-
-
-
Prev-Old
-
79.1 80.4 83.4 59.3 62.0 64.8 63.0
ICE-PL (Ours)
New
-
-
-
-
-
-
-
-
New
88.1 70.9 61.0 45.0 33.4 26.2 37.0 21.2
Acc-Old
-
-
-
-
-
-
-
-
Acc-Old
-
20.6 6.7
5.2 16.1 8.2
9.9
11.8
Prev-Old
-
-
-
-
-
-
-
-
Prev-Old
-
16.1 9.2 36.6 46.4 36.0 42.5 41.5
ICE-O (Ours)
New
55.6 69.0 76.3 61.0 64.2 62.7 64.0 68.8
New
87.8 89.1 89.3 71.3 74.9 74.5 72.2 70.7
Acc-Old
-
58.2 62.1 62.9 63.4 64.4 64.3 63.8 Acc-Old
-
82.5 82.1 78.5 74.3 72.9 70.6 67.9
Prev-Old
-
57.2 69.1 76.4 61.0 64.7 65.6 64.6 Prev-Old
-
81.9 86.3 87.1 69.8 73.7 71.7 69.4
ICE-PL&O (Ours)
New
55.6 65.0 73.3 50.1 57.1 49.0 51.4 56.7
New
87.8 87.9 89.1 66.5 68.9 61.3 63.2 58.9
Acc-Old
-
58.2 60.2 60.6 57.4 58.2 56.4 51.5 Acc-Old
-
80.4 80.1 74.0 67.4 63.8 57.2 53.2
Prev-Old
-
57.4 65.4 74.2 51.0 59.5 56.3 50.3 Prev-Old
-
79.7 84.9 85.0 65.1 68.1 59.2 60.2
Table 6 :
6Analysis of the performance (Macro-F1 %) on new and old classes on the class-incremental named entity detection and classification tasks on Few-NERD.TACRED (Detection)
TACRED (Classification)
Session
Type
1
2
3
4
5
6
7
8
9
10
Type
1
2
3
4
5
6
7
8
9
10
ER † (Wang et al., 2019)
New
29.2 19.8 23.2 28.2 27.1 10.2 47.5 10.6 36.6 24.1
New
93.9 53.9 43.6 50.9 33.9 22.2 48.9 21.6 43.0 27.3
Acc-Old
-
32.9 26.1 16.6 18.7 19.7 15.9 17.3 17.0 17.5 Acc-Old
-
69.5 61.4 49.3 36.4 41.9 39.1 34.9 41.1 40.1
Prev-Old
-
9.3 21.5 2.2 27.5 32.5 21.4 43.48 13.5 42.5 Prev-Old
-
63.8 68.9 54.6 56.3 49.5 43.2 61.3 31.3 66.4
KCN † (Cao et al., 2020)
New
29.2 19.8 22.1 28.3 27.6 8.2 38.8 11.3 34.9 21.7
New
95.3 56.4 40.1 49.0 33.9 20.8 45.0 23.5 49.0 21.7
Acc-Old
-
30.1 15.9 6.9 10.8 10.2 13.2
8.1
7.6
6.9 Acc-Old
-
65.1 49.4 37.0 35.7 28.0 30.0 30.2 32.6 28.8
Prev-Old
-
24.9 10.2 0.6 23.1 34.8 27.4 27.5 12.9 40.5 Prev-Old
-
57.6 43.7 46.9 58.6 43.4 32.0 59.4 32.9 68.3
KT † (Yu et al., 2021)
New
29.2 19.6 20.6 26.0 29.5 10.1 41.2 11.1 32.5 24.5
New
95.3 60.8 48.7 50.6 33.2 16.2 42.7 18.3 33.9 22.0
Acc-Old
-
30.2 11.7 10.0 10.2 9.6
9.6
6.8
7.6
7.9 Acc-Old
-
58.4 54.9 47.4 31.0 24.7 26.7 27.2 28.6 21.6
Prev-Old
-
25.3 9.6 12.1 23.0 34.5 26.1 24.9 14.9 41.6 Prev-Old
-
58.4 54.9 47.4 31.0 24.7 26.7 27.2 28.6 21.6
EMP † (Liu et al., 2022)
New
25.5 19.3 17.4 32.3 17.5 9.5 37.4
8.4
34.6 19.3
New
90.9 45.6 35.6 38.3 29.3 8.4 40.8 9.7 17.7 26.7
Acc-Old
-
27.5 19.9 16.4 11.7 17.0 17.0 15.5 12.8 16.6 Acc-Old
-
44.5 22.0 19.4 11.2 15.4 18.8 8.2 10.8 18.2
Prev-Old
-
22.9 21.9 5.9 22.8 31.9 21.2 34.1
7.1 45.8 Prev-Old
-
34.2 25.8 20.5 11.7 32.9 3.7 16.4 5.5 31.9
DRIFTED-BERT
New
28.7 16.0 19.3 20.2 21.8 11.3 43.2
8.6
40.7 20.0
New
93.6 57.4 35.6 32.4 29.9 13.0 36.3 7.3 18.3 12.3
Acc-Old
-
8.0
3.9
7.0
4.6
4.4
1.8
5.0
4.7
5.8 Acc-Old
-
18.6 5.7
4.6
2.1
4.6
6.4
3.1
5.5
6.9
Prev-Old
-
1.3
1.8 15.6 12.4 16.7 0.0
36.3
9.0 36.0 Prev-Old
-
0.0
1.9
6.3
3.5 22.5 17.4 13.8 10.8 47.1
ICE (Ours)
New
-
-
-
-
-
-
-
-
-
-
New
93.6 75.3 33.6 43.8 47.2 24.3 49.8 26.8 58.1 30.3
Acc-Old
-
-
-
-
-
-
-
-
-
-
Acc-Old
-
73.7 55.6 43.9 36.1 34.0 31.2 31.7 30.2 31.2
Prev-Old
-
-
-
-
-
-
-
-
-
-
Prev-Old
-
68.4 56.3 27.3 45.1 46.5 20.3 50.1 26.8 57.4
ICE-PL (Ours)
New
-
-
-
-
-
-
-
-
-
-
New
93.6 57.4 36.7 33.3 27.7 16.2 37.0 15.9 61.8 23.8
Acc-Old
-
-
-
-
-
-
-
-
-
-
Acc-Old
-
18.6 7.2
9.3
9.1
9.5
8.2 10.7 13.2 15.9
Prev-Old
-
-
-
-
-
-
-
-
-
-
Prev-Old
-
0.0
4.9 21.1 33.3 46.9 28.9 39.0 16.8 64.9
ICE-O (Ours)
New
28.7 16.4 19.3 28.5 29.4 20.2 42.2
7.3
38.2 26.1
New
92.7 78.0 45.6 61.3 61.0 34.5 64.2 33.5 70.0 33.8
Acc-Old
-
31.0 23.0 23.1 24.5 26.5 26.3 26.5 26.2 26.4 Acc-Old
-
86.4 71.3 60.4 58.0 55.0 50.8 48.0 46.8 45.0
Prev-Old
-
29.9 16.2 19.8 29.3 30.6 22.3 42.2
7.4 38.1 Prev-Old
-
84.2 73.6 44.7 61.1 54.5 30.0 63.9 33.2 69.9
ICE-PL&O (Ours)
New
28.7 17.2 18.1 20.3 16.5 7.7 35.1
6.4
34.8 15.2
New
92.7 67.9 50.0 55.8 43.5 19.8 53.8 24.8 57.0 21.4
Acc-Old
-
33.1 23.4 22.3 18.7 19.4 18.9 19.7 20.4 21.6 Acc-Old
-
81.5 62.0 50.4 38.9 31.1 28.1 29.2 27.2 26.5
Prev-Old
-
30.5 17.5 19.8 22.2 18.1 8.2
33.9
7.2 36.4 Prev-Old
-
78.6 67.2 49.1 46.3 50.9 21.5 53.2 24.2 61.0
Table 7 :
7Analysis of the performance (Macro-F1 %) on new and old classes on the class-incremental relation detection and classification tasks on TACRED.
Session is defined as an incremental learning stage to learn new classes with a model trained on the previous sessions.
AcknowledgmentsThis research is based upon work partially supported by the Amazon Research Award program and U.S. DARPA KMASS Program # HR001121S0034. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies, either expressed or implied, of DARPA or the U.S. Government. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright annotation therein.
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| {'fraction_non_alphanumeric': 0.05574523965856861, 'fraction_numerical': 0.06160210111621799, 'mean_word_length': 4.102928365610132, 'pattern_counts': {'":': 0, '<': 5, '<?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': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Class-incremental learning (CIL) aims to develop a learning system that can continually learn new classes from a data stream without forgetting previously learned classes. When learning classes incrementally, the classifier must be constantly updated to incorporate new classes, and the drift in decision boundary may lead to severe forgetting. This fundamental challenge, however, has not yet been studied extensively, especially in the setting where no samples from old classes are stored for rehearsal. In this paper, we take a closer look at how the drift in the classifier leads to forgetting, and accordingly, design four simple yet (super-) effective solutions to alleviate the classifier drift: an Individual Classifiers with Frozen Feature Extractor (ICE) framework where we individually train a classifier for each learning session, and its three variants ICE-PL, ICE-O and ICE-PL&O which further take the logits of previously learned classes from old sessions or a constant logit of an Other class as constraint to the learning of new classifiers. Extensive experiments and analysis on 6 class-incremental information extraction tasks demonstrate that our solutions, especially ICE-O, consistently show significant improvement over the previous state-of-the-art approaches with up to 44.7% absolute F-score gain, providing a strong baseline and insights for future research on class-incremental learning. 1', 'arxivid': '2305.16559', 'author': ['Minqian Liu [email protected] \nComputer Science Department Virginia Tech\n\n', 'Lifu Huang [email protected] \nComputer Science Department Virginia Tech\n\n'], 'authoraffiliation': ['Computer Science Department Virginia Tech\n', 'Computer Science Department Virginia Tech\n'], 'corpusid': 258947098, 'doi': '10.48550/arxiv.2305.16559', 'github_urls': ['https://github.com/VT-NLP/ICE.'], 'n_tokens_mistral': 25370, 'n_tokens_neox': 20801, 'n_words': 11379, 'pdfsha': '3b9abe8d06e7ae7c1ee3aeec2d10fca68f52923d', 'pdfurls': ['https://export.arxiv.org/pdf/2305.16559v1.pdf'], 'title': ['Teamwork Is Not Always Good: An Empirical Study of Classifier Drift in Class-incremental Information Extraction', 'Teamwork Is Not Always Good: An Empirical Study of Classifier Drift in Class-incremental Information Extraction'], 'venue': []} |
arxiv |
Non-Minimal Flavored S 3 ⊗ Z 2 Left-Right Symmetric Model
10 Apr 2017
Juan Carlos Gómez-Izquierdo [email protected]
Tecnologico de Monterrey
Campus Estado de MexicoAtizapan de Zaragoza
Estado de Mexico
Apartado Postal
52926Mexico
Instituto de Ciencias Nucleares
Universidad Nacional Autónoma de México
3000MéxicoD.FMéxico
Instituto de Física
Universidad Nacional Autónoma de México
01000MéxicoD.FMéxico
Non-Minimal Flavored S 3 ⊗ Z 2 Left-Right Symmetric Model
10 Apr 2017
We propose a non-minimal left-right symmetric model (LRSM) with Parity Symmetry where the fermion mixings arise as result of imposing an S 3 ⊗ Z 2 flavor symmetry, and an extra Z e 2 symmetry is considered to suppress some Yukawa couplings in the lepton sector. As a consequence, the effective neutrino mass matrix possesses approximately the µ − τ symmetry. The breaking of the µ − τ symmetry induces sizable non zero θ 13 , and the deviation of θ 23 from 45 • is strongly controlled by an free parameter and the complex neutrino masses. Then, an analytic study on the extreme Majorana phases is done since these turn out to be relevant to enhance or suppress the reactor and atmospheric angle. So that we have constrained the parameter space for the parameter and the lightest neutrino mass that accommodate the mixing angles. The highlighted results are: a) the normal hierarchy is ruled out since the reactor angle comes out being tiny, for any values of the Majorana phases; b) for the inverted hierarchy there is one combination in the extreme phases where the values of the reactor and atmospheric angles are compatible up to 2, 3 σ of C. L., but the parameter space is tight; c) the model favors the degenerate ordering for one combination in the extreme Majorana phases. In this case, the reactor and atmospheric angle are compatible with the experimental data for a large set of values of the free parameters. Therefore, this model may be testable by the future result that the Nova and KamLAND-Zen collaborations will provide. *
Introduction
Currently, we know that neutrinos oscillate and have a tiny mass. In the theoretical framework of three active neutrinos, the difference of the squared neutrino masses for normal (inverted) hierarchy are given by ∆m 2 21 10 −5 eV 2 = 7.60 +0. 19 −0. 18 , and |∆m 2 31 | 10 −3 eV 2 = 2.48 +0.05 −0.07 (2.38 +0.05 −0.06 ). Additionally, we have the values of the mixing angles sin 2 θ 12 /10 −1 = 3.23 ± 0.16, sin 2 θ 23 /10 −1 = 5.67 +0.32 −1.24 (5.73 +0.25 −0.39 ) and sin 2 θ 13 /10 −2 = 2.26 ± 0.12 (2.29 ± 0.12) [1].At present, there is no yet solid evidence on the Dirac CP-violating phase and the ordering that respects the neutrino masses. The Nova [2] and KamLAND-Zen [3] Collaborations can shed light on the hierarchy in the coming years.
In spite of the fact that the Standard Model (SM) works out almost perfectly, the neutrino experimental data can not be explained within this framework. If the neutrino sector opens the window to the new physics, then what is the new model and the extra ingredients that are needed to accommodate the masses and mixings ?. In this line of thought, a simplest route to include small neutrino masses and mixings to the SM is to add the missing right-handed neutrinos (RHN's) states to the matter content, and then invoking the see-saw mechanism [4][5][6][7][8][9][10]. However, we should point out that the RHN mass scale is introduced by hand with no relation whatsoever to the Higgs mechanism that gives mass to all other fields. Nonetheless, this problem may be alleviated if the minimal extension of the SM is replaced by the leftright symmetric model (LRSM) [7,[11][12][13][14] where the RHN's are already included in the matter content. Additionally, the see-saw mechanism comes in rather naturally in the context of leftright symmetric scenarios; aside from other nice features, as for instance the recovery of Parity Symmetry, and the appearance of right-handed currents at high energy, which also makes such extensions very appealing. Recently, the left-right scenarios have been revised [15][16][17][18][19][20][21][22][23][24][25] in order to make contact with the last experimental data of LHC. Moreover, the dark matter problem [26][27][28] and the diphoton excess anomaly [29][30][31][32][33] have been explored in this kind of scenarios.
Explaining the peculiar neutrino mixing pattern (besides the CKM mixing matrix) has been a hard task. Along this line, the mass textures have played an important role in trying to solve this puzzle [34]. In fact, discrete symmetries may be the missing ingredient to understand the mixings so that several groups have been proposed [35][36][37] to get in an elegant way the mass textures. In this line of thought, the S 3 flavor symmetry, in particular, is a good candidate to handle the Yukawa couplings for leptons and quarks; and this has been studied exhaustively in different frameworks . In most of these works, the meaning of the flavor has been extended to the scalar sector such that three Higgs doublets are required to accommodate the PMNS and CKM mixing matrices.
Although there are too many flavored models in the literature, the LRSM has received few attention in the context of the flavored puzzle [60][61][62]. It is not an easy task to study the mixings in the LRSM since the structure of the gauge group increases the Yukawa sector parameters compared to the SM. However, as was shown in the early works, Parity Symmetry might reduce substantially the gauge and Yukawa couplings; this last issue gives the opportunity to calculate the right-handed CKM matrix [21,22] which is crucial to study in great detail the W R gauge boson that comes out being a prediction of the LRMS. Then, it is fundamental to face the flavor puzzle in this kind of theoretical frameworks.
Therefore, we propose a non-minimal LRSM with Parity Symmetry where the fermion mixings arise as result of imposing an S 3 ⊗ Z 2 flavor symmetry, and an extra Z e 2 symmetry is considered to suppress some Yukawa couplings in the lepton sector. Additionally, a non conventional assignment is done for the matter content under the S 3 symmetry and this is the clear difference between the previous studies and this one. As a consequence, in the lepton sector, the effective neutrino mass matrix possesses approximately the µ−τ symmetry [63][64][65][66][67][68][69][70][71][72][73][74][75][76]. The breaking of the µ − τ symmetry induces sizable non zero θ 13 , and the deviation of θ 23 from 45 • is strongly controlled by an free parameter and the complex neutrino masses. Then, an analytic study on the extreme Majorana phases is done since these turn out to be relevant to enhance or suppress the reactor and atmospheric angle. Thus, we can constrain the parameter space for parameter and the lightest neutrino mass that accommodate the mixing angles. The highlighted results are: a) the normal hierarchy is ruled out since the reactor angle comes out being tiny, for any values of the Majorana phases; b) for the inverted hierarchy there is one combination in the extreme phases where the values of the reactor and atmospheric angles are compatible up to 2, 3 σ of C. L., but the parameter space is tight; c) the model favors the degenerate ordering for one combination in the extreme Majorana phases. In this case, the reactor and atmospheric angle are compatible with the experimental data for a large set of values of the free parameters. The quark sector will be discussed exhaustively in a future work, however, some preliminary results will be commented.
The paper is organized as follows: we present, in Sec. II, the matter content of the model and also their respective assignment under the S 3 symmetry. In addition, we briefly explain the scalar sector and argue about the need to include the Z e 2 symmetry. In Sec. III, the fermion mass matrices are obtained and we put attention on the lepton sector for getting the mixing matrices. We present, in Sec. IV, the PMNS matrix that the model predicts. Finally, we present an analytic study on the mixing angles and our results in Sec. V, and we close our discussion with a summary of conclusions.
Flavored Left-Right Symmetric Model
The minimal LRSM is based on the usual, SU (3) c ⊗ SU (2) L ⊗ SU (2) R ⊗ U (1) B−L , gauge symmetry where Parity Symmetry, P, is assumed to be a symmetry a high energy but it is broken at electroweak scale since there are no right-handed currents. The matter fields and their respective quantum numbers (in parenthesis) under the gauge symmetry are given by
Q (L,R) = u d (L,R) ∼ (3, (2, 1), (1, 2), 1/3), (L, R) = ν (L,R) ∼ (1, (2, 1), (1, 2), −1), Φ = φ 0 φ + φ − φ 0 ∼ (1, 2, 2, 0) ; ∆ (L,R) = δ + 2 δ ++ δ 0 − δ + 2 (L,R) ∼ (1, (3, 1), (1, 3), 2) .(1)
The gauge invariant Yukawa mass term is given by
−L Y =Q L y q Φ +ỹ qΦ Q R +L y Φ +ỹ Φ R + y LL ∆ L L c + y RRc ∆ R R + h.c.(2)
where the family indexes have been suppressed andΦ i = −iσ 2 Φ * i iσ 2 . Here, Parity Symmetry will be assumed in the above Lagrangian, then, this requires that Ψ iL ↔ Ψ iR , Φ i ↔ Φ † i and ∆ iL ↔ ∆ † iR for fermions and scalar fields, respectively. Thereby, the Yukawa couplings may reduce substantially and the gauge couplings too. In particular, for the former issue we have that y = y † ,ỹ =ỹ † and y R = y L . On the other hand, due to our purpose the scalar potential will be left aside. But, in the minimal LRMS the spontaneous symmetry breaking is as follows: Parity Symmetry is broken at the same scale where the ∆ R right-handed scale acquires its vacuum expectation value (vev). At the first stage, the RHN's are massive particles, then the rest of the particles turn out massive since the Higgs scalars get their vev's. Explicitly,
∆ L,R = 0 0 v L,R 0 , Φ = k 0 0 k , Φ = k * 0 0 k * .(3)
As result, the Yukawa mass term is given by
−L Y =q iL (Mq) ij q jR +¯ iL (M ) ij jR + 1 2ν iL (Mν ) ij ν c jL + 1 2ν c iR (M R ) ij ν jR + h.c.(4)
where the type I see-saw mechanism has been realized,
M ν = −M D M −1 R M T D ; so that the M L were neglected for simplicity.
In the present model, the Yukawa mass term will be controlled by the S 3 flavor symmetry. The non-Abelian group S 3 is the permutation group of three objects and this has three irreducible representations: two 1-dimensional, 1 S and 1 A , and one 2-dimensional representation, 2 (for a detailed study see [35]). The multiplication rules among them are
1 S ⊗ 1 S = 1 S , 1 S ⊗ 1 A = 1 A , 1 A ⊗ 1 S = 1 A , 1 A ⊗ 1 A = 1 A , 1 S ⊗ 2 = 2, 1 A ⊗ 2 = 2, 2 ⊗ 1 S = 2, 2 ⊗ 1 A = 2; a 1 a 2 2 ⊗ b 1 b 2 2 = (a 1 b 1 + a 2 b 2 ) 1 S ⊕ (a 1 b 2 − a 2 b 1 ) 1 A ⊕ a 1 b 2 + a 2 b 1 a 1 b 1 − a 2 b 2 2 .(5)
Having introduced briefly the gauge and the non-Abelian group, let us build the gauge and flavored Yukawa mass term. To do this, we will consider three Higgs bidoublets as well as three left-right triplets with the purpose of getting the mixing in the lepton sector. Here, we want to emphasize a clear difference between this model and the previous ones with the S 3 symmetry. In our model, the quark and lepton families have been assigned in a different way under the irreducible representation of S 3 . Explicitly, for the former and the Higgs sector respectively, the first and second family have been put together in a flavor doublet 2, and the third family is a singlet 1 S . On the contrary, for the latter sector, the first family is a singlet 1 S and the second and third families are put in a doublet 2. The advantage of making this choice is that the quark mass matrices may be put into two mass textures fashion that fit the CKM matrix very well; in the lepton sector, on the other hand, the appearance of the approximated µ − τ symmetry in the effective neutrino mass matrix is good signal to understand the mixings. Remarkably, Nova collaboration is testing the µ − τ symmetry and some results have been released [2]. The matter content of the model transforms in a not trivial way under the S 3 symmetry and this is displayed in the table below. Here, the Z 2 symmetry has been added in order to prohibit some Yukawa couplings in the lepton sector. Thus, the most general Yukawa mass Table 1: Non-minimal left-right model. Here, I = 1, 2 and J = 2, 3.
Matter Q I(L,R) Q 3(L,R) (L 1 , R 1 ) (L J , R J ) Φ I Φ 3 ∆ I(L,R) ∆ 3(L,R) S 3 2 1 S 1 S 2 2 1 S 2 1 S Z 2 1 1 1 −1 1 1 −1 1
term, that respects the S 3 ⊗ Z 2 flavour symmetry and the gauge group, is given as
−L Y = y q 1 Q 1L (Φ 1 Q 2R + Φ 2 Q 1R ) +Q 2L (Φ 1 Q 1R − Φ 2 Q 2R ) + y q 2 Q 1L Φ 3 Q 1R +Q 2L Φ 3 Q 2R + y q 3 Q 1L Φ 1 +Q 2L Φ 2 Q 3R + y q 4Q 3L [Φ 1 Q 1R + Φ 2 Q 2R ] + y q 5Q 3L Φ 3 Q 3R +ỹ q 1 Q 1L Φ 1 Q 2R +Φ 2 Q 1R +Q 2L Φ 1 Q 1R −Φ 2 Q 2R +ỹ q 2 Q 1LΦ3 Q 1R +Q 2LΦ3 Q 2R +ỹ q 3 Q 1LΦ1 +Q 2LΦ2 Q 3R +ỹ q 4Q 3L Φ 1 Q 1R +Φ 2 Q 2R +ỹ q 5Q 3LΦ3 Q 3R + y 1L1 Φ 3 R 1 + y 2 (L 2 Φ 2 +L 3 Φ 1 )R 2 + (L 2 Φ 1 −L 3 Φ 2 )R 3 + y 3 L 2 Φ 3 R 2 +L 3 Φ 3 R 3 +ỹ 1L1Φ3 R 1 +ỹ 2 (L 2Φ2 +L 3Φ1 )R 2 + (L 2Φ1 −L 3Φ2 )R 3 +ỹ 3 L 2Φ3 R 2 +L 3Φ3 R 3 + y L 1L1 ∆ 3L L c 1 + y L 2L1 [∆ 1L L c 2 + ∆ 2L L c 3 ] + y L 3 L 2 ∆ 1L +L 3 ∆ 2L L c 1 + y L 4 L 2 ∆ 3L L c 2 +L 3 ∆ 3L L c 3 + y R 1R c 1 ∆ 3R R 1 + y R 2R c 1 [∆ 1R R 2 + ∆ 2R R 3 ] + y R 3 R c 2 ∆ 1R +R c ∆ 2R R 1 + y R 4 R c 2 ∆ 3R R 2 +R c 3 ∆ 3R R 3 + h.c,(6)
In this flavored model, we have to keep in mind that Parity Symmetry will be assumed in the above Lagrangian in such a way the number of Yukawa couplings is reduced. More even, we stress that an extra symmetry Z e 2 is used to get a diagonal charged lepton and Dirac neutrino mass matrix whereas the Majorana mass matrices retain their forms. Explicitly, in the above Lagrangian, we demand that
L 3 ↔ −L 3 , R 3 ↔ −R 3 , ∆ 2L ↔ −∆ 2L , ∆ 2R ↔ −∆ 2R .(7)
so that the termsL 2 R 3 andL 3 R 2 are absent in the lepton sector. As was already commented, because of our interest in studying masses and mixings for fermions, the scalar potential will not be analyzed for the moment. We ought to comment that this study is not trivial since the scalar sector has been augmented, so that the potential is rather complicated, but this study has to be done eventually since is crucial for theoretical and phenomenological purpose. From Eq.(3) and Eq. (6), the mass matrices have the following structure
Mq = aq + b q bq cq bq aq − b q c q fq f q gq , M = a 0 0 0 b + c 0 0 0 b − c , M (L,R) = a (L,R) b (L,R) b (L,R) b (L,R) c (L,R) 0 b (L,R) 0 c (L,R) ,(8)
where the q = u, d and = e, ν D . Explicitly, the matrix elements for quarks and leptons are given as
au = y q 2 k 3 +ỹ q 2 k * 3 , b u = y q 1 k 2 +ỹ q 1 k * 2 , bu = y q 1 k 1 +ỹ q 1 k * 1 , cu = y q 3 k 1 +ỹ q 3 k * 1 , c u = y q 3 k 2 +ỹ q 3 k * 2 , fu = y †q 3 k 1 +ỹ †q 3 k * 1 , f u = y †q 3 k 2 +ỹ †q 3 k * 2 , gu = y q 5 k 3 +ỹ q 5 k * 3 , a d = y q 2 k 3 +ỹ q 2 k * 3 , b d = y q 1 k 2 +ỹ q 1 k * 2 , b d = y q 1 k 1 +ỹ q 1 k * 1 , c d = y q 3 k 1 +ỹ q 3 k * 1 ; c d = y q 3 k 2 +ỹ q 3 k * 2 , f d = y †q 3 k 1 +ỹ †q 3 k * 1 , f d = y †q 3 k 2 +ỹ †q 3 k * 2 , gu = y q 5 k 3 +ỹ q 5 k * 3 , a D = y 1 k 3 +ỹ 1 k * 3 , b D = y 3 k 3 +ỹ 3 k * 3 , c D = y 2 k 2 +ỹ 2 k * 2 , ae = y 1 k 3 +ỹ 1 k * 3 , be = y 3 k 3 +ỹ 3 k * 3 , ce = y 2 k 2 +ỹ 2 k * 2 , a (L,R) = y R 1 v 1(L,R) , b (L,R) = y R 2 v 2(L,R) b (L,R) = y R 2 v 3(L,R) , c (L,R) = y R 4 v 1(L,R) .(9)
where Parity Symmetry has been considered. Remarkably, we will end up having a complex symmetric (diagonal) quark (lepton) mass matrix if the vev's are complex; in the literature this scenario is well known as pseudomanifest left-right symmetry [77,78]. If the vev's are real, the quark (lepton) mass matrix is hermitian (real) and the number of CP phases are reduced, this framework is known as manifest left-right symmetry [77,79] . In this work, we will discuss only the first framework and the second one will be studied in an extended version of the model and its consequences on the quark sector.
Masses and Mixings
In principle, in the mass matrices, we can reduce a further the number of free parameters considering certain alignment in the vev's, see Eq. (9). So that, for the moment, we will assume that the vev's of Φ 1 and Φ 2 are degenerate. Explicitly, we demand that k 1 = k 2 ≡ k and
k 1 = k 2 ≡ k . Additionally, v 1R = v 2R = v R . Therefore, we have: Pseudomanisfest left-right theory. Mq = aq + bq bq cq bq aq − bq cq cq cq gq , M = a 0 0 0 b + c 0 0 0 b − c , M (L,R) = a (L,R) b (L,R) b (L,R) b (L,R) c (L,R) 0 b (L,R) 0 c (L,R) .(10)
Manifest left-right theory.
Mq = aq + bq bq cq bq aq − bq cq c * q c * q gq , M = a 0 0 0 b + c 0 0 0 b − c , M (L,R) = a (L,R) b (L,R) b (L,R) b (L,R) c (L,R) 0 b (L,R) 0 c (L,R) .(11)
As was already commented the full analysis of the quark masses and mixings will be left aside for this moment. However, we just make some comments. In the pseudomanifest framework, the M q mass matrix may be put into two mass textures fashion that fit the CKM matrix very well. In similar way, the manifest framework is tackled. For this case, the quark mixing matrix has fewer free parameters than the above framework since this is hermitian; the study, and its predictions on the mixing angles is work in progress.
Charged Leptons
The M e mass matrix is complex and diagonal then one could identify straight the physical masses, however, we will make a similarity transformation in order to prohibit a fine tuning in the free parameters. What we mean is the following, the M e mass matrix is diagonalized by U eL = S 23 P e and U eR = S 23 P † e , this is,
M e = diag.(|m e |, |m µ |, |m τ |) = U † eL M e U eR = P † m e P †
As result, one obtains that |m e | = |a e |, |m µ | = |b e − c e | and |m τ | = |b e + c e |.
Neutrinos
On the other hand, the M ν effective neutrino mass matrix is given as
Mν = X a 2 D −a D Y(b D + c D ) −a D Y(b D − c D ) −a D Y(b D + c D ) W(b D + c D ) 2 Z(b 2 D − c 2 D ) −a D Y(b D − c D ) Z(b 2 D − c 2 D ) W(b D − c D ) 2 where M −1 R ≡ X −Y −Y −Y W Z −Y Z W (13)
Now as hypothesis, we will assume that b D is larger than c D , in this way the effective mass matrix can be written as
Mν ≡ Aν −Bν (1 + ) −Bν (1 − ) −Bν (1 + ) Cν (1 + ) 2 Dν (1 − 2 ) −Bν (1 − ) Dν (1 − 2 ) Cν (1 − ) 2 (14) where A ν ≡ X a 2 D , B ν ≡ Ya D b D , C ν ≡ Wb 2 D and D ν ≡ Zb 2 D are complex. Besides, ≡ c D /b D is complex too.
Here, we want to stress that the last parameter will be considered as a perturbation to the effective mass matrix such that | | ≪ 1. To be more specific, | | ≤ 0.3 in order to break softly the µ − τ symmetry. So that, hereafter, we will neglect the 2 quadratic terms in the above matrix. Having done this, we go back to the effective neutrino mass matrix. In order to cancel the S 23 contribution that comes from the charged lepton sector, we make the following to M ν . We know thatM ν = diag.(m ν 1 , m ν 2 , m ν 3 ) = U † ν M ν U * ν , then U ν = S 23 U ν where the latter mixing matrix will be obtained below. Then,
M ν = U † ν M ν U * ν with Mν = S T 23 Mν S 23 ≈ Aν −Bν (1 − ) −Bν (1 + ) −Bν (1 − ) Cν (1 − 2 ) Dν −Bν (1 + ) Dν Cν (1 + 2 ) (15)
When the parameter is switched off the effective mass matrix, which is denoted by M 0 ν , possesses the µ − τ symmetry and this is diagonalized by
U 0 ν = cos θν e i(ην +π) sin θν e i(ην +π) 0 − sin θν √ 2 cos θν √ 2 − 1 √ 2 − sin θν √ 2 cos θν √ 2 1 √ 2 (16)
where the M 0 ν matrix elements are fixed in terms of the complex neutrinos physical masses, the θ ν free parameter and the η ν Dirac CP phase. To be more explicit,
Aν = (m 0 ν 1 cos 2 θν + m 0 ν 2 sin 2 θν )e 2i(ην +π) , −Bν = sin 2θν √ 8 (m 0 ν 2 − m 0 ν 1 )e i(ην +π) ; Cν = 1 2 (m 0 ν 1 sin 2 θν + m 0 ν 2 cos 2 θν + m 0 ν 3 ), Dν = 1 2 (m 0 ν 1 sin 2 θν + m 0 ν 2 cos 2 θν − m 0 ν 3 ).(17)
Including the parameter we can write the effective mass matrix as M ν = M 0 ν + M ν where the second matrix contains the perturbation, then, when we apply U 0
ν one gets M ν = U 0 † ν (M 0 ν + M ν )U 0 * ν . Explicitly Mν = Diag.(m 0 ν 1 , m 0 ν 2 , m 0 ν 3 ) + 0 − sin θν (m 0 ν 3 + m 0 ν 1 ) 0 0 cos θν (m 0 ν 3 + m 0 ν 2 ) − sin θν (m 0 ν 3 + m 0 ν 1 ) cos θν (m 0 ν 3 + m 0 ν 2 ) 0 (18)
The contribution of second matrix to the mixing one is given by
U ν ≈ N 1 0 −N 3 sin θr 1 0 N 2 N 3 cos θν r 2 N 1 sin θν r 1 −N 2 cos θν r 2 N 3 (19)
where we have defined the complex mass ratios r (1,2) ≡ (m 0 ν 3 + m 0 ν (1,2) )/(m 0 ν 3 − m 0 ν (1,2) ). Here, N 1 , N 2 and N 3 are the normalization factors which are given as N 1 = 1 + sin 2 θν |r 1 | 2 −1/2 , N 2 = 1 + cos 2 θν |r 2 | 2 −1/2 , N 3 = 1 + sin 2 θν |r 1 | 2 + cos 2 θν |r 2 | 2 −1/2 .
Finally, the effective mass matrix given in Eq. (14) is diagonalized approximately by U ν ≈ S 23 U 0 ν U ν . Therefore, the theoretical PMNS mixing matrix is written as
V P M N S = U † eL U ν ≈ P † e U 0 ν U ν .
PMNS Mixing Matrix
The PMNS mixing matrix is given explicitly as
V P M N S = P † e
cos θν N 1 e i(ην +π) sin θν N 2 e i(ην +π) sin 2θν N 3 2 (r 2 − r 1 ) e i(ην +π) − sin θν
√ 2 N 1 (1 + r 1 ) cos θν √ 2 N 2 (1 + r 2 ) − N 3 √ 2 [1 − r 3 ] − sin θν √ 2 N 1 (1 − r 1 ) cos θν √ 2 N 2 (1 − r 2 ) N 3 √ 2 [1 + r 3 ] (21)
where r 3 ≡ r 2 cos 2 θ ν + r 1 sin 2 θ ν . On the other hand, comparing the magnitude of entries V P M N S with the mixing matrix in the standard parametrization of the PMNS, we obtain the following expressions for the lepton mixing angles
sin 2 θ 13 = |V 13 | 2 = sin 2 2θν 4 N 2 3 | | 2 |r 2 − r 1 | 2 , sin 2 θ 23 = |V 23 | 2 1 − |V 13 | 2 = N 2 3 2 |1 − r 3 | 2 1 − sin 2 θ 13 , sin 2 θ 12 = |V 12 | 2 1 − |V 13 | 2 = N 2 2 sin 2 θν 1 − sin 2 θ 13 .(22)
As can be noticed, if vanishes, one would recover the exact µ − τ symmetry where θ 12 = 0 • and θ 23 = 45 • . Additionally, we have to point out that the reactor and atmospheric angles depend strongly on the neutrino masses ratios so that these angles are sensitive to the Majorana phases. At the same time, the reactor angle does not depend on the phase of the parameter , but on the other hand, the atmospheric one has a clear dependency on this phase.
Analytic Study and Results
In order to make an analytic study on the above formulas, let us emphasize that we are working in a perturbative regime which means that | | ≤ 0.3. Then N i normalization factors should be the order of 1 so that, as is usual in models where the µ − τ symmetry is broken softly, the solar angle is directly related to the free parameter θ ν , as can be seen in Eq. (22). Therefore, at the leading order we have that sin 2 θ 12 = sin 2 θ ν , then,
θ 12 = θ ν .(23)
Therefore, along the analytic study we will consider that sin θ ν ≈ 1/ √ 3 which is a good approximation to the solar angle. Additionally, we will analyze the extreme Majorana phases for the complex neutrino masses for each hierarchy. What we mean by extreme phases is that these can be either 0 or π. Explicitly, m 0 ν i = ±|m 0 ν i |, for i = 1, 2, 3, where |m 0 ν i | is the absolute mass. As we will see, these phases can be relevant to enhance or suppress the reactor and atmospheric angles. In the following, the lightest neutrino mass and the | | parameter will be constrained.
Normal hierarchy. From experimental data, the absolute neutrino masses are |m 0 ν 3 | = ∆m 2 31 + |m 0 ν 1 | 2 and |m 0 ν 2 | = ∆m 2 21 + |m 0 ν 1 | 2 . Now, the mass ratios r 1 , r 2 and r 3 can be approximated as follows 24) as results of this, we obtain
r 1 ≈ 1 + 2 m 0 ν 1 m 0 ν 3 ≈ 1, r 2 ≈ 1 + 2 m 0 ν 2 m 0 ν 3 , r 3 ≈ 1 + 2 m 0 ν 2 m 0 ν 3 cos 2 θν(sin 2 θ 13 ≈ sin 2 2θν | | 2 m 0 ν 2 m 0 ν 3 2 , sin 2 θ 23 ≈ 1 2 1 − 1 + 2 m 0 ν 2 m 0 ν 3 cos 2 θν 2 1 − sin 2 θ 13 .(25)
As can be noticed, if the strict normal hierarchy is assumed then the reactor angle comes out being very small since |m 0 ν 2 /m 0 ν 3 | 2 ≈ ∆m 2 21 /∆m 2 31 , and | | ≤ 0.3. This holds for any extreme Majorana phases in the neutrino masses and this result does not change substantially if the m 0 ν 1 is non-zero. Therefore, the normal spectrum is ruled out for | | ≤ 0.3. Inverted hierarchy. In this case, we have that |m 0 ν 2 | = ∆m 2 13 + ∆m 2 21 + |m 0 ν 3 | 2 and |m 0 ν 1 | = ∆m 2 13 + |m 0 ν 3 | 2 . The mass ratios r 1 , r 2 and r 3 are written approximately as angle as can be seen in Eq. (27)(28)(29)(30). Now, from the absolute value of the neutrino masses we have |m 0
ν 2 | ≈ |m 0 ν 1 |(1 + 2R 1 ), then |m 0 ν 2 | − |m 0 ν 1 | ≈ 2|m 0 ν 1 |R 1 , |m 0 ν 2 | + |m 0 ν 1 | ≈ 2|m 0 ν 1 | [1 + R 1 ] , |m 0 ν 1 ||m 0 ν 2 | ≈ |m 0 ν 1 | 2 [1 + 2R 1 ] ,(31)
where, R 1 ≡ ∆m 2 21 /4|m 0 ν 1 | 2 ≈ O(10 −3 ), if the |m 0 ν 3 | lightest neutrino mass is tiny. Therefore, for the Cases A and B, we have
sin 2 θ 13 ≈ 32 9 | | 2 R 2 1 m 0 ν 3 m 0 ν 1 2 , sin 2 θ 23 ≈ 1 2 1 + 1 ± 2 m 0 ν 3 m 0 ν 1 2 1 − sin 2 θ 13(32)
where the upper (lower) sign, in the atmospheric angle, stands for the Case A (Case B).
Here, we have to keep in mind that |m 0 ν 3 |/|m 0 ν 1 | < 1 so that we can conclude that the first two scenarios are ruled out since that the reactor angle is proportional to the small quantity (|m 0
ν 3 |/|m 0 ν 1 |)R 1 | |, where | | ≤ 0.3.
For the Case C ( Case D) the corresponding sign is the upper (lower), then the mixing angles are given as
sin 2 θ 13 ≈ 32 9 | | 2 m 0 ν 3 m 0 ν 1 2 (1 − R 1 ) 2 , sin 2 θ 23 ≈ 1 2 1 + 1 ± 2 3 m 0 ν 3 m 0 ν 1 2 1 − sin 2 θ 13 .(33)
From these formulas, in general, an | | large value will be needed to compensate the |m 0 ν 3 | lightest neutrino mass to get the allowed region for the reactor angle. But, the atmospheric angle prefers an | | small values. In addition, since that r 3 < 0, the complex parameter phase is taken to be α = 0 to increase the atmospheric angle value. In order to fix ideas, we obtain for the Case C: (a) if | | ≈ 0.3, it is required that |m 0 ν 3 |/|m 0 ν 1 | ≈ 0.26, to obtain sin 2 θ 13 ≈ 0.0229. As a consequence, we get sin 2 θ 23 ≈ 0.94 which is too large; (b) if | | ≈ 0.1, then we need that |m 0 ν 3 |/|m 0 ν 1 | ≈ 0.8 to get sin 2 θ 13 ≈ 0.0229, and therefore, sin 2 θ 23 ≈ 0.68, which is still large in comparison to the central value. For the Case D, the reactor angle has approximately the same values for | | ≈ 0.3, | | ≈ 0.1 and their respective |m 0 ν 3 |/|m 0 ν 1 | mass ratios as above case. Then, with these values of | |, we obtain sin 2 θ 23 ≈ 0.79 and sin 2 θ 23 ≈ 0.56, respectively. Notice that both values are approaching the allowed region for this mixing angle, then, this case is more favorable than the Case C. This happens since a large contribution of | |, in the atmospheric angle, is suppressed by r 3 which is minor than 1 and the reactor angle prefers an | | large values. Let us remark the following, if | | is tiny, we require that |m 0 ν 3 |/|m 0 ν 1 | neutrino mass ratio should be larger than 1 to enhance the reactor angle but this mass ratio violates the inverted ordering. This statement is valid for the Cases C and D. At the same time, if α = π is chosen in the atmospheric angle, this would be tiny for the same values of | | and the |m 0 ν 3 |/|m 0 ν 1 , as can be verified straight from Eq. (33). In order to get a complete view of the parameter space, let us show some plots for the reactor and atmospheric angles. We have considered the exact formulas given in Eq. (22), for the the observables as the θ 12 solar angle, ∆m 2 21 and ∆m 2 13 , their values were taken up to 3 σ. Then, the figure 1 shows the atmospheric angle versus the reactor one for the Case C and D. This scattering plots clearly support our analytic result on the Case C, this is, both mixing angles can not be accommodate simultaneously. In the Case D, the reactor angle is consistent with the experimental data but the atmospheric one is large but consistent up to 2 − 3 σ in its allowed region. In addition, for the Case D, the parameter space is shown in the figure 2. As can be seen, the atmospheric angle prefers small values for | | whereas the reactor one needs a large value, as was already pointed out. Moreover, the set of values for | | and |m 0 ν 3 | is tight. Degenerated hierarchy. In this case, |m 0
ν 3 | |m 0 ν 2 | |m 0 ν 1 | m 0 , with m 0 0.1 eV .
Then, the absolute neutrino masses can be written as |m 0 ν 3 | = ∆m 2 31 + m 2 0 ≈ m 0 (1 + ∆m 2 31 /2m 2 0 ) and |m 0 ν 2 | = ∆m 2 21 + m 2 0 ≈ m 0 (1 + ∆m 2 21 /2m 2 0 ). As in the inverted case, there are four independent cases for the signs which are shown below.
• Case A. If m 0 ν i > 0, r A 1 = |m 0 ν 3 | + m 0 |m 0 ν 3 | − m 0 , r A 2 = |m 0 ν 3 | + |m 0 ν 2 | |m 0 ν 3 | − |m 0 ν 2 | , r A 3 = r A 2 cos 2 θν + r A 1 sin 2 θν .(34)
• Case B. If If m 0 ν (2,1) > 0 and m 0 ν 3 < 0,
r B 1 = |m 0 ν 3 | − m 0 |m 0 ν 3 | + m 0 = 1 r A 1 , r B 2 = |m 0 ν 3 | − |m 0 ν 2 | |m 0 ν 3 | + |m 0 ν 2 | = 1 r A 2 , r B 3 = r B 2 cos 2 θν + r B 1 sin 2 θν .(35)
• Case C. If m 0 ν (3,2) > 0 and m 0 ν 1 = −m 0 ,
r C 1 = |m 0 ν 3 | − m 0 |m 0 ν 3 | + m 0 = 1 r A 1 , r C 2 = |m 0 ν 3 | + |m 0 ν 2 | |m 0 ν 3 | − |m 0 ν 2 | = r A 2 , r C 3 = r C 2 cos 2 θν + r C 1 sin 2 θν .(36)
• Case D. If m 0 ν 2 > 0 and m 0 ν (3,1) < 0,
r D 1 = |m 0 ν 3 | + m 0 |m 0 ν 3 | − m 0 = r A 1 , r D 2 = |m 0 ν 3 | − |m 0 ν 2 | |m 0 ν 3 | + |m 0 ν 2 | = 1 r A 2 , r D 3 = r D 2 cos 2 θν + r D 1 sin 2 θν .(37)
Notice that
|m 0 ν 3 | − m 0 ≈ 2m 0 R 2 , |m 0 ν 3 | + m 0 ≈ 2m 0 (1 + R 2 ) , |m 0 ν 3 | − |m 0 ν 2 | ≈ 2m 0 R 2 (1 − R 3 ) , |m 0 ν 3 | + |m 0 ν 2 | ≈ 2m 0 [1 + R 2 + R 4 ] ,(38)
with R 2 ≡ ∆m 2 31 /4m 2 0 , R 3 = ∆m 2 21 /∆m 2 31 and R 4 = ∆m 2
21 /4m 2 0 , where R 4 < R 3 R 2 . Thus, r A 1 ≈ (1 + R 2 )/R 2 and r A 2 ≈ r A 1 (1 + R 3 ).
To fix ideas on the order of magnitude of each defined quantity, we use the data for the inverted hierarchy and their respective central values. So that, R 2 ∼ 6 × 10 −2 , R 3 ∼ 3 × 10 −2 , R 4 ∼ 2 × 10 −3 and r A 1 ∼ 17 with m 0 = 0.1 eV . Indeed, R 2 and R 4 might be fairly small, and therefore r A 1 so large since that m 0 0.1 eV . Therefore, in the Cases A,
r A 2 − r A 1 ≈ r A 1 R 3 and r A 3 ≈ r A 1 then sin 2 θ 13 ≈ 2 9 | | 2 r A 1 R 3 2 , sin 2 θ 23 ≈ 1 2 1 − r A 1 2 1 − sin 2 θ 13 .(39)
In the Case B,
r B 2 − r B 1 ≈ −r A 1 /R 3 and r B 3 ≈ 1/r A 1 so that sin 2 θ 13 ≈ 2 9 | | 2 R 3 r A 1 2 , sin 2 θ 23 ≈ 1 2 1 − r A 1 2 1 − sin 2 θ 13 .(40)
Thus, in the former case if the reactor angle is fixed to its central value (sin 2 θ 13 ≈ 0.0229), with the above values for r A 1 and R 3 , we obtain that | | ≈ 0.5 which means a strong breaking of the µ−τ symmetry. As result, the atmospheric angle comes out begin too large. Analogously, for the second case one gets that | | ≈ 10 2 if the reactor angle is fixed to its central value. As consequence, the atmospheric angle is also quite large. Therefore, these two cases are excluded, the only feasible cases are the last ones.
For the Case C, from Eq.(36), we have r C 2 − r C 1 ≈ r A 1 (1 + R 3 ) and r C 3 ≈ r A 1 cos 2 θ ν , so that
sin 2 θ 13 ≈ 2 9 | | 2 r A 1 (1 + R 3 ) 2 , sin 2 θ 23 ≈ 1 2 1 − 2 3 r A 1 2 1 − sin 2 θ 13 .(41)
For the Case D, from Eq. (37), we obtain r D 2 − r D 1 ≈ −r A 1 and r D 3 ≈ r A 1 sin 2 θ ν . Then,
sin 2 θ 13 ≈ 2 9 | | 2 r A 1 2 , sin 2 θ 23 ≈ 1 2 1 − 1 3 r A 1 2 1 − sin 2 θ 13 .(42)
Roughly speaking, as in the inverted case, the reactor angle has approximately the same behavior for both cases but the atmospheric angle comes out being different. In here, on the other hand, notice that r 3 > 0 for both cases then if α = 0, the atmospheric angle would be smaller than 45 • which is far away from the experimental data, as can be verified from Eq.(41) and Eq. (42). In order to increase this value, it is needed that α = π. Additionally, because of r A 1 ≫ 1, the value of | | should be of the order of 10 −2 in order to not enhance too much the atmospheric angle, of course, we must be careful to not spoil the reactor angle or vice versa. Now, in Case C and D, if the reactor angle is fixed to its central value (sin 2 θ 13 ≈ 0.0229) then it is required that | | ∼ 2 × 10 −2 , so that one obtains sin 2 θ 23 ≈ 0.74 and sin 2 θ 23 ≈ 0.63, respectively. As can be seen, the favored case is the latter due to the r D 3 contribution, in the atmospheric angle, is minor than r C 3 such that the atmospheric angle is softly being deviated from 45 • . Now, an interesting fact is the following: if m 0 is increased to the allowed value, then r A 1 becomes quite large and therefore, a tiny | | value is needed to not deviate so much from 45 • the atmospheric angle and at the same time, to get an allowed region for the reactor angle. In this hierarchy, the µ − τ symmetry is being broken softly.
We will now explore the complete parameter space for both cases. The exact formulas for the mixing angles have been used with the respective extreme Majorana phases for each case, apart from the allowed values for ∆m 2 21 , ∆m 2 13 and θ ν the solar angle for the inverted ordering as a good approximation. Therefore, in figure 3, the atmospheric versus the reactor angle is show up to 3 σ. This panels allow to compare the two cases and these support our analytic result, in the Case D, both angles of interest are accommodated very well. In the figure 4, as can be seen, the parameter space is large where the atmospheric angle, and therefore the reactor one, is accommodated in good agreement with the experimental data. At the end of the day, the degenerate ordering is favored instead of the inverted case.
Conclusions
We have extended the scalar sector of the LRSM in order to get masses and mixings for fermions. In the lepton sector, neutrino masses and mixings have been studied in the limit of a slightly broken µ−τ symmetry, so that the reactor and atmospheric angles depend strongly on the free parameter, that characterizes the µ − τ symmetry breaking, and the neutrino masses. Due to this last fact, the mixing angles are sensitive to the extreme Majorana The main results are the following: (a) the model predicts a tiny value for the reactor angle in the normal hierarchy and this result holds for whatever extreme Majorana phases. Then, the normal ordering is completely ruled out for | | ≤ 0.3; (b) in the inverted hierarchy there is one combination in the extreme Majorana phases where the reactor and atmospheric angles are compatible up to 2−3 σ within the allowed region for the latter angle. This scenario is fairly constrained since the parameter space is so tight; (c) the degenerate ordering is the most viable scenario to accommodate simultaneously the reactor and atmospheric angles. In this case, there is one combination in the extreme Majorana phases where both angles are consistent with the current limits imposed by the experimental data for sin 2 θ 23 and sin 2 θ 13 . At the same time, a set of values for and the lightest neutrino mass was found such that the µ − τ symmetry is broken softly. Remarkably, the viable cases predict that θ 23 > 45 • .
For the moment, the quark sector has been left aside for a future work but we have pointed out that the mass matrices possess textures that might fit the CKM matrix. Although the model is quite elaborate, it is fairly predictive and testable by the future results that the Nova and KamLAND-Zen collaborations will provide.
e with m e = S T 23 M e S 23 . After factorizing the phases, we have m e = P eme P e where me = diag.(me, mµ, mτ ), Pe = diag.(e iηe , e iηµ , e iητ )
Figure 1 :
1sin 2 θ 23 versus sin 2 θ 13 . The left and right panels stand for the Case C and Case D, respectively. The dotdashed, dashed and thick lines stand for 1 σ, 2 σ and 3 σ, respectively for each case.
Figure 2 :
2Case D: Allowed region for sin 2 θ 23 . The dotdashed, dashed and thick lines stand for 1 σ, 2 σ and 3 σ .
Figure 3 :
3sin 2 θ 23 versus sin 2 θ 13 . The left and right panels stand for the Case C and Case D, respectively. The dotdashed, dashed and tick lines stand for 1 σ, 2 σ and 3 σ, respectively for each case.
Figure 4 :
4Case D: Allowed region for sin 2 θ 23 . The dotdashed, dashed and thick lines stand for 1 σ, 2 σ and 3 σ phases which may increase or decrease their respective values. Therefore, we have made an analytic study on the role that the extreme Majorana phases might have in each hierarchy. Additionally, the free parameter and the lightest neutrino mass have been constrained.
|1 + | 2 /2, which is not compatible with the observations. Nonetheless, this strict ordering allows us to infer that the | |e iα parameter magnitude has to be small in order to deviate sufficiently the atmospheric angle from 45 • , and the same time, this has to be enough large to enhance the reactor one. In here, the α associated phase determines if we are above or below of 45 • . Along this line, Nova experiment has discarded the lower octant[2]. On the contrary, if the constraint, on the lightest neutrino mass, is relaxed, the reactor angle comes out being non zero and the atmospheric one has an extra contribution, r 3 , which can enlarge or reduce the | | magnitude since this may be greater or minor than 1. So that, the factor | r 3 | might deviate drastically the atmospheric angle beyond of 45 • .Notice that, roughly speaking, the reactor angle turns out being equal for the Cases A and B, and also, for C and D. The key difference among them comes from the atmospheric
AcknowledgementsWe would like to thank Myriam Mondragón and Abdel Pérez-Lorenzana for their useful comments and discussion on the manuscript. This work was partially supported by a PAPIIT grant IN111115. The author thanks Red de Altas Energías-CONACYT for the financial support.
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| {'fraction_non_alphanumeric': 0.07240384128537915, 'fraction_numerical': 0.05697794130056877, 'mean_word_length': 3.3880619121676663, 'pattern_counts': {'":': 0, '<': 7, '<?xml version=': 0, '>': 8, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, '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 propose a non-minimal left-right symmetric model (LRSM) with Parity Symmetry where the fermion mixings arise as result of imposing an S 3 ⊗ Z 2 flavor symmetry, and an extra Z e 2 symmetry is considered to suppress some Yukawa couplings in the lepton sector. As a consequence, the effective neutrino mass matrix possesses approximately the µ − τ symmetry. The breaking of the µ − τ symmetry induces sizable non zero θ 13 , and the deviation of θ 23 from 45 • is strongly controlled by an free parameter and the complex neutrino masses. Then, an analytic study on the extreme Majorana phases is done since these turn out to be relevant to enhance or suppress the reactor and atmospheric angle. So that we have constrained the parameter space for the parameter and the lightest neutrino mass that accommodate the mixing angles. The highlighted results are: a) the normal hierarchy is ruled out since the reactor angle comes out being tiny, for any values of the Majorana phases; b) for the inverted hierarchy there is one combination in the extreme phases where the values of the reactor and atmospheric angles are compatible up to 2, 3 σ of C. L., but the parameter space is tight; c) the model favors the degenerate ordering for one combination in the extreme Majorana phases. In this case, the reactor and atmospheric angle are compatible with the experimental data for a large set of values of the free parameters. Therefore, this model may be testable by the future result that the Nova and KamLAND-Zen collaborations will provide. *', 'arxivid': '1701.01747', 'author': ['Juan Carlos Gómez-Izquierdo [email protected] \nTecnologico de Monterrey\nCampus Estado de MexicoAtizapan de Zaragoza\n\nEstado de Mexico\nApartado Postal\n52926Mexico\n\nInstituto de Ciencias Nucleares\nUniversidad Nacional Autónoma de México\n3000MéxicoD.FMéxico\n\nInstituto de Física\nUniversidad Nacional Autónoma de México\n01000MéxicoD.FMéxico\n'], 'authoraffiliation': ['Tecnologico de Monterrey\nCampus Estado de MexicoAtizapan de Zaragoza', 'Estado de Mexico\nApartado Postal\n52926Mexico', 'Instituto de Ciencias Nucleares\nUniversidad Nacional Autónoma de México\n3000MéxicoD.FMéxico', 'Instituto de Física\nUniversidad Nacional Autónoma de México\n01000MéxicoD.FMéxico'], 'corpusid': 119352650, 'doi': '10.1140/epjc/s10052-017-5094-0', 'github_urls': [], 'n_tokens_mistral': 22940, 'n_tokens_neox': 18828, 'n_words': 11367, 'pdfsha': '8d2da6ab52652cc8b8e239b2b4000f40d920eff7', 'pdfurls': ['https://arxiv.org/pdf/1701.01747v3.pdf'], 'title': ['Non-Minimal Flavored S 3 ⊗ Z 2 Left-Right Symmetric Model', 'Non-Minimal Flavored S 3 ⊗ Z 2 Left-Right Symmetric Model'], 'venue': []} |
arxiv |
SMALL TIME SHARP BOUNDS FOR KERNELS OF CONVOLUTION SEMIGROUPS
4 Mar 2014
Kamil Kaleta
Paweł Sztonyk
SMALL TIME SHARP BOUNDS FOR KERNELS OF CONVOLUTION SEMIGROUPS
4 Mar 2014Key-words: Lévy measureLévy processtempered processconvolution semigroupconvolution of measurestransition densityheat kernelsharp estimateexponential decay 2010 MS Classification: Primary 60G5160E07; Secondary 60J3547D0360J45
We study small time bounds for transition densities of convolution semigroups corresponding to purely jump Lévy processes in R d , d ≥ 1, including those with jumping kernels exponentially and subexponentially localized at infinity. For a large class of Lévy measures, non-necessarily symmetric and absolutely continuous with respect to the underlying measure, we find the optimal in time and space upper bound, for the corresponding transition kernels at infinity. In case of Lévy measures that are symmetric and absolutely continuous with respect to the Lebesgue measure, with densities g such that g(x) ≍ f (|x|) for nonincreasing profile functions f , we also prove the full characterization of the sharp two-sided transition densities bounds of the formThis is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to some interesting and surprising dichotomy of the decay properties at infinity for transition kernels of purely jump Lévy processes. All results are obtained by analytic methods, with no use of probabilistic arguments.
Introduction and statement of results
We study a convolution semigroup of probability measures {P t , t ≥ 0} on R d , d ∈ {1, 2, ...}, determined by their Fourier transforms F(P t )(ξ) = R d e iξ·y P t (dy) = exp(−tΦ(ξ)), t > 0, with the Lévy-Khintchine exponent of the form
Φ(ξ) = 1 − e iξ·y + iξ · y1 B(0,1) (y) ν(dy) − iξ · b, ξ ∈ R d ,
where ν is an infinite Lévy measure on R d , i.e., R d (1 ∧ |y| 2 ) ν(dy) < ∞ and ν(R d ) = ∞, and b ∈ R d is a drift term [17]. It is well known that there exists a Lévy process {X t , t ≥ 0} in R d with transition functions given by {P t , t ≥ 0} [32]. The densities of measures P t with respect to the Lebesgue measure are denoted by p t , whenever they exist. For some sufficient and necessary conditions on the existence of kernels p t we refer the reader to [24]. The problem of estimates of transition densities for jump Lévy processes has been intensively studied for many decades, mostly for stable processes [1,30,15,16,12,13,11,39,5]. The general method of estimating of the kernels of Lévy semigroups is based on their convolutional 0 K. Kaleta structure and construction. Recent papers [35,36,23,25,21,22] contain the estimates for more general classes of Lévy processes, including tempered processes with intensities of jumps lighter than polynomial. The paper [3] focuses on the estimates of densities for isotropic unimodal Lévy processes with Lévy-Khintchine exponents having the weak local scaling at infinity, while the papers [27,21] discusses the processes with higher intensity of small jumps, remarkably different than stable one. In [8,9,20] the authors investigate the case of more general, non-necessarily space homogeneous, symmetric jump Markov processes with jump intensities dominated by those of isotropic stable processes. Estimates of kernels for processes which are solutions of SDE driven by Lévy processes were obtained in [29]. For estimates of derivatives of Lévy densities we refer the reader to [34,2,33,21,26,22]. In [18] the authors gave a very interesting geometric interpretation of the transition densities for symmetric Lévy processes.
In the present paper, we focus on some special type of the small time bounds of the densities p t . Before we state our main results, we first need to introduce some necessary auxiliary notation and set the framework for our study. Denote Ψ(r) = sup |ξ|≤r Re (Φ(ξ)) , r > 0.
We note that Ψ is continuous and non-decreasing and sup r>0 Ψ(r) = ∞, since ν(R d ) = ∞. Let Ψ −1 (s) = sup{r > 0 : Ψ(r) = s} for s ∈ (0, ∞) so that Ψ(Ψ −1 (s)) = s for s ∈ (0, ∞) and Ψ −1 (Ψ(s)) ≥ s for s > 0. To shorten the notation below, we set A substantial part of our work concerns a large class of Lévy measures that are non-necessarily symmetric and absolutely continuous with respect to the Lebesgue measure. However, the sharpness of our results is most evident if the Lévy measure ν has the density g(x) = g(−x) such that g(x) ≍ f (|x|), x ∈ R d , for some nonincreasing function f : [0, ∞) → (0, ∞].
With this framework, our investigations may be seen as the study of the following type of the small time estimates of the densities p t (x) (for simplicity we assume here that b = 0). There are constants c 1 , C 1 , C 2 ∈ (0, 1], c 2 , C 3 , C 4 ≥ 1, θ > 0 and t 0 > 0 such that
C 1 [h(t)] −d ≤ p t (x) ≤ C 3 [h(t)] −d , t ∈ (0, t 0 ], |x| ≤ θh(t),(2)
and
C 2 t f (c 2 |x|) ≤ p t (x) ≤ C 4 t f (c 1 |x|), t ∈ (0, t 0 ], θh(t) ≤ |x|,(3)
where f is the profile of the density g of ν. We say that this two-sided bound is sharp when c 1 = c 2 = 1. Clearly, this is irrelevant when the profile function has a doubling property, e.g. when f (r) = r −d−β . However, for tempered processes, if the tail of the Lévy measure decays at infinity faster than polynomial (e.g. f (r) ≍ e − √ r or f (r) ≍ r −d−β e −r as r → ∞), then this is of great importance. In the above estimates, we do not require the optimality of constants C 1 , C 2 , C 3 and C 4 .
The small time bounds as in (2)-(3) are known to hold for a wide class of Lévy and more general Markov processes. Most of available results imply the estimates of the transition densities in the sharp form when x is small, and not sharp, with c 1 < 1 < c 2 , for large x (see e.g. [3,8,9,35,36,25,21,22]). Beside some degenerate examples, in general, the Lévy measures satisfy a kind of the doubling condition around zero. This property is inherited by the profile function and, therefore, we often have f (c|x|) ≍ f (|x|) for any fixed c > 0 and for all small x. In many cases, the small time bounds of the densities p t (x) for small x can be derived from the properties of the corresponding Lévy-Khintchine exponent. Indeed, very often, by the Fourier transform, the asymptotics of Φ at infinity directly translates into the asymptotics of p t and ν at zero (see e.g. [3]).
For large x this picture is usually dramatically different. As we will see below, in this case the asymptotic behaviour of p t (x) strongly depends on the subtle convolutional properties of the corresponding Lévy measures. If the tail of the Lévy measure is lighter than polynomial at infinity, then we can expect that lim r→∞ f (cr)/f (r) = ∞, for all c ∈ (0, 1). In this case, the Lévy-Khintchine exponent vanishes at zero quadratically and the sharp bounds of p t (x) for large x and small t can not be derived from it. Furthermore, if we have the upper bound in (3) with f (c 1 |x|) for some c 1 ∈ (0, 1), then f (c 1 |x|) can not be directly replaced by cf (|x|) for any constant c. Of course, this does not mean that in this case the bound with the best possible rate f (|x|) cannot hold. Unfortunately, in most cases it is too difficult to settle whether the worse rate in (3) is only a consequence of the flaw of the method or assumptions, or whether, perhaps, the bound of the form (3) with the exact rate f (|x|) does not hold for large x. It is known that the Lévy measure ν(dx) = g(x) dx is a vague limit of measures P t (dx)/t = (p t (x)/t) dx as t → 0 + outside the origin, which may cause the false intuition that for small t the both functions p t (x) and tg(x) should share exactly the same asymptotic properties. As we will show below, although sometimes sharp bounds in (3) seem to be possible or even evident, they surprisingly do not hold in general. Therefore it is quite reasonable to ask when exactly these bounds are satisfied in the sharpest form.
To the best of our knowledge, there is no a comprehensive argument or result, working in a satisfactory generality including tempered processes with jump intensities exponentially localized at infinity, which ultimately explains and settles when exactly the sharp small time bounds for densities of purely jump Lévy processes are satisfied. The following Theorem 1 definitively resolves this problem for convolution semigroups built on symmetric Lévy measures that are absolutely continuous with respect to the Lebesgue measure with densities comparable to nonincreasing profiles. It gives the full characterization of sharp bounds (2)-(3) with the exact rate f (|x|) for all x ∈ R d . For later use we denote g r (·) = g(·)1 B(0,r) c (·), r > 0. Theorem 1. Let ν(dx) = g(x)dx be a Lévy measure such that ν(R d ) = ∞, g(y) = g(−y) and there is a nonincreasing function f :
[0, ∞) → (0, ∞] such that g(x) ≍ f (|x|), x ∈ R d . Moreover assume that b ∈ R d .
Then the following two conditions (1.1) and (1.2) are equivalent. (1.1) There exist r 0 > 0 and constants L 1 , L 2 > 0 such that the following two estimates
(a) g r 0 * g r 0 (x) ≤ L 1 g(x), |x| ≥ 2r 0 , (b) Ψ(1/|x|) ≤ L 2 |x| d g(x), |x| ≤ 2r 0 , hold. (1.
2) There exist t 0 , θ > 0 and the constants C 1 − C 4 such that for every t ∈ (0, t 0 ] the transition densities p t exist and satisfy
C 1 [h(t)] −d ≤ p t (x + tb) ≤ C 3 [h(t)] −d , t ∈ (0, t 0 ], |x| ≤ θh(t),
and
C 2 t g(x) ≤ p t (x + tb) ≤ C 4 t g(x), t ∈ (0, t 0 ], |x| ≥ θh(t).
Both conditions (a) and (b) in (1.1) are rather local in the sense that they refer to the different ranges of x. The proof of Theorem 1, which consists of the two parts, for small and for large
x separately, also reflects this property. In particular, the next result states that the condition (1.1) (b) in fact characterizes the bounds (1.2) for small x. It is worth to point out that under our assumptions the estimate opposite to (1.1)(b) always holds true (as a consequence of the first bound in (4)).
Theorem 2. Let ν(dx) = g(x)dx be a Lévy measure such that ν(R d ) = ∞, g(y) = g(−y) and there is a nonincreasing function f :
[0, ∞) → [0, ∞] such that g(x) ≍ f (|x|), x ∈ R d . Moreover assume that b ∈ R d .
Then the following two conditions (2.1) and (2.2) are equivalent. (2.1) There exist r 0 > 0 and a constant L 2 > 0 such that
Ψ(1/|x|) ≤ L 2 |x| d g(x), |x| ≤ 2r 0 .
(2.2) There exist t 0 , θ, R > 0 such that θh(t 0 ) ≤ R and the constants C 1 − C 4 such that for every t ∈ (0, t 0 ] the transition densities p t exist and satisfy
C 1 [h(t)] −d ≤ p t (x + tb) ≤ C 3 [h(t)] −d , t ∈ (0, t 0 ], |x| ≤ θh(t),
and
C 2 t g(x) ≤ p t (x + tb) ≤ C 4 t g(x), t ∈ (0, t 0 ], θh(t) ≤ |x| ≤ R.
It was recently proved in [3] for a class of isotropic unimodal Lévy processes (i.e. g(x) and p t (x) are assumed to be strictly radial and unimodal functions) that the estimates p t (x) ≍ [h(t)] −d ∧ tΨ(1/|x|)|x| −d for small t and small x are equivalent to the property that the corresponding Lévy-Khintchine exponent has Matuszewska indices strictly between 0 and 2 at infinity. As shown there, in this class of processes, the latter property yields (2.1). This reformulation of the condition (2.1) in terms of the Lévy-Khintchine exponent easily extends to our settings (see Lemma 5). However, in general, the functions g(x) and p t (x) corresponding to convolution semigroups investigated in the present paper are not isotropic and unimodal. Therefore, the proof of Theorem 2 requires more general methods than those known and available for isotropic unimodal case. Due to possible applications, it is worth to point out that our both conditions (2.1) and (2.2) in fact imply the two-sided bound in the minimum form p t (x + tb) ≍ [h(t)] −d ∧ tg(x) (see further discussion in Proposition 1 and Remark 1).
The characterization of (1.2) (in fact, the second bound in (1.2)) for large x in terms of the convolution condition (1.1) (a) is given in Theorem 3 below. This result can be seen as the key and main ingredient of Theorem 1. It was obtained independently of Theorem 2 under the following regularity condition (E) on Φ which is essentially more general than (1.1) (b).
(E) There exist a constant L 0 > 0 and t p > 0 such that
R d e −t Re(Φ(ξ)) |ξ| dξ ≤ L 0 [h(t)] −d−1 , t ∈ (0, t p ].
The condition (E) gives not only the existence of densities p t ∈ C 1 b (R d ) for t ∈ (0, t p ], but it also provides a necessary regularity of the small jump part of the process (see Preliminaries). Here we investigate the small time properties of the densities p t and it is intuitively clear that the study of the second bound in (1.2) for large x also should require some regularity of the Lévy-Khintchine exponent Φ for large arguments. One can verify that if there is α > 0, r 0 > 0 and a constant C ∈ (0, 1] such that Ψ(λr) ≥ Cλ α Ψ(r), λ ≥ 1, r > r 0 , then (E) holds for all t ∈ (0, 1/Ψ(r 0 )) (see Lemma 5). On the other hand, it is clear that (E) excludes the symbols that varies slowly (e.g. logarythmically) at infinity. In this case the integral on the left hand side is not finite for small t.
Theorem 3 below gives the characterization of the sharp small time bounds of densities for big spatial arguments in terms of the decay of convolution of the Lévy measures at infinity. Theorem 3. Let ν(dx) = g(x)dx be a Lévy measure such that ν(R d ) = ∞, g(y) = g(−y) and there is a nonincreasing function f :
[0, ∞) → (0, ∞] such that g(x) ≍ f (|x|), x ∈ R d .
Moreover assume that b ∈ R d and that (E) holds with some t p > 0.
Then the following two conditions (3.1) and (3.2) are equivalent. (3.1) There exist r 0 > 0 and a constant L 1 > 0 such that
g r 0 * g r 0 (x) ≤ L 1 g(x), |x| ≥ 2r 0 .
(3.2) There exists t 0 ∈ (0, t p ], R > 0 and the constants C 2 , C 4 such that we have
C 2 t g(x) ≤ p t (x + tb) ≤ C 4 t g(x), t ∈ (0, t 0 ], |x| ≥ R.
In particular, if (3.1) is true for some r 0 , then (3.2) holds with R = 4r 0 and t 0 := t p ∧ 1 Ψ(1/r 0 ) . If (3.2) is true for some t 0 and R, then (3.1) holds for r 0 = R/2.
Note that Theorems 1 and 2 do not require imposing the assumption (E) a priori. Indeed, any of the equivalent conditions (2.1) and (2.2) (respectively (1.1) (b) and (1.2) for small x) implies that the Lévy-Khintchine exponent Φ satisfies (E) (Lemma 5). In fact, the condition (E) is more general and covers an essentially larger class of semigroups than (2.1) (cf. Examples 1 and 2(2)). In particular, the statement of Theorem 3 and the argument in its proof are completely independent of Theorem 2 and bounds (2.1)-(2.2).
Theorem 3 determines when exactly the sharp two-sided bounds as in (3) (with c 1 = c 2 = 1) are satisfied for large x. In particular, it shows that such bounds hold for a large class of symmetric tempered Lévy processes for which they were not known before. Here the most interesting examples include processes with jump intensities exponentially and suboexponentially localized at infinity, even if the intensities of small jumps are remarkably different from stable one, whenever the regularity condition (E) is satisfied (see Corollary 6 and Example 2). Such tempered processes are important from the mathematical physics point of view (see e.g. [7]) and, as we will see in the sequel, they seem to be quite interesting in the present context. For instance, if we consider the class of Lévy processes with Lévy measures ν(dx) = g(x)dx such that g(x) = g(−x) ≍ |x| −δ e −m|x| , δ ≥ 0, m > 0, for large x (this covers important families of tempered Lévy processes such as relativistic stable or Lamperti ones), then Proposition 2 states that the convolution condition (3.1) holds true only exactly in two cases, when β ∈ (0, 1) and δ ≥ 0 or when β = 1 and δ > (d + 1)/2. Theorem 3 (see also Corollary 6) thus immediately settles that the two sided sharp bounds of the form (3.2) are satisfied for these two ranges of parameters only. In particular, they cannot hold when β > 1 or when β = 1 and δ ∈ (0, (d + 1)/2). This a little bit surprising dichotomy property was not known before (see further discussion in Example 5).
The study of the small time bounds in Theorem 2 and the lower bound in Theorem 3 is based on an application of the results obtained recently in [21] and on some new tricky ideas. However, the most critical part of the paper is the proof of the upper bound in Theorem 3. In fact, the primary motivation of our investigations was to understand and explain when exactly the upper bound as in (3.2) can be expected to hold and how it can be described by the detail and direct properties of the corresponding Lévy measure. The answer we give is that it is enough to know how fast the tail (or rather profile) of a single convolution of the Levy measures (restricted to the complement of some neighborhood of the origin) decays at infinity. This fits very well the convolutional structure of the semigroup {P t : t ≥ 0}, but it is a little unexpected that under the condition (3.1) the decay properties of all n-th convolutions of Lévy measures appearing in the construction is decided exactly by the decay of the first one (we briefly recall the construction in Preliminaries). Note that the condition (3.1) has been recently discovered in [19] in a completely different context as a powerful tool to study the estimates of the eigenfunctions and some ultracontractivity properties of the Feynman-Kac semigroups for Lévy processes. Moreover, similar convolution conditions, especially for the tails of measures, have been widely studied on the real line and the halfline in the context of various types of subexponentiality. It is known for many years that these properties play an important role in the study the relation between one-dimensional infinitely divisible distributions and their Lévy measures (see e.g. [38,40] and references therein). We would also like to mention that we obtained recently in [20] the upper bound for densities of Feller semigroups with jump kernels absolutely continuous with respect to the Lebesgue measure with densities that are dominated by some radial functions satisfying the condition as in (3.1). However, the argument in this paper requires some additional smoothness of the majorizing functions and the additional regularity of the intensity of small jumps, which is in fact assumed to be of the stable type.
The second important question we address in the present paper is about the generality in which such type of convolution condition on Lévy measure implies the sharp small time upper bound of the corresponding density p t (x) for large x similar to (3.2). We show that it holds true in much more general case, extending far beyond the settings of Theorems 1 and 3. Below we proceed in the general framework, under which we also worked in our recent paper [21]. We consider a large class of Lévy measures satisfying the following localization (domination) condition from above.
ν(A) ≤ L 3 f (dist(A, 0))(diam(A)) γ , for every A ∈ B(R d ) with dist(A, 0) > 0.
Here diam(A) is the diameter and dist(A, 0) is the distance to 0 of the set A ⊂ R d and B(R d ) denotes the Borel sets in R d . For comparison, in [22] the author consider a different type of a localization condition, which is based on the estimate of the tail of the Lévy measure by the tail of some multidimensional (subexponential) distribution. Note that the condition (D) covers a large class of symmetric and asymmetric Lévy measures which are not absolutely continuous with respect to Lévy measures, including some product and discrete Lévy measures and those with tails very fast decaying at infinity. Under the following convolution condition (C), naturally generalizing (3.1), we obtain the sharpest possible upper bound of p t (x) for small t and large x which can be given by using the majorant f satisfying the localization condition (D).
(C) There exist the constants L 1 , L 4 > 0 and r 0 > 0 such that for every |x| ≥ 2r 0 and r ∈ (0, r 0 ]
|x−y|>r 0 , |y|>r f (|y − x|) ν(dy) ≤ L 1 Ψ(1/r) f (|x|) and f (r) ≤ L 4 Ψ(1/r)r −γ ,
with f and γ given by the domination condition (D).
Theorem 4. Let ν be a Lévy measure such that ν(R d ) = ∞ and let the assumptions (E), (D) and (C) be satisfied for some t p > 0, the function f , the parameter γ and some r 0 > 0. Then there is a constant C 5 > 0 and R > 0 such that
p t (x + tb h(t) ) ≤ C 5 t [h(t)] γ−d f (|x|), |x| > 4r 0 , t ∈ (0, t 0 ],
where t 0 := t p ∧ 1 Ψ(1/r 0 ) , and b r is given by (1). The proof of Theorem 4 is critical for the whole paper. Its key argument are some sharp estimates of the n-th convolutions of restricted Lévy measures (Lemma 2) which are based on our new convolution condition (C) and were not known before. One can check that under the assumptions of Theorem 3 both conditions (D) and (C) hold with γ = d and f being the profile of the density g. In this case, the convolution condition (C) directly reduces to assumption (3.1) (see Lemma 3). Theorem 3 is thus the direct corollary from Theorem 4. Note that in light of the general property in (4) below, the second inequality in (C) is only a technical assumption saying that the profile f is not too rough around zero.
We close the introduction by a brief discussion of the sharpness of our new convolution assumption in (C) compared to the condition (P) introduced recently in [21] together with (D) as a key assumption to study the upper bound for transition densities.
(P) There exists a constant M > 0 such that
|y|>r f (s ∨ |y| − |y|/2) ν(dy) ≤ Mf (s)Ψ(1/r), s > 0, r > 0,
with f given by (D). Note that the structure of the condition (P) is much more isotropic than that of (C) and, therefore, it is often more convenient to check. Under (D), the condition (P) allowed us to get the result (see [21,Theorem 1]) which, in particular, implies the upper bound as in Theorem 4, but with the rate f (|x|/4) instead of f (|x|) and with some additional exponentially-logarithmic correction term. At the stage of the paper [21], it was completely unclear whether the sharpest possible upper bound with f (|x|) cannot be obtained under the condition (P). In Proposition 3, although both conditions have completely different structure, we prove that (C) always implies the inequality in (P) for large s and small r, but the converse implications is not true. This in fact means that the condition (P) is too weak to guarantee the optimal rate in the estimate of p t (x) for small t and big x in general. More precisely, it holds for a larger class than the convolution condition (C) and give some bounds for densities, but it cannot be used to derive the sharp bound as in Theorem 4 with the exact rate f (|x|) imposed by the localization condition (D). This is illustrated by Example 5.
The structure of the paper is as follows. In Preliminaries we collect all facts needed in the sequel and briefly recall the construction of the semigroup {P t : t ≥ 0}. Based on that, we precisely explain what is the main object of our study in this paper. In Section 3 we investigate the consequences of the condition (C) and estimate the convolutions of Lévy measures. In Sections 4 we prove Theorems 4 and 3 involving the bounds for p t (x) for large x. Section 5 concerns small time bounds for small x. It includes the proof of Theorem 2, the discussion of further implications and formal proof of Theorem 1. In Section 6 we illustrate our results by various examples, including two less regular cases (Examples 3 and 4), and discuss the convolution condition with respect to some various typical profiles of Lévy densities (Proposition 2 and Corollary 6). In Subsection 6.3, we also illustrate the sharpness of our convolution condition (C) compared with (P).
Preliminaries
We use c, C, L (with subscripts) and M to denote finite positive constants which may depend only on ν, b, and the dimension d. Any additional dependence is explicitly indicated by writing, e.g., c = c(n). We write f (
x) ≍ g(x) whenever there is a constant c such that c −1 f (x) ≤ g(x) ≤ cf (x).
We will need the following preparation. As usual we divide the Lévy measure in the two parts. For r > 0 we denotẽ
ν r (dy) = 1 B(0,r) (y)ν(dy) andν r (dy) = 1 B(0,r) c (y) ν(dy).
In terms of the corresponding Lévy process,ν r is related to the jumps which are close to the origin, whileν r represents the large jumps. Note that there exist the constants L 5 , L 6 such that for every r > 0
(4) |ν r | ≤ L 5 Ψ(1/r) and Ψ(2r) ≤ L 6 Ψ(r),
which follows from [21, Proposition 1] or [14]. We now briefly recall the construction of the semigroup {P t , t ≥ 0}. For the restricted Lévy measures we consider the two semigroups of measures
{P r t , t ≥ 0} and {P r t , t ≥ 0} such that F(P r t )(ξ) = exp t e iξ·y − 1 − iξ · y ν r (dy) , ξ ∈ R d , and F(P r t )(ξ) = exp t (e iξ·y − 1)ν r (dy) , ξ ∈ R d ,
respectively. We have
|F(P r t )(ξ)| = exp −t |y|<r (1 − cos(y · ξ)) ν(dy) = exp −t Re(Φ(ξ)) − |y|≥r (1 − cos(y · ξ)) ν(dy) ≤ exp(−t Re(Φ(ξ))) exp(2tν(B(0, r) c )), ξ ∈ R d ,
and, therefore, by (E), for every r > 0 and t ∈ (0, t p ] the measuresP r t are absolutely continuous with respect to the Lebesgue measure with densitiesp r
t ∈ C 1 b (R d ). We have P t =P r t * P r t * δ tbr , and p t =p r t * P r t * δ tbr , t > 0, where b r is defined by (1), and P r t = exp(t(ν r − |ν r |δ 0 )) = ∞ n=0 t n (ν r − |ν r |δ 0 )) n * n! (5) = e −t|νr| ∞ n=0 t nνn * r n! , t ≥ 0 .
As usual, below we will useP r t ,p r t andP r t with r = h(t) and for simplification we will writẽ
P t =P h(t) t ,p t =p h(t) t andP t =P h(t) t
. As proven in [21,Lemma 8], if ν(R d ) = ∞ and (E) holds with t p > 0 then there exist constants C 6 , C 7 and C 8 such that
p t (x) ≤ C 6 [h(t)] −d exp −C 7 |x| h(t) log 1 + C 8 |x| h(t) , t ∈ (0, t p ], x ∈ R d .
Therefore, we always have
p t (x + tb h(t) ) = (p t * P t )(x) ≤ C 6 [h(t)] −d R d G((y − x)/h(t))P t (dy), t ∈ (0, t p ], x ∈ R d ,(6)
with
G(s) := e −C 7 s log(1+C 8 s) , s ≥ 0.
The main objective of the present paper is to find and study the precise estimates of convolutions ν n * r and the measureP t , and, in consequence, also the optimal upper bound for the integral on the right hand side of (6) when x is large. This will be achieved in the next two sections.
Convolutions of Lévy measures
In this section we prove the sharp upper bounds for n-th convolutions of Lévy measures, which are basic for our further investigations.
First we discuss some decay properties of nonincreasing functions f satisfying our new convolution condition (C). They will be very important below. Lemma 1. Let ν be a Lévy measure such that ν(R d ) = ∞ and let f : [0, ∞) → (0, ∞] be a nonincreasing function satisfying the first inequality in (C). Then the following holds.
(a) We have f (s − r 0 ) ≤ C 9 f (s), s ≥ 3r 0 , with
C 9 := inf c(x 0 , ε) : 0 = x 0 ∈ R d , 0 < ε < |x 0 | ≤ r 0 /2 ≥ 1,(7)
where c(x 0 , ε) :=
L 2 Ψ(1/(|x 0 | − ε)) ν(B(x 0 , ε)) ⌈r 0 /(|x 0 |−ε)⌉ . (b)
There is a constant C 10 := C 10 (r 0 ) > 0 such that
G(s/(2r 0 )) ≤ C 10 f (s), s ≥ r 0 .
Proof. To prove (a) first observe that by the assumption ν(
R d ) = ∞, there is 0 = x 0 ∈ R d and 0 < ε such that ε < |x 0 | ≤ r 0 /2 and ν(B(x 0 , ε)) > 0. Let r ε := |x 0 | − ε. For s ≥ 2r 0 let R d ∋ x s := (s/|x 0 |)x 0 .
For all s ≥ 2r 0 , by monotonicity of f and (C), we have
L 2 Ψ(1/r ε )f (s) = L 2 Ψ(1/r ε )f (|x s |) ≥ |xs−y|>r 0 , |y|>rε f (|y − x s |) ν(dy) ≥ B(x 0 ,ε) f (|y − x s |) ν(dy) ≥ f (|x s | − |x 0 | + ε)ν(B(x 0 , ε)) = ν(B(x 0 , ε))f (s − r ε ), which gives f (s − r ε ) ≤ (L 2 Ψ(1/r ε ))/ν(B(x 0 , ε))f (s)
, for all s ≥ 2r 0 . The inequality in (a) follows from this with constant C 9 given by (7), for all s ≥ 3r 0 . Clearly, C 9 ≥ 1, since f is nonincreasing.
We now show (b). Let n r 0 := inf n ∈ N : (1 + C 8 (n + 2)/2) (C 7 (n+2)/2) > C n 9 /f (2r 0 ) . First we prove that the inequality G((n + 2)r 0 /(2r 0 )) = G((n + 2)/2) ≤ f ((n + 2)r 0 ), n ≥ n r 0 , holds. If we suppose that this is not true, then there is n ≥ n r 0 such that
f ((n + 2)r 0 ) < G((n + 2)/2) = e −(C 7 (n+2)/2) log(1+C 8 (n+2)/2) = (1 + C 8 (n + 2)/2) −(C 7 (n+2)/2) < f (2r 0 )/C n 9 . However, by (a) we have 0 < f (2r 0 ) ≤ C n 9 f ((n + 2)r 0 ), for every n ∈ N.
This gives a contradiction. We thus proved that the inequality in (b) holds with the constant C 10 = C 9 for all s ≥ (n r 0 +2)r 0 , and, therefore, it also holds with C 10 :
= C 9 ∨[f ((n r 0 +2)r 0 )] −1 for all s ≥ r 0 .
The following lemma yields the sharpest upper bound for the convolutionsν * n r given by the profile function f localizing the Lévy measure from above in (D). By sharpest bound we mean here the estimate with the exact rate f (·) instead of f (c ·) for some c ∈ (0, 1). Such bounds were not known before and it is a little bit surprising or even unexpected that the single estimate from (D) extends to all convolutions via the condition (C). Weaker, not sharp, versions of Lemma 2 (b) with rates f (c dist(A, 0)) for some c ∈ (0, 1) were studied before (see e.g. [21,Lemma 9]). However, our present result is based on a completely different argument using our new convolution condition (C), which proved to be the optimal assumption to study such bounds. Lemma 2 will be a key argument in proving Theorem 4.
Lemma 2. Let ν be a Lévy measure such that ν(R d ) = ∞ and let the assumptions (D) and (C) be satisfied for some function f , the parameter γ and some r 0 > 0. Then the following hold.
(a) There is a constant C 11 = C 11 (r 0 ) such that
|x−y|>r 0 f (|y − x|)ν * n r (dy) ≤ (C 11 Ψ (1/r)) n f (|x|), |x| ≥ 3r 0 , r ∈ (0, r 0 ], n ∈ N. (8) (b) For every n ∈ N and every bounded A ∈ B(R d ) such that dist(A, 0) ≥ 3r 0 − r 0 /2 n we have (9)ν n * r (A) ≤ C n 12 [Ψ(1/r)] n−1 f (dist(A, 0)) (diam(A)) γ , r ∈ (0, r 0 ], with a constant C 12 := C 12 (r 0 , ⌈diam(A)/r 0 ⌉).
Proof. First we consider (a). We prove that (8) holds with C 11 given by (10). For n = 1 it is just the assumption (C). Assume now that (8) is true for some natural n and all x ∈ R d such that |x| ≥ 3r 0 . We will show that it holds also for n + 1. For every r ∈ (0,
r 0 ] and x ∈ R d with |x| ≥ 3r 0 we have |x−y|>r 0 f (|y − x|)ν * (n+1) r (dy) = |x−z|<3r 0 |(x−z)−y|>r 0 f (|(x − z) − y|)ν * n r (dy)ν r (dz) + |x−z|≥3r 0 |(x−z)−y|>r 0 f (|(x − z) − y|)ν * n r (dy)ν r (dz) = I 1 + I 2 .
To estimate I 1 we consider two cases. When 3r 0 ≤ |x| < 5r 0 , then simply
I 1 ≤ f (r 0 )|ν * n r ||ν r |[f (5r 0 )] −1 f (|x|) ≤ f (r 0 )L n+1 5 [f (5r 0 )] −1 f (|x|) (Ψ (1/r)) n+1 .
If now |x| ≥ 5r 0 , then by (D) and Lemma 1 (a) we get
ν r (B(x, 3r 0 )) ≤ L 3 [Ψ (1/r 0 )] −1 (3r 0 ) γ Ψ (1/r) f (|x| − 3r 0 ) ≤ L 3 C 3 9 [Ψ (1/r 0 )] −1 (3r 0 ) γ f (|x|)Ψ (1/r)
, and, consequently, in this case,
I 1 ≤ f (r 0 )|ν * n r |ν r (B(x, 3r 0 )) ≤ L 3 C 3 9 L n 5 [Ψ (1/r 0 )] −1 (6r 0 ) γ f (r 0 )f (|x|)Ψ (1/r) n+1 .
To estimate I 2 it is enough to observe that by induction hypothesis we have
|(x−z)−y|>r 0 f (|(x − z) − y|)ν * n r (dy) ≤ (C 11 Ψ (1/r)) n f (|x − z|), r ∈ (0, r 0 ],
and, in consequence, by assumption (C),
I 2 ≤ (C 11 Ψ (1/r)) n |x−z|>r 0 f (|x − z|)ν r (dz) ≤ L 1 (C 11 ) n (Ψ (1/r)) n+1 f (|x|),
for every r ∈ (0, r 0 ]. Hence, (8) holds for n + 1, every r ∈ (0, r 0 ] and every x ∈ R d such that |x| ≥ 3r 0 with constant
C 11 = L 5 f (r 0 )[f (5r 0 )] −1 + L 3 C 3 9 f (r 0 ) [Ψ (1/r 0 )] −1 (6r 0 ) γ + L 1 ∨ L 5 ,(10)
and proof of (a) is complete.
We now show (b). We prove the desired bound with constant C 12 given by (11). When n = 1 then our claim follows directly from (D). Suppose now that (9) is true for some n ∈ N, all bounded sets A ∈ B(R d ) such that dist(A, 0) ≥ 3r 0 − r 0 /2 n and every r ∈ (0, r 0 ]. We check (9) for n + 1. To shorten the notation let δ A := dist(A, 0). We consider two cases: 3r 0 − r 0 /2 n+1 < δ A < 6r 0 and δ A ≥ 6r 0 . Let first 3r 0 − r 0 /2 n+1 < δ A < 6r 0 . Let
ν (n+1) * r (A) = ν r (A − y)ν n * r (dy) = |y|<δ A −r 0 /2 n+1 + |y|≥δ A −r 0 /2 n+1 =: I 11 + I 12 .
By (D), the second part of (C) and (4), we have
I 11 ≤ L 3 |y|<δ A −r 0 /2 n+1 f (δ A − |y|)(diam(A − y)) γνn * r (dy) ≤ L 3 f (r 0 /2 n+1 )|ν n * r |(diam(A)) γ ≤ L 3 L 4 (2 n+1 /r 0 ) γ Ψ(2 n+1 /r 0 )|ν n * r |(diam(A)) γ ≤ L 3 L 4 [(r 0 /2) γ f (6r 0 )] −1 Ψ(2/r 0 )(2 γ L 5 L 6 ) n f (δ A )(Ψ(1/r)) n (diam(A)) γ .
To estimate I 12 we just use the induction hypothesis and Lemma 1 (a). Indeed, we have By exactly the same argument as for I 12 , we get I 22 ≤ L 5 C 3 9 C n 12 f (δ A )(Ψ(1/r) n (diam(A)) γ . It is enough to estimate I 21 . By (D) and Lemma 1 (a), we have
I 12 = ν n * r ((A − y) ∩ B(0, δ A − r 0 /2 n+1 ) c )ν r (dy) ≤ C n 12 f (δ A − r 0 /2 n+1 )(Ψ(1/r)) n−1 (diam(A)) γ |ν r | ≤ L 5 C 9 C n 12 f (δ A )(Ψ(1/r)) n (diam(A)) γ . Let now δ A >I 21 ≤ L 3 (diam(A)) γ |y|<δ A −3r 0 f (dist(A, y))ν n * r (dy) ≤ L 3 (diam(A)) γ C ⌈diam(A)/r 0 ⌉ 9 |y−x A |>r 0 f (|y − x A |)ν n * r (dy),
with some x A ∈ R d such that |x A | = δ A . Thus, by (8), we conclude that
I 21 ≤ L 3 C ⌈diam(A)/r 0 ⌉ 9
C n 11 f (δ A )(Ψ(1/r)) n (diam(A)) γ and, therefore, (9) holds with
C 12 := max L 3 , L 3 L 4 [(r 0 /2) γ f (6r 0 )] −1 Ψ(2/r 0 ) + L 5 C 3 9 + L 3 C ⌈diam(A)/r 0 ⌉ 9 , 2 γ L 5 L 6 , C 11 ,(11)
which completes the proof of the lemma.
We now show that under the assumption that the Lévy measure has a density which is comparable to some radially nonincreasing profile, the condition (3.1) of Theorem 3 is in fact equivalent to (C). Proof. Clearly, we only need to show that the condition (3.1) of Theorem 3 implies (C). For every r ∈ (0, r 0 ) and |x| ≥ 2r 0 , by (3.1) and the monotonicity of f and Ψ, we have
|x−y|>r 0 , |y|>r f (|y − x|)g(y)dy ≤ c |x−y|>r 0 , |y|>r 0 f (|y − x|)g(y)dy + r<|y|≤r 0 f (|y − x|) g(y)dy ≤ c 1 (Ψ(1/r 0 )f (|x|) + f (|x| − r 0 )ν(B(0, r) c )) ≤ c 2 Ψ(1/r)(f (|x|) + f (|x| − r 0 )).
If now |x| ≥ 4r 0 , then by the comparability g(x) ≍ f (|x|) > 0 and by similar argument as in Lemma 1 (a) based on (3.1), we get
f (|x|) ≥ c 3 B((2r 0 /|x|)x,r 0 ) f (|y − x|)f (|y|)dy ≥ c 4 f (|x| − r 0 ) B((2r 0 /|x|)x,r 0 ) f (|y|)dy ≥ c 5 f (3r 0 )r d 0 f (|x| − r 0 )
and the first inequality in (C) is satisfied. If |x| ∈ [2r 0 , 4r 0 ], then by strict positivity and monotonicity of f , f (|x| − r 0 ) ≤ c 6 f (|x|), and the first bound in (C) follows again. To show the second part of (C) for γ = d we observe that by (4) we have
c 7 r d f (r) ≤ r/2≤|y|<r
g(y)dy ≤ ν(B(0, r/2) c ∩ B(0, r)) ≤ L 5 Ψ(2/r) ≤ L 5 L 6 Ψ(1/r), r > 0.
Proofs of Theorems 4 and 3
We start with the following lemma which is a corollary from the estimates of the n-th convolutions of Lévy measures proven in the previous section.
f (|y − x|)P t (dy) ≤ e C 11 f (|x|), |x| ≥ 3r 0 , t ∈ (0, t 0 ]. (b) For every bounded A ∈ B(R d ) such that dist(A, 0) ≥ 3r 0 we havē P t (A) ≤ e C 12 t f (dist(A, 0))(diam(A)) γ , t ∈ (0, t 0 ].
Proof. Statements (a) and (b) are direct consequences of (5) and estimates (a) and (b) in Lemma 2, respectively.
We are now ready to prove Theorem 4.
Proof of Theorem 4. By (6), we only need to estimate the integral
I := R d G((y − x)/h(t))P t (dy)
for all |x| ≥ 4r 0 and t ∈ (0, t 0 ], where t 0 := 1/Ψ(1/r 0 ) ∧ t p (recall that t p is given in (E)).
By Lemma 1 (b) (as a consequence of (C)), we have G(s/(2h(t 0 ))) ≤ G(s/2r 0 ) ≤ C 10 f (s), s ≥ r 0 .
Let now t ∈ (0, t 0 ] and |x| ≥ 4r 0 . By (13), we have G(|y −x|/h(t)) ≤ G(r 0 /(2h(t)))G(|y −x|/(2h(t 0 ))) ≤ C 10 G(r 0 /(2h(t)))f (|y −x|), |y −x| > r 0 , and, consequently, we get
I = |y−x|≤r 0 G(|x − y|/h(t))P t (dy) + |y−x|>r 0 G(|x − y|/h(t))P t (dy) ≤ |y−x|≤r 0 G(|x − y|/h(t))P t (dy) + C 10 G(r 0 /(2h(t))) |y−x|>r 0 f (|x − y|)P t (dy).
Denote the two integrals above by I 1 and I 2 , respectively. We first estimate I 1 . By Fubini, we have (B(x, r 0 )) .
I 1 = |y−x|≤r 0 G(|x−y|/h(t)) 0 dsP t (dy) = 1 0 1 {y∈R d : G(|x−y|/h(t))>s,|y−x|≤r 0 }Pt (dy)ds = 1 0P t B(x, r 0 ∧ h(t)G −1 (s)) ds = 1 G(r 0 /h(t))P t B(x, h(t)G −1 (s)) ds + G(r 0 /h(t))P t
Applying now Lemma 4 (b) to the both members above, we get
I 1 ≤ ce C 12 t[h(t)] γ f (|x| − r 0 ) 1 0 G −1 (s) γ ds + tG(r 0 /h(t))f (|x| − r 0 )r γ 0 ,
with C 12 = C 12 (r 0 , 1) and finally, by using Lemma 1 (a) and noting that 1 0 (G −1 (s)) γ ds < ∞, we obtain
I 1 ≤ c 1 t[h(t)] γ f (|x|), t ∈ (0, t 0 ],
where c 1 = c 1 (r 0 ). It is enough to estimate I 2 and G(r 0 /(2h(t))). We deduce directly from Lemma 4 (a) that
I 2 ≤ e C 11 f (|x|), t ∈ (0, t 0 ].
Also, it follows from [17, Lemma 3.6.22] that Ψ(r) ≤ 2Ψ(1)(1 + r 2 ), r > 0, and, in consequence,
t[h(t)] γ = [h(t)] γ Ψ 1 h(t) ≥ c 2 [h(t)] γ 1 + 1 [h(t)] 2 = c 2 [h(t)] γ+2 1 + [h(t)] 2 ≥ c 3 1 + r 2 0 [h(t)] γ+2 ≥ c 4 G(r 0 /(2h(t))),
where c 4 = c 4 (Ψ, r 0 ). Finally, we obtain
p t (x + tb h(t) ) ≤ C 6 [h(t)] −d I ≤ c 5 [h(t)] −d (I 1 + G(r 0 /(2h(t)))I 2 ) ≤ c 6 t [h(t)] γ−d f (|x|),
with c 6 = c 6 (Ψ, r 0 ), for t ∈ (0, t 0 ] and |x| ≥ 4r 0 . This completes the proof.
p t (x + tb) ≥ ctf (|x| + c 1 h(t 0 )), |x| ≥ h(t 0 ), t ∈ (0, t 0 ].
(the constant C 6 in the estimate (7) of [21] may be assumed to be smaller than 1). By Lemma 1 (a), we conclude that for every |x| ≥ 4r 0 and t ∈ (0, t 0 ] we have
p t (x + tb) ≥ c 2 tg(x),
which completes the proof of the first implication.
To prove the opposite implication we assume that the estimates (3.2) hold. Let r 0 = R/2. By the both bounds in (3.2) and by the semigroup property, we have
g r 0 * g r 0 (x) ≤ c 3 2 t 0 2 |y−x|>r 0 , |y|>r 0 p t 0 /2 x − y + t 0 2 b p t 0 /2 y + t 0 2 b dy ≤ c 4 2 t 0 2 p t 0 (x + t 0 b) ≤ c 5 t 0 g(x),
for all |x| ≥ 2r 0 . The proof is complete.
Proofs of Theorems 2 and 1, and related results
In Lemma 5 below we collect some basic properties of the Lévy-Khintchine exponents corresponding to the Lévy measures investigated in Theorems 1-3. In particular, we show that the condition (2.1) implies (E). This will be used in the proofs of Theorem 2 and Proposition 1 below.
Lemma 5. Let ν(dx) = g(x)dx be a Lévy measure such that ν(R d ) = ∞, g(y) = g(−y) and there is a nonincreasing function f : [0, ∞) → [0, ∞] such that g(x) ≍ f (|x|), x ∈ R d . Moreover assume that b ∈ R d . Then the following hold.
(a) There exists a constant C 13 such that
C 13 Ψ(|ξ|) ≤ Re Φ(ξ) ≤ Ψ(|ξ|), ξ ∈ R d \ {0} . (b)
The condition (2.1) is equivalent to the property that there are α 1 , α 2 ∈ (0, 2), C 14 , C 15 > 0 and s 0 > 0 such that
C 14 λ α 1 Ψ(s) ≤ Ψ(λs) ≤ C 15 λ α 2 Ψ(s), λ ≥ 1, s ≥ s 0 ,(14)
that is, Ψ has weak lower and upper scaling properties with indices α 1 and α 2 at infinity (see e.g.
Ψ(λs) ≥ C 16 λ α Ψ(s), λ ≥ 1, s ≥ s 0 ,(15)
then the condition (E) holds for all t ∈ (0, 1/Ψ(s 0 )).
Proof. For ξ ∈ R d we define Φ 0 (ξ) = Φ 0 (|ξ|) := R d (1 − cos(ξ · y))f (|y|)dy. It is known that there is an isotropic unimodal Lévy process in R d with the Lévy-Khintchine exponent Φ 0 and the Lévy measure ν 0 (dy) = f (|y|)dy (for the formal definition and further details on unimodal Lévy processes we refer the reader to [37]). By comparability g(x) ≍ f (|x|), x ∈ R d , we have
Re Φ(ξ) ≍ Φ 0 (ξ), ξ ∈ R d .(16)
This and [14, Proposition 1] yields
Ψ(|ξ|) = sup |z|≤|ξ| Re Φ(z) ≍ sup |z|≤|ξ| Φ 0 (z) =: Ψ 0 (|ξ|) ≍ Φ 0 (ξ), |ξ| > 0.(17)
The both properties (16) and (17) give the assertion (a) of the lemma.
We now prove (b). Suppose first that (2.1) holds. Then, by the inequality Ψ(1/|x|) ≤ L 2 |x| d g(x), |x| ≤ 2r 0 , and by (4) and the same argument as in (12), we get
Ψ 0 (1/r) ≍ r d f (r), r ∈ (0, 2r 0 ],
and finally, we derive from [3, Theorem 26] that the function Ψ 0 has the property that there are α 1 , α 2 ∈ (0, 2), c 1 , c 2 > 0 and s 0 > 0 such that
c 1 λ α 1 Ψ 0 (s) ≤ Ψ 0 (λs) ≤ c 2 λ α 2 Ψ 0 (s), λ ≥ 1, s ≥ s 0 .(18)
By (17) also the function Ψ has the scaling property as in (18). The converse implication in (b) uses exactly converse argument and it is omitted.
Since by (a) we have Re Φ(ξ) ≍ Ψ(|ξ|) for ξ ∈ R d \ {0}, the property in assertion (c) can be established by following the argument (estimate) in [3, Lemma 16].
Proof of Theorem 2. We first consider the implication (2.1) ⇒ (2.2). Assume (2.1) and note that by Lemma 5 the condition (E) is satisfied with some t p > 0. Moreover, observe that by (2.1), (4) and (12), and the monotonicity of f , we have f (r) ≍ Ψ(1/r)r −d for r ∈ (0, r 0 ] and, consequently, the doubling property f (r) ≍ f (2r) holds for all r ∈ (0, r 0 /2]. Thus, by [21,Theorem 2] we obtain that there are t 0 ∈ (0, t p ] and θ > 0 such that θh(t 0 ) ≤ R := r 0 /2, for which the both lower bounds in (2.2) hold. To prove the upper bound define f 0 (r) := f (r) ∨ f (r 0 /2), r > 0. Since f (r) ≤ f 0 (r) for r > 0, and f 0 (r) has a doubling property for all r > 0, also the assumptions of [21,Theorem 1] are satisfied with such profile function f 0 and we get
p t (x + tb) ≤ ch(t) −d min 1, th(t) d f 0 (|x|) + e −c 1 |x| h(t) log(1+c 2 |x| h(t) ) , |x| > 0, t ∈ (0, t p ],
with some constants c, c 1 , c 2 > 0. In particular,
p t (x + tb) ≤ ch(t) −d min 1, th(t) d f (|x|) + e −c 1 |x| h(t) log(1+c 2 |x| h(t) ) , |x| ∈ (0, r 0 /2], t ∈ (0, t p ].
It is enough to estimate the exponentially-logarithmic member in the above estimate. By (2.1), for |x| ∈ (0, r 0 /2] and t ∈ (0, t p ], we have
th(t) d f (|x|) ≥ L −1 2 h(t) d |x| d Ψ 1 |x| Ψ 1 h(t) . If |x| ≤ h(t), then we easily have th(t) d f (|x|) ≥ L −1 2 ≥ L −1 2 e −c 1 |x| h(t) log(1+c 2 |x| h(t) )
. When |x| > h(t), then by the doubling property of Ψ in (4), we also get
th(t) d f (|x|) ≥ L −1 2 h(t) d |x| d Ψ 1 |x| Ψ |x| h(t) 1 |x| ≥ L −1 2 L −1 6 |x| h(t) −d−log 2 L 6 ≥ c 3 e −c 1 |x| h(t) log(1+c 2 |x| h(t) ) .
In particular, we see that the both upper bounds in (2.2) also hold for R = r 0 /2, t ∈ (0, t 0 ] and the same θ. We now show the opposite implication. Assume that (2.2) holds and let
r(t) := 2π R d e −t Re Φ(ξ) dξ − 1 d = [p t (tb)] − 1 d , t > 0.
By the first two-sided bound in (2.2), we have c −1 4 h(t) ≤ r(t) ≤ c 4 h(t), t ∈ (0, t 0 ], with some constant c 4 ≥ 1. Since the function r(t) is continuous in (0, t 0 ], it is also onto the interval (0, r(t 0 )]. Moreover, by the both bounds in (2.2), for every t ∈ (0, t 0 ], we have
[h(t)] −d ≤ c 5 tf (θh(t)) = c 5 f (θh(t)) Ψ(1/h(t)) ,
and by the comparability of r(t) and h(t), we also get
[h(t)] −d ≥ (c −2 4 θ) d [c −1 4 θr(t)] −d , t ∈ (0, t 0 ], and f (θh(t)) Ψ(1/h(t)) ≤ f (c −1 4 θr(t)) Ψ((c −2 4 θ)/(c −1 4 θr(t))) , t ∈ (0, t 0 ]. Therefore (c −2 4 θ) d [c −1 4 θr(t)] −d ≤ c 5 f (c −1 4 θr(t)) Ψ((c −2 4 θ)/(c −1 4 θr(t))) , t ∈ (0, t 0 ].
Let now r 0 := (2c 4 ) −1 θr(t 0 ) and note that for every r ∈ (0, 2r 0 ] there is t ∈ (0, t 0 ] such that r = c −1 4 θr(t). We conclude that by doubling property and monotonicity of Ψ it holds that
Ψ(1/r) ≤ c 6 f (r)r d , r ∈ (0, 2r 0 ],
which is exactly (2.1).
The following proposition may be seen as the complement to Theorem 2. A one of its important consequences is that the bounds from (2.2) always imply the two-sided bound in the minimum form as in (2.3) below, while the converse implications holds true under the assumption (E). Proposition 1. Let ν(dx) = g(x)dx be a Lévy measure such that ν(R d ) = ∞, g(y) = g(−y) and there is a nonincreasing function f :
[0, ∞) → [0, ∞] such that g(x) ≍ f (|x|), x ∈ R d . Moreover assume that b ∈ R d .C 17 min [h(t)] −d , tg(x) ≤ p t (x + tb) ≤ C 18 min [h(t)] −d , tg(x) , t ∈ (0, t 0 ], |x| ≤ R.
(2.4) There exist t 0 > 0, R > 0 and a constant C 19 such that we have
p t (x + tb) ≤ C 19 t g(x), t ∈ (0, t 0 ], |x| ≤ R.
Then the following implications holds. (4) for t ∈ (0, t 0 ], by the same argument as in (12) and the doubling property of Ψ, we get 1/t = Ψ(1/h(t)) ≍ Ψ(1/(θh(t))) ≍ f (θh(t))[θh(t)] d , t ∈ (0, t 0 ]. This clearly gives that there are constants c 1 , c 2 ≥ 1 such that [21,Lemma 7] gives that there are θ, c 3 > 0 such that
(2.1) ⇐⇒ (2.2) ⇐⇒ [(E) ∧ (2.3)] ⇐⇒ [(E) ∧ (2.4)][h(t)] −d ≤ c 1 tg(x), |x| ≤ θh(t), t ∈ (0, t 0 ], and tg(x) ≤ c 2 [h(t)] −d , |x| ≥ θh(t), t ∈ (0, t 0 ]. Withp t (x + tb) ≥ c 3 [h(t)] −d , |x| ≤ θh(t), t ∈ (0, t p ].
By this estimate, the upper bound in (2.3) or (2.4) and the doubling property of Ψ we thus get
[h(t)] −d ≤ c 4 tf (θh(t)) = c 4 f (θh(t)) Ψ(1/h(t)) ≍ f (θh(t)) Ψ(1/(θh(t)))
, t ∈ (0, t 0 ]. Now, by using a similar argument as in the second part of the proof of Theorem 2, we obtain that there is r 0 , c 5 > 0 such that
Ψ(1/r) ≤ c 5 f (r)r d , r ∈ (0, 2r 0 ],
and (2.1) holds. The proof is complete.
The discussion of essentiality of the condition (E) in the above proposition continues in Remark 1 in the last section. We close this section by giving the formal proof of Theorem 1.
Proof of Theorem 1. First observe that by monotonicity and strict positivity of functions f and Ψ one can easily extend the estimate (1.1) (b) to r ∈ (0, R] for every R > 2r 0 , possibly with a worse constant L 2 dependent on f (R) and Ψ(1/R). In particular, it holds for r ∈ (0, 16r 0 ]. Moreover, note that by Lemma 5 the condition (1.1) (b) implies (E). Therefore, the implication (1.1) ⇒ (1.2) is a direct corollary from (the proofs of) Theorems 2 and 3. Indeed, (1.2) is a conjuction of (2.2) and (3.2) with the same R = 4r 0 . Consider now the implication (1.2) ⇒ (1.1). If the both bounds in (1.2) hold, then the condition (1.1) (b) follows from Theorem 2 with some r 0 > 0. Therefore, as mentioned above, by Lemma 5 also the assumption (E) is satisfied, and the condition (1.1) (a) can be directly derived from Theorem 3 with the same r 0 .
Further results, discussion and examples
In this section we discuss our results and some of their consequences in a more detail. In particular, we illustrate them by several examples.
Symmetric and absolutely continuous Lévy measures.
We now illustrate the outcomes of our study with the Lévy measures with densities that are comparable to some specific profile functions. Our Theorem 1 is a consequence of the two separate results for small and large x given in Theorems 2 and 3, respectively. Therefore, for more transparency, below we discuss these two theorems separately.
As we mentioned in Introduction, the case of small x is somewhat better explored. In the example below we test Theorems 2 on some Lévy measures with the three different types of singularity at zero. Throughout this subsection we always assume that f : [0, ∞) → (0, ∞] is a nonincreasing profile function such that f · 1 [0,1] = κ with κ : [0, 1] → (0, ∞] satisfying the conditions κ(0) = ∞ and κ(1) < ∞. Example 1. Consider the following three types of Lévy measures ν(dx) = g(x)dx with g(x) = g(−x) ≍ f (|x|), x ∈ R d , where the corresponding small jump profiles κ are as follows.
(1) κ(r) = r −d (low intensity of small jumps) or (2) κ(r) = r −d−α 1 log (1 + 1/r) α 2 if α 1 ∈ (0, 2), α 2 ≥ 0 and κ is nonincreasing function such that κ(r) = r −d−α 1 log (1 + 1/r) α 2 for r close to zero if α 1 ∈ (0, 2), α 2 < 0 or (3) κ(r) = r −d−2 log (1 + 1/r) −2 (high intensity of small jumps). By direct calculations based on [21, Proposition 1], one can show that for the above three cases we have:
(1) Ψ(r) ≍ log(1 + r), h(t) ≍ e −1/t , (2) Ψ(r) ≍ r α 1 (log(1 + r)) −α 2 , h(t) ≍ t 1 α 1 log 1 + 1 t α 2 α 1 ,
(3) Ψ(r) ≍ r 2 log(1 + r) (not r 2 (log(1 + r)) 2 ), h(t) ≍ t 1 2 log 1 + 1 t −1 2 , whenever r ≥ 1 and t ∈ (0, 1/Ψ(1)]. We thus see that the condition (2.1) is satisfied in the case (2) with r 0 = 1/2, but it fails for (1) and (3). Theorem 2 states that the small time sharp two sided bounds for small x of the forms (2.1) are true for convolution semigroups with Lévy measures as in (2) only. In the remaining cases, they should be expected in a different form. Note that in some sense the singularities at zero as in (1) and (3) are borderline for the Lévy measures that are required to satisfy ν({x : |x| < 1}) = ∞ and |x|<1 |x| 2 ν(dx) < ∞. Informally speaking, this means that the condition (2.1) is typical for the measures having some balance between these two integrability properties (cf. [3]). For some available results on the estimates for transition densities corresponding to Lévy processes with slowly varying characteristic exponent as in (1), we refer the reader to [4, pages 117-118]. Sharp small time estimates for small x for processes with high intensity of small jumps as in (3) are still an open and very interesting problem (see [27,28] and the recent discussion in [21, page 22]).
The next remark is devoted to Proposition 1. (2.1). Indeed, for example the transition densities of the isotropic geometric α-stable processes, α ∈ (0, 2), in R d fulfil the estimates (see [4,Theorem 5.52]) We now illustrate our Theorem 3. To shorten the formulations below, first we set some useful notation. Recall that by κ : [0, 1] → (0, ∞] we denote a nonincreasing function such that κ(1) < ∞, κ(0) = ∞. In the sequel we assume that
p t (x) ≍ t |x| d−tα , |x| ≤ 1, t ∈ (0, 1 ∧ d/(2α)),m ≥ 0, β > 0, 0 < c ≤ κ(1)e m(19)
and δ ≥ 0 when m > 0 and δ > d when m = 0.
Below we will consider the profiles f := f κ,m,β,δ,c , where
f κ,m,β,δ,c (s) = 1 [0,1] (s) · κ(s) + c 1 (1,∞) (s) · e −ms β s −δ , s ≥ 0.(21)
In general, a wider range of δ can be considered in (21). However, we want f κ,m,β,δ,c to be a strictly positive and nonincreasing profile function for the sufficiently regular density of the Lévy measure. Therefore, in the remaining we always restrict our attention to the settings given by (19)- (20).
Consider the following convolution condition similar to (2.1) for functions f κ,m,β,δ,c .
(F) There exists a constant C = C(m, β, δ) > 0 such that |y−x|>1, |y|>1 f κ,m,β,δ,c (|y − x|)f κ,m,β,δ,c (|y|)dy ≤ Cf κ,m,β,δ,c (|x|), |x| ≥ 2.
We will need the following proposition which gives the characterization of (F) in terms of defining parameters β and δ. Proof. We first prove that the given restrictions of parameters imply the condition (F). Since the case m = 0 and δ > d is obvious, once we consider the case m > 0, β ∈ (0, 1) and δ ≥ 0. To shorten the notation let f := f κ,m,β,δ,c . We start by justifying that for every β ∈ (0, 1) and η ≥ 0 there is s 0 ≥ 1 such that
u β + v β ≥ (u + v) β + η log(u ∧ v), u, v ≥ s 0 .(22)
This estimate is a consequence of the standard inequality
(u ∨ v) β ≥ (u + v) β − β(u ∧ v)/(u ∨ v) 1−β , u, v > 0,
and the fact that for any η ≥ 0 we can find s 0 ≥ 1 such that
(1 − β)s β ≥ η log s, s ≥ s 0 .
Indeed, with these inequalities, for every u, v ≥ s 0 , we get
u β + v β = (u ∨ v) β + (u ∧ v) β ≥ (u + v) β − β(u ∧ v) (u ∨ v) 1−β + (1 − β)(u ∧ v) β + β(u ∧ v) β ≥ (u + v) β + η log(u ∧ v) + β(u ∧ v) (u ∧ v) 1−β − β(u ∧ v) (u ∨ v) 1−β ≥ (u + v) β + η log(u ∧ v),
which is exactly (22). Let now η := ((d + 1 − δ)/m) ∨ 0 and find s 0 ≥ 1 such that the inequality in (22) holds (recall δ ≥ 0). When s 0 > 1 and |x| ∈ [2, 2s 0 ) then the inequality in (F) holds by integrability and monotonicity properties of function f . Therefore we consider only the case |x| ≥ 2s 0 . With this we have
|y−x|>1, |y|>1 f (|y − x|)f (|y|)dy ≤ 2 1<|y|≤s 0 + s 0 <|y|≤|y−x| =: 2(I 1 + I 2 ).
Since there is c 1 > 0 such that f (s − s 0 ) ≤ c 1 f (s) for every s > 2s 0 , we get
I 1 ≤ f (|x| − s 0 ) |y|>1 f (|y|)dy ≤ c 2 f (|x|).
By the inequality (22) with η := ((d + 1 − δ)/m) ∨ 0 applied to u = |y − x| and v = |y|, we get
I 2 =c 3 s 0 <|y|≤|y−x| e −m[(|y−x|) β +|y| β ] (|y − x||y|) −δ dy ≤ c 4 e −m|x| β |x| −δ 1<|y|≤|y−x| |y| −δ−mη dy ≤ c 5 e −m|x| β |x| −δ 1<|y| |y| −d−1 dy ≤ c 6 f (|x|),
which completes the proof of the first implication for β ∈ (0, 1) and δ ≥ 0.
We now consider the most interesting case when m > 0, β = 1 and δ > (d + 1)/2. We will prove that with this range of parameters (F) also holds true. When d = 1, then this is an easy exercise. We consider only the case d ≥ 2. Observe that the condition (F) is in fact isotropic in a sense that it depends on the norm of x only. Therefore we may and do assume that x = (x 1 , 0, ..., 0) with x 1 > 2. Let
I := |y|>1, |y−x|>1 f (|y − x|)f (|y|)dy = |y|>1, y 1 <1 + 1≤y 1 ≤x 1 −1 + |y−x|>1, y 1 >x 1 −1 = I 1 + I 2 + I 3 .
Note that both integrals I 1 and I 3 are bounded above by f (|x| − 1) |y|>1 f (|y|)dy ≤ c 7 f (|x|). Thus, it suffices to estimate the integral I 2 . Let y = (y 1 , ..., y d ). By using spherical coordinates for d − 1 integrals with respect to dy 2 ...dy d , we get
I 2 = c 8 1≤y 1 ≤x 1 −1 e −m(|y−x|+|y|) (|x − y||y|) −δ dy ≤ c 9 |x| −δ ∞ 0 x 1 /2 1 e −m( √ (x 1 −s) 2 +r 2 + √ s 2 +r 2 )
r d−2 (s 2 + r 2 ) δ/2 dsdr.
One can directly check that for (s, r) ∈ [1,
x 1 /2] × [0, ∞) we have (x 1 − s) 2 + r 2 + √ s 2 + r 2 = x 1 + r 2 1 (x 1 − s) 2 + r 2 + x 1 − s + 1 √ s 2 + r 2 + s ≥ x 1 + r 2 √ s 2 + r 2 + s ≥ x 1 + r 2 √ s 2 + sr 2 + s(23)
and, since |x| = x 1 , in consequence,
I ≤ c 10 e −m|x| |x| −δ x 1 /2 1 ∞ 0 e − mr 2 √ s 2 +sr 2 +s r d−2 (s 2 + r 2 ) δ/2 drds =: c 11 e −m|x| |x| −δ J(x 1 ).
It is enough to prove that the function J(x 1 ) given by the double integral above is bounded for all x 1 > 2. By using the substitution r = √ su, for every x 1 > 2 we get
J(x 1 ) = x 1 /2 1 ∞ 0 e − mu 2 √ 1+u 2 +1 s d−1 2 u d−2 s δ (1 + u 2 /s) δ/2 duds ≤ ∞ 0 e − mu 2 √ 1+u 2 +1 u d−2 du · ∞ 1 s d−1 2 −δ ds.
We see that the first integral on the right hand side of the above inequality is always finite, while the second one is convergent whenever δ > (d + 1)/2. This completes the proof of the first implication.
To justify the opposite implication, we show that if m > 0, β > 1, δ ≥ 0 or if m > 0, β = 1 and δ ∈ [0, (d + 1)/2], then the condition (F) does not hold. Let m > 0 and suppose first that β > 1 and δ ≥ 0. For |x| ≥ 3 we have
|y−x/2|<1 f (|y − x|)f (|y|)dy ≥ c 12 |x| −2δ e −m[(|x|/2+1) β +(|x|/2+1) β ] = c 12 |x| −2δ e −2m(|x|/2+1) β ,
and, consequently,
|y−x|>1, |y|>1 f (|y − x|)f (|y|)dy f (|x|) ≥ c 13 |x| −δ e m(|x| β −2(|x|/2+1) β ) → ∞, as |x| → ∞,
which shows that (F) cannot hold. Suppose now that β = 1 and δ ∈ [0, (d + 1)/2]. We will show that also in this case (F) does not hold. As before, we consider only the case d ≥ 2. With no loss of generality we assume that x = (2n, 0, ..., 0), for natural n ≥ 2. By using spherical coordinates for d − 1 integrals with respect to dy 2 ...dy d , where y = (y 1 , ..., y d ), we get
I := |y|>1, |y−x|>1 f (|y − x|)f (|y|)dy ≥ 1<|y 1 |<n e −m(|y−x|+|y|) (|x − y||y|) −δ dy ≥ c 14 |x| −δ ∞ 0 n 1 e −m( √ (2n−s) 2 +r 2 + √ s 2 +r 2 ) r d−2 (s 2 + r 2 ) δ/2 dsdr.
The same argument as in (23) yields that for all (s,
r) ∈ [1, n] × [0, ∞) we have (2n − s) 2 + r 2 + √ s 2 + r 2 = 2n + r 2 1 (2n − s) 2 + r 2 + 2n − s + 1 √ s 2 + r 2 + s ≤ 2n + 2r 2 √ s 2 + r 2 + s ≤ 2n + r 2 s
and, in consequence,
I ≥ c 14 e −m|x| |x| −δ n 1 ∞ 0 e − mr 2 s r d−2 (s 2 + r 2 ) δ/2 drds.
Denote the last double integral by I n . It is enough to show that I n → ∞ as n → ∞. By using the substitution r = √ su, we get
I n = n 1 ∞ 0 e −mu 2 s d−1 2 u d−2 s δ (1 + u 2 /s) δ/2 duds ≥ ∞ 0 e −mu 2 u d−2 (1 + u 2 ) δ/2 du · n 1 s d−1 2 −δ ds.
Since I n ≥ c 15 n 1 s d−1 2 −δ ds, we see that I n → ∞ as n → ∞ whenever δ ≤ (d + 1)/2, which completes the proof of the proposition.
The next corollary shows that for symmetric Lévy processes with jump intensities comparable to radially nonincreasing functions f κ,m,β,δ,c the restriction of parameters given by Proposition 2 in fact characterizes the two sided bounds as in Theorem 3. Note that for the class of convolution semigroups considered in Corollary 6 the regularity condition (E) in fact depends only on the type of singularity of the function κ at zero. Corollary 6. Assume that (19)- (20) hold. Let ν(dy) = g(y)dy be a Lévy measure such that ν(R d ) = ∞ and g(x) = g(−x) ≍ f κ,m,β,δ,c (|x|), x ∈ R d , and let b ∈ R d . Assume, moreover, that the assumption (E) is satisfied with some t p > 0. Then
p t (x + tb) ≍ t f κ,m,β,δ,c (|x|), |x| ≥ R, t ∈ (0, t 0 ],(24)
for some R > 0 and t 0 ≤ t p if and only if the one of the conditions (a), (b) or (c) in Proposition 2 holds.
Proof. The result is a direct consequence of Theorem 3 and Proposition 2. Indeed, under the assumption of the theorem, the condition (3.1) in Theorem 3 is equivalent to (F).
The above result applies to a large class of purely jump symmetric Lévy processes including the wide range of subordinate Brownian motions and more general unimodal Lévy processes. It not only gives a sharp bound for the decay of the corresponding transition density at infinity for small time, but also settles when exactly such a bound holds true. The most interesting examples are tempered Lévy processes with jump intensities exponentially and suboexponentially localized at infinity. We cover a big subclass of tempered stable processes and others, even with more general intensities of small jumps that are remarkably different than stable one, whenever the condition (E) is satisfied.
Let us now briefly test our Theorem 3 with some exact examples of Lévy processes with Lévy measures absolutely continuous with respect to Lebesgue measure.
Example 2.
(1) One can see that in the case of the relativistic α-stable process with parameter ϑ > 0 (α ∈ (0, 2), κ(r) = r −d−α , m = ϑ 1/α , β = 1, δ = (d + α + 1)/2) [10] and the subexponentially and exponentially tempered α-stable process (α ∈ (0, 2), κ(r) = r −d−α , m > 0, β ∈ (0, 1], δ = d + α) [31] the condition(3.1) holds, and the corresponding densities satisfy the two-sided small time sharp bounds as in (3.2). (2) It is useful to see some other examples of κ different from r −d−α for which the background regularity assumption (E) in Theorem 3 (Corollary 6) is satisfied. One can check that it still holds true when κ is as in (2) and (3) of Example 1. In particular, this shows that the condition (E) covers a larger class of semigroups than (2.1). On the other hand, as we mentioned in Introduction, (E) does not hold when the characteristic exponent Φ slowly varies at infinity. For instance, it fails for (1) in Example 1. (3) When m > 0, β = 1 and δ = 0 (e.g. Lamperti stable process [6]), then the convolution condition (3.1) (or, simply, (F)) does not hold and Theorem 3 (Corollary 6) states that the optimal bounds for large x of the transition densities have to be of different form (cf. Example 5).
6.2.
More general Lévy measures. We now illustrate our Theorem 4 by discussing examples of (non-necessarily symmetric) Lévy processes with more general Lévy measures that are not absolutely continuous with respect to Lebesgue measure. where the parameters α 1 and α 2 satisfy
(i) α 1 ∈ (0, 2), α 2 ∈ {−2, 0, 2}, or (ii) α 1 = α 2 = 2.
Note that for such choice of α 2 the function κ is decreasing on the whole interval (0, 1). Let ν be a Lévy measure such that
ν(A) ≍ S µ(dθ) 1 0 1 A (sθ)κ(s)ds + c ∞ 1 1 A (sθ)s −δ e −ms β ds ,
where c = κ(1)e m and µ is a nondegenerate measure on S (sometimes called spectral measure) such that µ(S ∩ B(θ, ρ)) ≤ c 1 ρ γ−1 , θ ∈ S, ρ > 0,
for some γ ∈ [1, d]. Let moreover b ∈ R d .
Estimates of transition densities of convolution semigroups corresponding to such Lévy measures have been studied in [35,36,21], but the optimal small time bounds for large x were still an open problem. Available results allow to get the upper bound for large x with the rate |x| 1−γ−δ e −m|c 2 x| β , for some constant c 2 ∈ (0, 1), and they cannot answer the question whether the correct rate is given by |x| 1−γ−δ e −m|x| β . In particular, [21, Theorem 1] yields
p t (x + tb h(t) ) ≤ c 3 t [h(t)] γ−d |x| 1−γ−δ e −m| x 4 | β , t ∈ (0, 1], |x| ≥ R > 0, with h(t) ≍ t 1 α 1 log 1 + 1 t −α 2 α 1 for α 1 ∈ (0, 2), α 2 ∈ {−2, 0, 2} t 1 2 log 1 + 1 t −1 2 for α 1 = α 2 = 2 , t ∈ (0, 1].(25)
On the other hand, one can derive from [21, Theorem 2] that if, furthermore, µ is symmetric and for some finite set D 0 = {θ 1 , θ 2 , ..., θ n } ⊂ S, n ∈ N, and the positive constants c 4 , ρ 0 we have µ(S ∩ B(θ, ρ)) ≥ c 4 ρ γ−1 , θ ∈ D 0 , ρ ∈ (0, ρ 0 ], then there is R ≥ 1 such that
p t (x + tb h(t) ) = p t (x + tb) ≥ c 5 t [h(t)] γ−d |x| 1−γ−δ e −m|x| β , t ∈ (0, 1], x ∈ D,
where D = {x ∈ R d : x = rθ, r ≥ R, θ ∈ D 0 }. Therefore, it is reasonable to ask what is the sharpest possible upper bound for the decay rate at infinity (cf. Example 5 below).
Our present Theorem 4 gives an answer to this problem. Indeed, whenever β ∈ (0, 1) and δ ≥ 0, or β = 1 and, at least, δ > 1, then it states that
p t (x + tb h(t) ) ≤ c 6 t [h(t)] γ−d |x| 1−γ−δ e −m|x| β , t ∈ (0, 1], |x| ≥ 4.
We now verify all assumptions of Theorem 4 in this case. Denote q(r) := κ(r)1 0<r≤1 + cr −δ e −mr β 1 r>1 , and consider first the integral
and, consequently, by checking that (15) holds, we can verify that the assumption (E) is satisfied for all t ∈ (0, 1]. Moreover, by direct calculation using (26) and the asymptotic formulas above, one can derive the asymptotics for h(t), t ∈ (0, 1], as in (25).
Also, it can be verified that for every A ∈ B(R d ) with δ A := dist(A, 0) > 0 we have
ν(A) ≤ c 8 δ 1−γ A κ(δ A )1 0<δ A ≤1 + cδ −δ A e −mδ β A 1 δ A >1 [diam(A)] γ ,
which means that the assumption (D) holds with the function f (r) := κ(r)r 1−γ 1 0<r≤1 + cr 1−γ−δ e −mr β 1 r>1 and the given γ. By Lemma 7 below also the convolution condition in (C) for r 0 = 1 and such f is satisfied when β ∈ (0, 1) and δ ≥ 0, or β = 1 and δ > 1. The second part of (C) is an easy consequence of (26). Indeed, for r ∈ (0, 1] we have
f (r)r γ = rκ(r) ≍ r −α 1 log 1 + 1 r −α 2 = r −2 r 2−α 1 log 1 + 1 r −α 2 ≍ Ψ(1/r) for α 1 ∈ (0, 2) and f (r)r γ = rκ(r) ≍ r −2 log 1 + 1 r −2 < r −2 log 1 + 1 r −1 ≍ Ψ(1/r) for α 1 = 2.
This completes the verification of assumptions of Theorem 4.
We now prove the auxiliary lemma which was needed in the previous example.
Lemma 7. Let m > 0, β ∈ (0, 1] and δ ≥ 0 and let ν be a Lévy measure such that for every
A ∈ B(R d ) with dist(A, 0) > 1 ν(A) ≍ S ∞ 1 1 A (sθ)s −δ e −ms β dsµ(dθ),
where µ is a measure on S such that there is γ ∈ [1, d] for which µ(S ∩ B(θ, ρ)) ≤ cρ γ−1 , θ ∈ S, ρ > 0, with some constant c > 0. Let f (r) = r 1−γ−δ e −mr β for r > 1. When β ∈ (0, 1) and δ ≥ 0 or β = 1 and, at least, δ > 1, then there is a constant L 1 > 0 such that we have
|y−x|>1, |y|>r f (|y − x|)ν(dy) ≤ L 1 f (|x|)Ψ(1/r), r ∈ (0, 1], |x| > 2.
Proof. By (4), for |x| ≥ 2 and r ∈ (0, 1], we have Let first β ∈ (0, 1) and δ ≥ 0. Denote η := 2/m and find s 0 ≥ 1 for which the inequality (22) holds with such η. Note that for |x| ∈ [2, 2s 0 ] the desired inequality easily follows from properties of the function f and it is enough to consider |x| > 2s 0 . Let
I = S s 0 1 + ∞ s 0 1 B(x,s 0 )∩B(x,1) c (sθ) + ∞ s 0 1 B(x,s 0 ) c (sθ) f (|sθ − x|)f (|sθ|)s γ−1 dsµ(dθ) =: I 1 + I 2 + I 3 .
By the fact that f (|x| − s 0 ) ≤ c 2 f (|x|) for |x| > 2s 0 , we get
I 1 ≤ c 2 f (|x| − s 0 )µ(S) s 0 1 f (s)s γ−1 ds ≤ c 3 f (|x|)Ψ(1/r), r ∈ (0, 1].
Since S ∞ 1 1 B(x,s 0 )∩B(x,1) c (sθ)s γ−1 dsµ(dθ) ≤ c 4 < ∞ for every |x| > 2s 0 , we also get
I 2 ≤ f (1)f (|x| − s 0 ) S ∞ 1 1 B(x,s 0 )∩B(x,1) c (sθ)s γ−1 dsµ(dθ) ≤ c 5 f (|x|)Ψ(1/r), r ∈ (0, 1].
Thus, it is enough to estimate I 3 . To this end, we use the inequality (22) for u = |x − sθ| and v = |sθ|. Similarly as in the first part of the proof of Proposition 2, we get
I 3 ≤ c 6 e −m|x| β S ∞ s 0 1 B(x,s 0 ) c (sθ)(|sθ − x| ∧ |sθ|) −2 (|sθ − x||sθ|) 1−γ−δ s γ−1 dsµ(dθ)
and one can directly show that the last double integral is bounded by c 7 |x| 1−γ−δ for all |x| ≥ 2s 0 . Therefore, finally we obtain that I 3 ≤ c 8 f (|x|)Ψ(1/r) for all r ∈ (0, 1], which completes the proof of the lemma for β ∈ (0, 1). Let now β = 1. In this case, we directly have
I ≤ c 9 e −m|x| S ∞ 1 1 B(x,1) c (sθ)(|sθ − x||sθ|) 1−γ−δ s γ−1 dsµ(dθ)
and when, at least, δ > 1, the last double integral again is bounded by c 10 |x| 1−γ−δ for all |x| ≥ 2. Therefore, again, the desired bound holds and the proof of the lemma is complete.
The next example is devoted to purely discrete Lévy measures. Clearly, ν is a not necessarily symmetric purely atomic Lévy measure. By [21,Proposition 1] we can easily check that in this case Re Φ(ξ) ≍ |ξ| 2 ∧ |ξ| α/q , ξ ∈ R d \ {0}, and h(t) ≍ t q/α , t ∈ (0, 1]. Thus the assumption (E) is satisfied for all t ∈ (0, 1]. Moreover, one can see that also the domination property (D) holds exactly with the function f and γ = 0.
Some upper bound of transition densities in this case have been recently obtained: it follows from [21, Theorem 1] that p t (x + tb h(t) ) ≤ ct − dq α (1 ∧ tf (|x|/4)) , x ∈ R d , t ∈ (0, 1],
On the other hand, under the additional assumption that ν is symmetric (i.e., for every v ∈ {v n : n = 1, .., k 0 } we have −v ∈ {v n : n = 1, .., k 0 }), by [21,Theorem 2] we also obtain p t (x + tb h(t) ) = p t (x + tb) ≥ c 1 t − dq α (1 ∧ tf (|x|)) , x ∈ A q , t ∈ (0, 1]. As in the previous example, arguments in [21] and other available results allow to get the upper bound for large x with the rate |x| −δ e −m|c 2 x| β , for some constant c 2 ∈ (0, 1), but not exactly |x| −δ e −m|x| β . Our present Theorem 4 says that in this case the optimal rate for large x is indeed given by |x| −δ e −m|x| β for all m > 0, β ∈ (0, 1], δ > 0 and α ∈ (0, 2q). In particular, we have p t (x + tb h(t) ) ≤ c 3 t − dq α (1 ∧ tf (|x|)) , x ∈ R d , t ∈ (0, 1]. To justify this, it is enough to check the convolution condition in (C). As before, it suffices to see that for some constant c 4 > 0
y∈Aq∩B(0,1) c ∩B(x,1) c f (|x − y|)f (|y|) ≤ c 4 f (|x|), |x| ≥ 2.
The left hand side is not larger than 6.3. Sharpness of the convolution condition (C). We now compare the convolution condition in (C) for small r with the assumption (P) proposed recently in [21]. In Proposition 3 and Example 5 below we show that (C) always implies the inequality in (P) for the same range of r ∈ (0, r 0 ] and s ≥ 8r 0 , but there are Lévy measures and corresponding functions f satisfying (P), for which (C) fails. Note that the following result does not require the condition (D).
Proof. We use the standard covering argument. First we introduce the two types of covers which will be used below. Let and by (27) and monotonicity of Ψ, we finally obtain It remains to estimate I 2 (s). This will be done by using the second cover related to n 1 (d). With this we have The proof is complete.
I 2 (s) ≤
We now give some examples for which the opposite implication in Proposition 3 does not hold.
Example 5. Let ν(dy) = g(y)dy be a Lévy measure such that g(y) = g(−y) ≍ |y| −d−α (1 + |y|) d+α−δ e −m|x| β for y ∈ R d \ {0}, where m > 0, β ∈ (0, 1], α ∈ (0, 2), δ ≥ 0 (for simplicity we assume here that b = 0). One can directly check that for this range of parameters the condition (P) is always satisfied for all r > 0 and s > 0 while, as proved in Proposition 2, the condition (C) does not hold when β = 1 and δ ≤ (d + 1)/2. By using [21, Theorems 1-2] (cf. [22,8]) we obtain that for all m > 0, β ∈ (0, 1], α ∈ (0, 2) and δ ≥ 0 it holds that
c 1 t −d/α ∧ tg(x) ≤ p t (x) ≤ c 2 t −d/α ∧ tg(x/4) , x ∈ R d , t ∈ (0, 1].
Similarly as in the examples discussed in the previous sebsection, this bound may suggest that the correct decay rate of p t (x) at infinity is again given exactly by g(x). However, our Theorem 3 (Corollary 6) states that when β = 1 and δ ≤ (d + 1)/2 then g(x/4) in the upper bound for large x cannot be replaced by c 3 g(x) for any c 3 > 0. This means that in this case the lower estimate is too weak and the correct two-sided bound for densities p t (x) is of different form! The case β = 1 and δ = (d + 1)/2 determines some kind of 'phase transition' in the dynamics of the convolution semigroups (P t ). This subtle dichotomy phenomena cannot be seen from the previously known results on the asymptotic behaviour of the kernels of jump processes (see e.g. [35,36,9,8,25,21,22]), but also it cannot be definitively explained at this stage of our study. We close the discussion by recalling that this range of parameters also involves some well know and important classes of tempered Lévy processes such as Lamperti stable ones and others, for which the optimal bounds of densities are still an open problem.
was supported by the National Science Center (Poland) post-doctoral internship grant on the basis of the decision No. DEC-2012/04/S/ST1/00093. P. Sztonyk was supported by the National Science Center (Poland) grant on the basis of the decision No. DEC-2012/07/B/ST1/03356.
r<|y|<1 y ν(dy) if r ≤ 1, b + 1<|y|<r y ν(dy) if r > 1.
( D )
DThere exist a nonincreasing function f : [0, ∞) → (0, ∞], a parameter γ ∈ [0, d], and a constant L 3 > 0 such that
6r 0 and ν (n+1) * r (A) = ν r (A − y)ν n * r (dy) = |y|<δ A −3r 0 + |y|≥δ A −3r 0 =: I 21 + I 22 .
Lemma 3 .
3Let ν(dx) = g(x)dx be a Lévy measure such ν(R d ) = ∞ and let f : [0, ∞) → (0, ∞] be a nonincreasing function such that g(x) ≍ f (|x|), x ∈ R d . Then the condition (3.1) of Theorem 3 is equivalent to (C).
Lemma 4 .
4Let ν be a Lévy measure such that ν(R d ) = ∞ and let the assumptions (D) and (C) be satisfied for some function f , the parameter γ and some r 0 > 0. Recall thatP t =P h(t) t and let t 0 := 1 Ψ(1/r 0 ) . The following hold. (a) We have |x−y|>r 0
Proof of Theorem 3 .
3First note that under the assumption g(x) ≍ f (|x|), x ∈ R d , the condition (D) holds with f and γ = d. Consider the implication (3.1) ⇒ (3.2). The upper bound in (3.2) is a direct consequence of Lemma 3 and Theorem 4 with t 0 := t p ∧ 1/Ψ(1/r 0 ) and R = 4r 0 . The lower bound follows from [21, Theorem 2] under the condition (3.1). Indeed, by [21, Theorem 2] we have
If there is α > 0, s 0 > 0 and a constant C 16 > 0 such that
Proof.
Recall that by Theorem 2 the conditions (2.1) and (2.2) are equivalent, and by Lemma 5 the condition (2.1) implies (E). We also see that the condition (2.3) implies (2.4) (in fact with no use of (E)). To complete the proof, we show the implications (2.2) ⇒ (2.3), [(E) ∧ (2.3)] ⇒ (2.1) and [(E) ∧ (2.4)] ⇒ (2.1). Let t 0 := sup {t ∈ (0, t p ] : h(t) ≤ r 0 }. By taking r = θh(t) in (2.1) and
Remark 1 .
1It is reasonable to ask how far are the bounds in (2.3) and (2.4) from (2.2) and (2.1) or, in other words, how essential is the condition (E) in the implications of Proposition 1. Consider the following observations. (1) First note that the condition (E) does not imply any of the conditions (2.3) and (2.4). Counterexamples are here the Lévy process with high intensities of small jumps. For instance, if the Lévy measure has the profile for small x as in Example 1 (3), then (E) holds true, but the estimates in (2.3) and (2.4) fail. (2) Also, when (E) does not hold, (2.4) does not imply any of the conditions (2.2) and
which directly implies the upper bound in(2.4). However, in this case both conditions (2.1) and (2.2) fail. Since Φ(ξ) = log(1 + |ξ| α ), also the assumption (E) does not hold.(3) Since, in general, (E) does not imply (2.3), one can ask if there are examples of processes with densities satisfying (2.3), for which (E) and both conditions (2.1) and (2.2) are not true. However, it seems to be very difficult to indicate or construct such an example. This problem remains open.
Proposition 2 .
2Let (19)-(20) be satisfied. Then the condition (F) holds if and only if (a) m = 0 and δ > d or (b) m > 0, β ∈ (0, 1) and δ ≥ 0 or (c) m > 0, β = 1 and δ > (d + 1)/2.
Example 3 .
3(Product Lévy measures) Let m > 0, β ∈ (0, 1] and δ ≥ 0 and let κ(r) := r −1−α 1 log 1 + 1 r −α 2 for r ∈ (0, 1] and κ(0) = ∞,
α 1 = α 2 = 2,while for r > 1 we get g(r) = g(1) + r 1 s 2−δ e −ms β ds ≍ c 7 . Thus, by [21, Corollaries 2 and 3] Re(Φ(ξ)) ≍ |ξ| 2 g(1/|ξ|), ξ ∈ R d \ {0} ,
f1
(|y − x|)ν(dy) ≤ cf (|x| − 1)Ψ(1/r) + |y−x|>1, |y|>1 f (|y − x|)ν(dy)≤ c 1 f (|x|)Ψ(1/r) + I, B(x,1) c (sθ)f (|sθ − x|)f (|sθ|)s γ−1 dsµ(dθ).
Example 4 .
4(Discrete Lévy measures) Let {v n : n = 1, .., k 0 } be a family of k 0 ∈ N, k 0 ≥ d, vectors in R d such that lin {v n : n = 1, ..,k 0 } = R d and let b ∈ R d . For q > 0 denote A q = x ∈ R d : x = 2 qn v k ,where n ∈ Z, k = 1, ..., k 0 and f (s) := 1 [0,1] (s) · s −α/q + e m 1 (1,∞) (s) · e −ms β s −δ , s > 0, with m > 0, β ∈ (0, 1], δ > 0 and α ∈ (0, 2q). Let ν(dy) := R d f (|y|)δ Aq (dy) = y∈Aq f (|y|).
4 e −m|x| β |x| −δ ,which completes the justification.
Proposition 3 .
3Let ν be a Lévy measure and let f : [0, ∞) → (0, ∞] be a nonincreasing function such that there is r 0 > 0 and a constant L 1 satisfying |x−y|>r 0 , |y|>r f (|y − x|) ν(dy) ≤ L 1 Ψ(1/r)f (|x|), |x| ≥ 2r 0 , r ∈ (0, r 0 ]. (27) Then there exists an absolute constant M > 0 such that |y|>r f s ∨ |y| − |y| 2 ν(dy) ≤ MΨ(1/r)f (s), s ≥ 8r 0 , r ∈ (0, r 0 ].
n 0 (d) := inf {n ∈ N : ∃ (x k ) n k=1 ⊂ B(0, 1) c s.t. ∀ 1 ≤ i ≤ n ∃ 1 ≤ j ≤ n, 1/4 < |x i − x j | < 3/8 and B(0, 2) ∩ B(0, 1) c ⊂ n k=1B(x k , 1/2) since for y ∈ B(0, s) c ∩ B(sx k , s/2) we have |y − sx k | 0 , r 0 <|y−sx k | f (|y − sx k |) ν(dy)
f (|sx k |) ≤ c 2 f (s)Ψ(1/r).
n 1 k=1
1{y: 4r 0 <|y|<s}∩Γsz k f (s − |y|/2) ν(dy).Whenever |y| > 4r 0 and y ∈ Γ sz k for some k ∈ {1, ..., n 1 }, we haves − |y|/2 = s − (3/4)|y| + |y|/4 > (4r 0 )/4 + (s − |y|) + |y|/4 ≥ |(s + r 0 )z k − sz k | + |sz k − (|y|/s)(sz k )| + |(|y|/s)(sz k ) − y| ≥ |(s + r 0 )z k − y|and, finally,I 2 (s) ≤ n 1 k=1 {y:4r 0 <|y|<s}∩Γsz k f (|(s + r 0 )z k − y|) ν(dy) ≤ n 1 k=1 |y|>r 0 , |(s+r 0 )z k −y|>r 0 f (|(s + r 0 )z k − y|) ν(dy).One more use of(27) and monotonicity of Ψ givesI 2 (s) ≤ L 1 n 1 k=1f (|(s + r 0 )z k |)Ψ(1/r 0 ) ≤ c 3 f (s)Ψ(1/r).
these inequalities the two sided bound in (2.3) is a direct consequence of (2.2).Proofs of implications [(E) ∧ (2.3)] ⇒ (2.1) and [(E) ∧ (2.4)] ⇒ (2.1) are exactly the
same. Indeed,
Also, for 0 = z ∈ R d we denoteWith this, we may and do choose a finite sequence of points (x k ) n 0 k=1 ⊂ B(0, 1) c such that for every s > 0 we have B(0, 2s) ∩ B(0, s) c ⊂ n 0 k=1 B(sx k , s/2) and for every x i ∈ (x k ) n 0 k=1 there is another x j ∈ (x k ) n 0 k=1 such that s/4 < |sx i − sx j | < 3s/8. Similarly, we may and do find a sequence of points (z k ) n 1 k=1 ⊂ S such that for every s > 0 we have B(0, s) ⊂ n 1 k=1 Γ sz k . To verify the latter assertion it is enough to see that for everyWe now apply the both covers above to estimate the integral on the left hand side of (28). Let r ∈ (0, r 0 ] and s ≥ 8r 0 be fixed. We have Hence, it is enough to estimate I 2 (s) and I 3 (s). We first consider I 3 (s). By using the first cover related to n 0 (d) introduced above, we haveRecall that the above cover is choosen in a way that for every x i ∈ (x k ) n 0 k=1 we may find another x j ∈ (x k ) n 0 k=1 such that B(sx i , s/8) ⊂ B(sx j , s/2)\B(sx j , s/8). By this fact we may be sure that where n k − 1 ≥ 0 is a number of small balls B(sx i , s/8) covered by B(sx k , s/2)\B(sx k , s/8) for given k (we assume that each ball B(sx i , s/8) is covered once). Clearly, n 0 k=1 n k = 2n 0 . Now
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. Wyb. Wyspiańskiego. 27Warszawa and Institute of Mathematics and Computer Science, Wrocław University of TechnologyPoland E-mail address: [email protected], [email protected] and Institute of Mathematics and Computer Science, Wrocław University of Tech- nology, Wyb. Wyspiańskiego 27, 50-370 Wrocław, Poland E-mail address: [email protected], [email protected]
. Wybrzeże Wyspiańskiego. 27Paweł Sztonyk, Institute of Mathematics and Computer Science, Wrocław University of TechnologyPoland E-mail address: [email protected]ł Sztonyk, Institute of Mathematics and Computer Science, Wrocław University of Technology, Wybrzeże Wyspiańskiego 27, 50-370 Wrocław, Poland E-mail address: [email protected]
| {'fraction_non_alphanumeric': 0.11822383431814193, 'fraction_numerical': 0.044439244135026035, 'mean_word_length': 3.2457029309488328, 'pattern_counts': {'":': 0, '<': 58, '<?xml version=': 0, '>': 194, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 6, '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 study small time bounds for transition densities of convolution semigroups corresponding to purely jump Lévy processes in R d , d ≥ 1, including those with jumping kernels exponentially and subexponentially localized at infinity. For a large class of Lévy measures, non-necessarily symmetric and absolutely continuous with respect to the underlying measure, we find the optimal in time and space upper bound, for the corresponding transition kernels at infinity. In case of Lévy measures that are symmetric and absolutely continuous with respect to the Lebesgue measure, with densities g such that g(x) ≍ f (|x|) for nonincreasing profile functions f , we also prove the full characterization of the sharp two-sided transition densities bounds of the formThis is done for small and large x separately. Mainly, our argument is based on new precise upper bounds for convolutions of Lévy measures. Our investigations lead to some interesting and surprising dichotomy of the decay properties at infinity for transition kernels of purely jump Lévy processes. All results are obtained by analytic methods, with no use of probabilistic arguments.', 'arxivid': '1403.0912', 'author': ['Kamil Kaleta ', 'Paweł Sztonyk '], 'authoraffiliation': [], 'corpusid': 118637834, 'doi': '10.1007/s11854-017-0023-6', 'github_urls': [], 'n_tokens_mistral': 34505, 'n_tokens_neox': 29994, 'n_words': 17248, 'pdfsha': '4641839c12e207fbb1a61ace2463836ed5a6fd3c', 'pdfurls': ['https://arxiv.org/pdf/1403.0912v2.pdf'], 'title': ['SMALL TIME SHARP BOUNDS FOR KERNELS OF CONVOLUTION SEMIGROUPS', 'SMALL TIME SHARP BOUNDS FOR KERNELS OF CONVOLUTION SEMIGROUPS'], 'venue': []} |
arxiv |
An Analysis of the Johnson-Lindenstrauss Lemma with the Bivariate Gamma Distribution
May 24, 2023
Jason Bernstein
Lawrence Livermore National Laboratory Livermore
94550CA
Alec M Dunton
Lawrence Livermore National Laboratory Livermore
94550CA
Benjamin W Priest
Lawrence Livermore National Laboratory Livermore
94550CA
An Analysis of the Johnson-Lindenstrauss Lemma with the Bivariate Gamma Distribution
May 24, 2023
Probabilistic proofs of the Johnson-Lindenstrauss lemma imply that random projection can reduce the dimension of a data set and approximately preserve pairwise distances. If a distance being approximately preserved is called a success, and the complement of this event is called a failure, then such a random projection likely results in no failures. Assuming a Gaussian random projection, the lemma is proved by showing that the no-failure probability is positive using a combination of Bonferroni's inequality and Markov's inequality. This paper modifies this proof in two ways to obtain a greater lower bound on the no-failure probability. First, Bonferroni's inequality is applied to pairs of failures instead of individual failures. Second, since a pair of projection errors has a bivariate gamma distribution, the probability of a pair of successes is bounded using an inequality from[Jensen, 1969]. If n is the number of points to be embedded and µ is the probability of a success, then this leads to an increase in the lower bound on the no-failure probability of 1 2 n 2 (1 − µ) 2 if n 2 is even and 1 2 n 2 − 1 (1 − µ) 2 if n 2 is odd. For example, if n = 10 5 points are to be embedded in k = 10 4 dimensions with a tolerance of ϵ = 0.1, then the improvement in the lower bound is on the order of 10 −14 . We also show that further improvement is possible if the inequality in[Jensen, 1969]extends to three successes, though we do not have a proof of this result.
Notation n ∈ N + : Number of points to be embedded x j ∈ R d : jth data point d ∈ N + : Dimension of each x j k ∈ N + : Dimension of embedding space R ∈ R d×k : Random projection matrix; entries are i.i.d. N (0, 1) w i = x j − x j ′ : Difference between x j and x j ′ w i = w i /∥w i ∥ 2 : Unit-vector in direction of w i ϵ ∈ (0, 1)
: Tolerance on projection error Γ(k/2, 2)
: Gamma distribution with shape parameter k/2 and scale parameter 2 Z i = R Tw i : Random projection, has the N (0 k , I k ) distribution V i = ∥Z i ∥ 2 2 : Projection error, has the Γ(k/2, 2) distribution
p(V i ≤ v i )
: Cumulative distribution function of a projection error S i = {|V i − k| ≤ kϵ} : The ith projection error divided by k is close to one, a 'success'
V i and V i ′ H(v i , v i ′ )
: Joint cumulative distribution function of V i and V i ′ ρ 2 ii ′ = (w i ·w i ′ ) 2 : Correlation of V i and V i ′ I k : Identity matrix of dimension k × k 0 k : Zero vector of length k ⊗ : Kronecker product Subscripts j, j ′ = 1, . . . , n : Indices for individual points i, i ′ = 1, . . . , n 2 : Indices for pairs of points ℓ = 1, . . . , k : Index over elements of Z i Acronyms JL : Johnson-Lindenstrauss i.i.d.
: Independent and identically distributed
Introduction
The Johnson-Lindenstrauss (JL) lemma says, informally, that the dimension of a data set can be reduced by projection such that distances between points are almost preserved [Johnson and Lindenstrauss, 1984]. Review papers on the lemma and its extensions include [Nelson, 2020, Ghojogh et al., 2021, Freksen, 2021 and additional overviews can be found in [Vempala, 2005, Matoušek, 2013, Vershynin, 2018. In this paper, the terms success and failure are used to indicate that a distance is preserved, or not, by a projection. With this terminology, the JL lemma implies that for an arbitrary data set, there is a projection that results in no failures. Details of the lemma include choosing a sufficiently large embedding dimension, precisely defining what it means for a distance to be preserved, and picking a class of projections. Due in part to the importance of dimension reduction in data science, research is active on the lemma and related ideas [Larsen andNelson, 2017, Matoušek, 2008].
A common approach to proving the JL lemma is to show that the no-failure probability is positive for a particular class of random projections [Frankl and Maehara, 1988, Indyk and Motwani, 1998, Dasgupta and Gupta, 2003. Note for context that if the projection is random, then the successes and failures are random, and so there is an associated probability of having no failures. The difficulty with this approach is that the joint distribution of the failures is not available in closed-form, so it is not immediate that the no-failure probability is positive. However, following [DasGupta, 2008], the no-failure probability can be shown to be positive in essentially two steps. First, Bonferroni's inequality, or union bounding, is used to bound the probability of at least one failure, where the upper bound is the sum of the individual failure probabilities. Second, an embedding dimension is chosen such that this upper bound is less than one. This embedding dimension is found using Markov's inequality and the chi-square distribution of the projection errors. As discussed in detail later, a projection error quantifies how much the random projection changes the distance between two points and determines whether a failure or success occurs. Applying De Morgan's laws with this choice of embedding dimension then implies that the no-failure probability is positive, which proves the JL lemma. Overall, this approach bounds the no-failure probability by the sum of the individual failure probabilities and therefore depends on the marginal distribution of the failures.
The contribution of this paper is an improvement to the lower bound on the no-failure probability that appears in the JL lemma. Our approach is to apply Bonferroni's inequality to pairs of failures, rather than to each failure individually as in [Dasgupta and Gupta, 2003]. This leads to computing the joint success probability, or probability that two successes occur, which is a function of the joint distribution of the corresponding projection errors. The joint distribution is the bivariate gamma distribution studied in [Kibble, 1941], whose properties have been well-studied and are applicable here. In particular, an inequality that applies to this distribution from [Jensen, 1969] is used to bound each joint success probability. Bounding joint success probabilities, rather than probabilities of individual successes, leads to a greater lower bound on the no-failure probability. The improvement in the lower bound on the nofailure probability is typically small for embedding dimensions and data set sizes that arise in practice. For example, if n = 10 5 points are embedded in k = 10 4 dimensions with tolerance ϵ = 0.1, then the lower bound on the no-failure probability increases by approximately 10 −14 . This improvement was not found to lead to a smaller embedding dimension, which would be desirable from a data compression perspective, but this work may still be a step in that direction.
In the JL lemma, Bonferroni's inequality leads to a conservative bound on the probability of at least one failure, as noted in [Li et al., 2006b] and [Li et al., 2006a, footnote 7], for example. The source of this conservativeness is that the inequality is applied to each failure separately, since this ignores information contained in the joint distribution of the failures. Applying Bonferroni's inequality to individual failures therefore leads to a smaller bound on the no-failure probability than necessary. Our approach is to apply Bonferroni's inequality to pairs of failures instead, which allows information from the bivariate distributions of the failures to be used. Combining Bonferroni's inequality in this way with an inequality for bivariate gamma distributions from [Jensen, 1969] leads to a greater lower bound on the no-failure probability. Sharper Bonferroni inequalities have also been developed that rely on probabilities of pairs of events [Hunter, 1976, Worsley, 1982, but they are not considered here. It is also worth noting that much of the statistical work on Bonferroni's inequality, and variations of the inequality, has been related to multiple hypothesis testing. Somewhat analogously to the situation here, Bonferroni's inequality can lead to conservative hypothesis tests, as noted in [Bland and Altman, 1995] and [Wasserman, 2004, ch. 10], for example.
As previously noted, the bivariate gamma distribution is the joint distribution of two projection errors resulting from a Gaussian random projection. For our purposes, the distribution depends on the embedding dimension and a correlation parameter that is the square of the dot product of the points to be embedded. These dot products can often not be computed exactly due to computational considerations, and so the bound discussed here is independent of the dot products. The distribution is studied in [Wicksell, 1933, Kibble, 1941, Krishnaiah and Rao, 1961, Moran, 1967, Jensen, 1969, Iliopoulos et al., 2005 and generalized to higher-dimensions in [Krishnamoorthy and Parthasarathy, 1951]. [Balakrishna and Lai, 2009, sec. 8.1] and [Kotz et al., 2004, sec. 48, sec. 2.3] review the distribution and provide expressions for its density, distribution, and moment generating function. More generally, [Kotz et al., 2004, ch. 48] and [Balakrishna and Lai, 2009, ch. 8] collect these properties for several bivariate distributions that have marginal gamma distributions. Following [Balakrishna and Lai, 2009], the particular bivariate gamma distribution that occurs here can be called the Kibble-Wicksell bivariate gamma distribution. We call this distribution the bivariate gamma distribution for simplicity.
The paper is organized as follows. Section 2 gives background on Gaussian random projections and shows that the bivariate gamma distribution is the joint distribution of two projection errors. Section 3 then uses the inequality from [Jensen, 1969] for bivariate gamma distributions to get an improved lower bound on the no-failure probability. Section 4 provides additional discussion of our analysis and notes limitations of our approach. Last, Section 5 concludes the paper and considers possible future work.
Random Projections and Projection Errors
This section reviews linear, Gaussian random projections and shows that pairs of projection errors have the bivariate gamma distribution. In particular, Section 2.1 describes linear, Gaussian random projections and shows that the resulting projection errors are marginally gamma distributed. Section 2.2 describes the bivariate gamma distribution of a pair of projection errors, and Section 2.3 reviews an inequality for this distribution that we apply to the JL lemma. While the material may not be new to some readers, reviewing this section may still be useful since it introduces notation and terminology used in later sections.
Linear, Gaussian Random Projection
This paper considers random projections that can be represented as a matrix of independent random variables, each having the N (0, 1) distribution. If the matrix is R ∈ R d×k and w ∈ R d , then R T w ∈ R k is the random projection of w and is a Gaussian random variable by linearity. The point R T w is referred to as the embedding of w and k is referred to as the embedding dimension. Random projections have several favorable properties that make them useful for dimension reduction, in which case k << d. For example, they do not depend on the data directly and they are not likely to significantly change distances between points if the embedding dimension is sufficiently large. More detailed overviews of random projections can be found in [Nelson, 2020] and [Ghojogh et al., 2021], for example, and additional references are given in Section 1.
For notation, let x j and x j ′ be two of the n points to be embedded, and let i = i(j, j ′ ) be an index over pairs of points. The difference between the ith pair of points is denoted w i = x j − x j ′ , so that w i = w i /∥w i ∥ 2 is the difference between the pair of points scaled to have length one. The ith random projection is defined as
Z i = R Tw i ,(1)
and is Gaussian distributed by linearity,
Z i ∼ N (0 k , I k ).(2)
Since the elements of the random projection vector are independent, N (0, 1) random variables, it follows that the ith projection error, defined as
V i = ∥Z i ∥ 2 2 ,(3)
has the chi-square distribution with k degrees of freedom. Equivalently, V i has the gamma distribution with shape parameter k/2 and scale parameter 2, denoted V i ∼ Γ(k/2, 2). The random variable V i is called a projection error because it quantifies how much the random projection changes the distance between two points. Note that the mean of a projection error is E[V i ] = k and the variance of a projection error is Var[V i ] = 2k. It is common to see the projection matrix defined instead as R/ √ k, which leads to the projection errors having mean one and variance 2/k. The different scalings lead to different interpretations of the projection errors but do not change our results. The scaling employed here is used since it simplifies the modeling of the projection errors with the bivariate gamma distribution.
The important point is that if V i /k is close to one, then the random projection, scaled by 1/ √ k, approximately preserved the distance between the points x j and x j ′ . Hence, if we define a success as the event that a projection error is close to its mean, then ideally the success probability is high. The notion of a projection error being close to its mean is formalized by defining the ith success as
S i = {V i ∈ [k(1 − ϵ), k(1 + ϵ)]},(4)
where ϵ ∈ (0, 1) is called the tolerance since it determines the limits on the allowed projection error. The probability of a success, p(S i ), depends on the embedding dimension and the tolerance, but not the points to be embedded or index i, and so is a constant,
µ = p(V i ≤ k(1 + ϵ)) − p(V i ≤ k(1 − ϵ)).(5)
By the weak law of large numbers, the success probability converges to one as k → ∞. A failure is defined as the complement of a success. The ith failure is denoted F i = S c i and the probability of this outcome is p(F i ) = 1 − µ by (5). Figure 1 shows how the success probability changes with the embedding dimension and tolerance. We see that the success probability converges to one as k → ∞ for fixed ϵ and is positive for all possible values of the embedding dimension. The probabilities are computed using the Γ(k/2, 2) distribution function.
Bivariate Gamma Distribution
The approach taken here depends on bounding probabilities of the form p(S i ∩ S i ′ ), which is a function of the joint distribution of V i and V i ′ . We refer to these probabilities as joint success probabilities, in contrast to the marginal success probability µ = p(S i ) defined in Section 2.1. A brief review of the main points of this section is now provided to help guide the discussion:
1. The joint distribution of two projection errors, V i and V i ′ , is the bivariate gamma distribution described in [Kibble, 1941]. This distribution depends on k and the correlation of V i and V i ′ , which is the squared-dot product ofw i andw i ′ , or (w i ·w i ′ ) 2 ;
2. Joint success probabilities can thus be computed if w i and w i ′ are known, but this may not be possible if the data has not been collected yet or n is large, for example;
3. Instead, the joint success probabilities are bounded below using an inequality from [Jensen, 1969]. The inequality for our purposes and in our notation is p(
S i ∩ S i ′ ) ≥ µ 2 .
The remainder of this section provides additional details on these points. To identify the joint distribution of the projection errors V i and V i ′ , it is helpful to start with the joint distribution of the random projections Z i and Z i ′ . By linearity and independence of the elements of R, the two random projections are jointly Gaussian distributed,
Z i Z i ′ ∼ N 0 k 0 k , I k ρ ii ′ I k ρ ii ′ I k I k ,(6)
where I k is the identity matrix of dimension k × k and the correlation parameter in the covariance matrix is
ρ ii ′ =w i ·w i ′ .(7)
The correlation between Z i and Z i ′ is due to each being a function of the same random matrix R. The covariance matrix in (6) can also be written as
Σ ii ′ ⊗ I k ,(8)
where
Σ ii ′ = 1 ρ ii ′ ρ ii ′ 1 (9)
is a correlation matrix and ⊗ is the Kronecker product. Hence, the random projections Z i and Z i ′ have the same distribution as k independent samples from a bivariate normal distribution with covariance matrix Σ ii ′ . The full joint distribution of all n 2 random projections is given in Appendix A.3 for completeness. Joint distributions of random projections are also used for estimation in [Li et al., 2006b], [Li et al., 2006a], and [Kang, 2021], for example.
The Gaussian distribution of the random projections in (6) determines the joint distribution of the projection errors V i and V i ′ . Specifically, the joint distribution of V i and V i ′ is the bivariate gamma distribution discussed in [Kibble, 1941] and reviewed in [Balakrishna and Lai, 2009, sec. 8.2]. Furthermore, since V i and V i ′ are each the sum of k independent, squared, standard normal random variables, they are both marginally Γ(k/2, 2) distributed. The projection errors are independent if and only if ρ ii ′ = 0 since Gaussian random variables are independent if and only if they are uncorrelated. This correlated gamma distribution characterization of V i and V i ′ leads to their joint distribution being called the bivariate gamma distribution.
The correlation of V i and V i ′ is computed from the moment generating function of the bivariate gamma distribution with parameters k and ρ ii ′ . A derivation of the moment generating function is given in [Kibble, 1941] and [Krishnamoorthy and Parthasarathy, 1951], and Appendix A.1 for completeness. To derive the correlation of V i and V i ′ , note first from (6) that
V i V i ′ d = k ℓ=1 Z 2 iℓ Z 2 i ′ ℓ ,(10)
where Z iℓ and Z i ′ ℓ are the ℓth elements of Z i and Z i ′ , respectively, and Cov
[Z iℓ , Z i ′ ℓ ] = ρ ii ′ . It is shown in Appendix A.1 that Cov[Z 2 iℓ , Z 2 i ′ ℓ ] = 2ρ 2 ii ′ , which with (10) implies that the covariance of V i and V i ′ is Cov[V i , V i ′ ] = 2kρ 2 ii ′ .(11)
The correlation of V i and V i ′ is thus
Cor[V i , V i ′ ] = Cov[V i , V i ′ ] (Var[V i ] Var[V i ′ ]) 1/2 (12) = ρ 2 ii ′ (13) since Var[V i ] = Var[V i ′ ] = 2k. It follows that V i and V i ′ are positively correlated if ρ ii ′ ̸ = 0, or equivalently if w i and w i ′ are not orthogonal.
The bivariate gamma density function of the projection errors is derived in [Kibble, 1941]. Specifically, [Kibble, 1941, eqn. 12] gives the density of V i /2 and V i ′ /2, so a scaling by two is applied to get the joint density of V i and V i ′ as
h(v i , v i ′ ) = 1 4Γ(k/2)(1 − ρ 2 ii ′ ) v i v i ′ 4ρ 2 ii ′ (k/2−1)/2 exp − v i + v i ′ 2(1 − ρ 2 ii ′ ) I k/2−1 v i v i ′ ρ 2 ii ′ 1 − ρ 2 ii ′ ,(14)
where I k/2−1 (·) is the modified Bessel function of the first kind and order k/2 − 1; see also the last equation in [Kibble, 1941, sec. 5.4] for a similar expression. The scaling by one-half appears to be a difference between the bivariate gamma distribution and random projection literature, which we follow the latter on and do not scale. If ρ ii ′ = 0, then V i and V i ′ are independent and the joint density function is the product of the marginal Γ(k/2, 2) density functions of V i and V i ′ . This can be obtained from the joint density (14) using the identity and Stegun, 1964, eqn. 9.6.7]. [Iliopoulos et al., 2005] gives expressions for the conditional means and variances of the distribution. Figure 2 shows the bivariate gamma density function as the correlation, ρ 2 ii ′ , is varied from 0.1 to 0.9 in increments of 0.1. The density is symmetric in v i and v i ′ and becomes increasingly concentrated along the v i = v i ′ axis as ρ 2
I k/2−1 v i v i ′ ρ 2 ii ′ 1 − ρ 2 ii ′ ∼ 1 Γ(k/2) 1 2 v i v i ′ ρ 2 ii ′ 1 − ρ 2 ii ′ (k/2−1) (15) as ρ ii ′ → 0 [Abramowitz
ii ′ increases. Recall as well that the marginal expectations and variances are equal to k and 2k since the projection errors have the Γ(k/2, 2) distribution. Figure 3 shows the integration domain for computing the joint success probability. The domain is
square since S i ∩ S i ′ is equivalent to V i ∈ [k(1 − ϵ), k(1 + ϵ)] and V i ′ ∈ [k(1 − ϵ), k(1 + ϵ)]
, and so the joint success probability is the integral
p(S i ∩ S i ′ ) = k(1+ϵ) k(1−ϵ) k(1+ϵ) k(1−ϵ) h(v i , v i ′ ) dv i dv i ′ .(16)
Alternatively, if the joint cumulative distribution function of V i and V i ′ is denoted
H(v i , v i ′ ) = p(V i ≤ v i , V i ′ ≤ v i ′ ),(17)
then the joint success probability can be written as
p(S i ∩ S i ′ ) = H(k(1 + ϵ), k(1 + ϵ)) − 2H(k(1 + ϵ), k(1 − ϵ)) + H(k(1 − ϵ), k(1 − ϵ)),(18)
where the middle term follows from the symmetry of the distribution function, [Balakrishna and Lai, 2009, sec. 8.2.2] and the references therein. The joint success probability can thus, in principle, be computed by numerically integrating the density in (14) or evaluating (18). However, numerically integrating the density function (14) may be difficult when k is large, and so an approximation of the distribution function is needed in this case. To this end, the central limit theorem implies that the distribution of V i and V i ′ , suitably scaled, converges to a bivariate Gaussian distribution as k → ∞. The bivariate gamma distribution H(v i , v i ′ ) can therefore be quickly approximated using a bivariate normal approximation for sufficiently large k. Details on this approach are provided in Appendix A.2.
H(v i , v i ′ ) = H(v i ′ , v i ). Formulas for H(v i , v i ′ ) are given in
Bounding the Joint Success Probability
An inequality for bivariate gamma distributions from [Jensen, 1969] provides a lower bound on the joint success probabilities. The inequality applies here since, by (6), Z i and Z i ′ are jointly Gaussian distributed and each has expected value 0 k and a diagonal covariance matrix. Under these conditions, [Jensen, 1969] showed that if a ≥ b ≥ 0, then
p(V i ∈ [a, b], V i ′ ∈ [a, b]) ≥ p(V i ∈ [a, b])p(V i ′ ∈ [a, b]),(19)
where V i = ∥Z i ∥ 2 2 and V i ′ = ∥Z i ′ ∥ 2 2 . Taking a = k(1 − ϵ) and b = k(1 + ϵ) in this inequality, it follows that the joint success probabilities are bounded below by µ 2 , or
p(S i ∩ S i ′ ) ≥ µ 2 ,(20)
with equality if and only if ρ ii ′ = 0. The equality case is verified by noting that V i and V i ′ are independent if and only if ρ ii ′ = 0, in which case p(S i ∩ S i ′ ) = p(S i )p(S i ′ ) = µ 2 . Furthermore, the joint success probability is always positive since µ > 0 for all tolerances and embedding dimensions, which implies that two successes are never mutually exclusive. It should also be noted that the inequality in [Jensen, 1969] applies to a broader range of bivariate gamma distributions than the particular distribution considered here. Specifically, the inequality applies even if the covariance matrix of Z i and Z i ′ does not have the form ρ ii ′ I k , though this full generality is not needed here. The inequality on joint success probabilities (20) is equivalent to
p(S i ′ |S i ) ≥ µ(21)
since p(S i ∩ S i ′ ) = p(S i ′ |S i )µ by Bayes' rule, meaning that the conditional success probability p(S i ′ |S i ) can be greater than the success probability. In words, this inequality says that knowledge of one success may increase the probability of a second success. This suggests another interpretation of (20), which is that the success-indicator random variables, I[S i ] and I[S i ′ ], are non-negatively correlated. An upper bound on the joint success probability is also available. Since {S i ∩S i ′ } ⊆ S i and p(S i ) = µ, it follows that µ is an upper bound on p(S i ∩ S i ′ ), or
p(S i ∩ S i ′ ) ≤ µ.(22)
Note that if ρ 2
ii ′ = 1, thenw i =w i ′ orw i = −w i ′ , implying V i = V i ′ and thus p(S i ∩ S i ′ ) = p(S i ) = µ.
Hence, equality occurs in (22) if ρ 2
ii ′ = 1, or w i and w i ′ point in the same or opposite directions. In summary, µ 2 ≤ p(S i ∩ S i ′ ) ≤ µ, with the lower and upper bounds obtained when ρ ii ′ = 0 and ρ 2 ii ′ = 1, respectively. Figure 4 shows how the joint success probability changes as a function of the correlation, ρ ii ′ , and embedding dimension, k. For all k, the joint success probability equals µ 2 and µ when ρ ii ′ = 0 and ρ 2 ii ′ = 1, respectively, and these are the lower and upper bounds on the probability. The bounds vary with the embedding dimension, and for example are (0.18, 0.42) when k = 10 3 and (0.85, 0.92) when k = 10 4 . To create this plot, a Gaussian approximation of the joint success probability was made when k > 300 to avoid issues with numerically integrating the density in (14). The appendix provides details on this approximation, which is justified by the central limit theorem as noted at the end of Section 2.2.
We continue with reviewing the JL lemma, which shows that the no-failure probability is positive for a particular embedding dimension. The inequality in (20) is then used to improve the lower bound on the no-failure probability.
Bounding the No-Failure Probability
The JL lemma gives a lower bound on the embedding dimension, as a function of n and ϵ, that ensures the no-failure probability is positive,
p ∩ ( n 2 ) i=1 S i > 0.(23)
The no-failure probability can quickly be shown to be positive if the successes are mutually independent or if all ρ 2 ii ′ = 1. That is, if the successes are mutually independent, or all ρ ii ′ = 0, then the no-failure Figure 4: The joint success probability as a function of the embedding dimension and the correlation ρ ii ′ . The probability equals µ when ρ 2 ii ′ = 1 and equals µ 2 when ρ ii ′ = 0. The tolerance here is ϵ = 0.025.
probability is µ ( n 2 ) and positive since µ > 0 for all k and ϵ. Alternatively, if all ρ 2 ii ′ = 1, then the no-failure probability equals µ and is again positive for all values of k and ϵ. However, if some ρ 2 ii ′ / ∈ {0, 1}, then the distribution of the number of successes is not available in closed-form and the no-failure probability is not clearly positive for a given k and ϵ; recall from Section 2.2 that the random projections are correlated due to the random matrix R, which implies that the successes are dependent as well. An approach to showing that the no-failure probability is positive is therefore needed that allows for dependent successes.
In this section, we first review the approach in [Dasgupta and Gupta, 2003] that uses Bonferroni's inequality to get a lower bound on the no-failure probability. The approach is then adapted to find a greater lower bound on the no-failure probability using the lower bound on the joint success probability given in (20). The new lower bound is greater than the standard lower bound for a fixed embedding dimension, though the difference depends on the embedding dimension as discussed later in this section.
We begin by reviewing the proof of the JL lemma in [Dasgupta and Gupta, 2003] to show that the no-failure probability is positive if the embedding dimension is sufficiently large. The first step is to bound the probability of one or more failures using Bonferroni's inequality,
p ∪ ( n 2 ) i=1 F i ≤ ( n 2 ) i=1 p(F i )(24)= n 2 (1 − µ),(25)
which shows that the probability of at least one failure is bounded above by the expected number of failures. Taking complements and using De Morgan's laws, it follows that the no-failure probability satisfies
p ∩ ( n 2 ) i=1 S i ≥ 1 − n 2 (1 − µ).(26)
The inequality in (26) can be written in an equivalent form that is commonly-used,
p ∩ ( n 2 ) i=1 S i ≥ ( n 2 ) i=1 p(S i ) − n 2 − 1 ,(27)
where the substitution p(S i ) = µ is made to complete the equivalence [Casella and Berger, 2021, eqn. 1.2.10]. We refer to the lower bound in (26) as the marginal lower bound since it is derived with the marginal distribution of the projection errors. Furthermore, since µ → 1 as k → ∞ for fixed ϵ, an embedding dimension exists such that the marginal lower bound is positive; see [Dasgupta and Gupta, 2003, eqn. 2.1] for a lower bound on the embedding dimension that implies the no-failure probability is positive. Note as well that since the marginal lower bound is obtained by applying Bonferroni's inequality to each failure separately, any possible dependency between failures is ignored. A greater lower bound on the no-failure probability is found by repeating the steps above, but applying Bonferroni's inequality to pairs of failures instead of each failure individually. The inequality (20) from [Jensen, 1969] is then applied to each of the joint success probabilities appearing in the resulting inequality. Bounding the probability of pairs of failures instead of single failures allows the joint distribution of the projection errors to be used in bounding the no-failure probability. The number of failures to consider, n 2 , is assumed to be even initially to simplify the notation and derivation of the lower bound. Proceeding as for the marginal lower bound, the first step in deriving the improved lower bound is to get an upper bound on the probability of at least one failure,
p(∪ ( n 2 ) i=1 F i ) = p(∪ 1 2 ( n 2 ) i=1 {F 2i−1 ∪ F 2i }) (28) ≤ 1 2 ( n 2 ) i=1 p(F 2i−1 ∪ F 2i ) (29) = 1 2 ( n 2 ) i=1 (1 − p(S 2i−1 ∩ S 2i )) (30) ≤ 1 2 ( n 2 ) i=1 (1 − µ 2 ) (31) = 1 2 n 2 (1 − µ 2 ),(32)
where (29)
p ∩ ( n 2 ) i=1 S i ≥ 1 − 1 2 n 2 (1 − µ 2 ).(33)
The lower bound in (33) is referred to as the bivariate lower bound since it is derived with the bivariate gamma distribution of the projection errors. As with the marginal lower bound, an embedding dimension that makes the left side of (33) positive can be found since µ → 1 as k → ∞. Note as well that the bivariate lower bound is a worst-case bound on the no-failure probability in that it corresponds to the case where all w i · w i ′ = 0. If instead at least one w i · w i ′ ̸ = 0, then p(S i ∩ S i ′ ) > µ 2 since ρ 2 ii ′ > 0 and the inequality in (33) is strict.
The bivariate lower bound is strictly greater than the marginal lower bound for all embedding dimensions and tolerances. This is verified by subtracting the marginal lower bound in (26) from the bivariate lower bound in (33) and noting that the difference, ∆(n, µ) = 1 2
n 2 (1 − µ) 2 ,(34)
is positive for all µ, though it does converge to zero as the embedding dimension increases since µ → 1 as k → ∞. In general, k will be taken to be large enough that the marginal lower bound is positive, ensuring that the no-failure probability is positive. This makes the success probability µ close to one, which leads to a small difference between the bivariate and marginal lower bounds. For example, the difference is approximately ∆(n, µ) ≈ 10 −14 when n = 10 5 , k = 10 4 , and ϵ = 0.1, as noted in Section 1. If the number of failures is odd, then one failure cannot be paired off with another failure and the upper bound (32) and difference (34) are modified accordingly. Specifically, if n 2 is odd, then the upper bound on the no-failure probability given in (32) is 1 2 n 2 − 1 (1 − µ 2 ) + (1 − µ). In this expression, the first summand corresponds to n 2 − 1 pairs of failures and the second summand corresponds to the failure that cannot be paired off with another failure. The difference between the bivariate lower bound and the marginal lower bound is given by (34) with the n 2 replaced by n 2 − 1, which is positive. Going forward, the number of failures to consider is assumed to be even for simplicity.
A greater lower bound on the no-failure probability than the bivariate lower bound is available if the points to be embedded have a certain geometry. In particular, since each p(S 2i−1 ∩ S 2i ) ≤ µ by (22), it follows by summing over joint success probabilities that
1 − 1 2 n 2 (1 − µ) ≥ 1 − 1 2 ( n 2 ) i=1 (1 − p(S 2i−1 ∩ S 2i )) ,(35)
where equality holds if and only if ρ 2 2i−1,2i = 1 for all i and the lower bound is from (30). Hence, if the points can be indexed such that ρ 2 2i−1,2i = 1 for all i, then the no-failure probability is greater than or equal to the left side of (35). However, there is no guarantee in general that the left side of (35) is a lower bound on the no-failure probability, whereas the bivariate lower bound always holds.
To summarize this section, the no-failure probability can be shown to be positive using the marginal or bivariate distributions of the projection errors. The two lower bounds on the no-failure probability are derived similarly using Bonferroni's inequality and De Morgan's laws. However, the bivariate lower bound is greater than the marginal lower bound by an amount that depends on the number of points to be embedded and the success probability. The difference between the bounds is due to a lower bound on the joint success probabilities that is obtained only when the projection errors are independent. If the projection errors are correlated, then the bivariate lower bound is conservative by an amount that increases with the magnitudes of the correlations. Hence, accounting for possible dependency or correlation between successes leads to an improved lower bound on the no-failure probability.
Discussion
This section discusses several additional points related to our approach and main result. First, Section 4.1 considers whether the approach can be extended by applying Bonferroni's inequality to more than two successes at a time. Section 4.2 then discusses whether the improved lower bound on the no-failure probability affects selection of the embedding dimension. Section 4.3 gives a data-dependent lower bound on the no-failure probability that improves on the bivariate lower bound. Unfortunately the datadependent lower bound cannot be computed when n is large since it is a summation over n 2 terms, though it could potentially be estimated. Last, Section 4.4 notes limitations of our approach.
Generalizing the Derivation
A natural follow-up question to this work is whether the derivation can be generalized by applying Bonferroni's inequality to three or more successes at a time. The answer depends on whether the inequality in [Jensen, 1969] extends to higher-dimensional gamma distributions, which we do not know. However, the inequality for three successes,
p(S 3i−2 ∩ S 3i−1 ∩ S 3i ) ≥ µ 3 .(36)
can be shown to hold in a few simple cases, where we take i = 1 for simplicity:
1. If S 1 , S 2 , S 3 are independent, then ρ 12 = ρ 23 = ρ 31 = 0 and p(S 1 ∩ S 2 ∩ S 3 ) = µ 3 ;
2. If ρ 2 12 = ρ 2 23 = ρ 2 31 = 1, then p(S 1 ∩ S 2 ∩ S 3 ) = µ and (36) holds since µ > µ 3 ;
3. If S 1 and S 2 are conditionally independent given S 3 , then p(S 1 ∩ S 2 |S 3 ) = p(S 1 |S 3 )p(S 2 |S 3 ) and
p(S 1 ∩ S 2 ∩ S 3 ) = p(S 1 ∩ S 2 |S 3 )p(S 3 ) (37) = p(S 1 |S 3 )p(S 2 |S 3 )p(S 3 ) (38) ≥ µ 3 ,(39)
where the second equality follows from conditional independence and the inequality follows from (21) and p(S 3 ) = µ. The condition for S 1 and S 2 to be conditionally independent is derived in Section A.3 as ρ 12·3 = 0,
where ρ 12·3 is the partial correlation coefficient of Z 1 and Z 2 given Z 3 and defined in (72).
It seems plausible that (36) holds in general since, by (20), it is equivalent to
p(S 3 |S 1 ∩ S 2 ) ≥ µ,(41)
which says that the probability of a success conditional on two successes is at least as great as the unconditional probability of a success. However, we do not have a proof of (36) for arbitrary correlations. We can quantify how the lower bound on the no-failure probability changes if the inequality in [Jensen, 1969] extends to any three successes, or if (36) always holds. Redoing the derivation of the bivariate lower bound with (36) yields the following lower bound on the no-failure probability,
p(∩ ( n 2 ) i=1 S i ) ≥ 1 − 1 3 n 2 (1 − µ 3 ).(42)
The difference between this lower bound and the bivariate lower bound is n 2
1 3 µ 3 − 1 2 µ 2 + 1 6 ,(43)
which is positive, decreasing over µ ∈ [0, 1], and converges to zero as µ → 1. Hence, if the inequality in [Jensen, 1969] generalized to (36), then the lower bound on the no-failure probability would be improved relative to the bivariate lower bound. We leave the tasks of proving or disproving (36) and determining if there is further value in extending this argument to more than three successes as future work.
Selection of Embedding Dimension
A practical question is whether the smallest embedding dimension that leads to a positive no-failure probability is greater for the bivariate or marginal lower bounds. The bivariate lower bound is positive when
µ > 1 − 2 n 2 −1 ,(44)
and the marginal lower bound is positive when
µ > 1 − n 2 −1 ,(45)
so the goal is to compare the smallest values of k that satisfy these inequalities. To investigate this question, we compute the smallest embedding dimension that leads to a positive no-failure probability for five ϵ ∈ [0.01, 0.2] and two-hundred n ∈ [10 4 , 10 8 ]. The desired embedding dimensions are computed using bisection, and found to be the same for both bounds and all combinations of n and ϵ. That is, the bivariate lower bound does not lead to a smaller embedding dimension than the marginal lower bound, which is reasonable considering the small difference between the bounds. Figure 5 shows the computed embedding dimensions. In contrast to this numerical approach, the embedding dimension is often selected using an analytical lower bound derived with Markov's inequality [Dasgupta and Gupta, 2003, eqn. 2
.1], k ≥ 24 log(n) 3ϵ 2 − 2ϵ 3 .(46)
Previously, [Li et al., 2006a] and [Rojo and Nguyen, 2010] found that embedding dimensions found numerically can be approximately 13-40% lower than those found with analytical bounds. For the values of n and ϵ considered in this section, the value of k found numerically is approximately 80-92% of the value given by (46). [Li et al., 2006a] and [Rojo and Nguyen, 2010] appear to have considered different values of n and ϵ, which may account for the difference in these percentages.
Data-dependent Lower Bound on the No-Failure Probability
Random projections are used for dimension reduction in part because they are independent of the data, unlike matrix factorization methods like singular value decomposition, for example. Furthermore, their analysis is also usually independent of the data in that the embedding dimension and lower bounds on the no-failure probability do not depend on the data. Both the marginal lower bound and the bivariate lower bound derived in Section 3 are independent of the data being embedded, for example. However, a possibly greater data-dependent lower bound than the bivariate lower bound is given by (30),
p(∩ ( n 2 ) i=1 S i ) ≥ 1 − 1 2 ( n 2 ) i=1 (1 − p(S 2i−1 ∩ S 2i )) ,(47)
where the embedding dimension can be selected to ensure that the lower bound is positive. A similar data-dependent lower bound on the no-failure probability is not available from the derivation of the marginal lower bound. Actually computing this lower bound requires knowing the ρ ii ′ , and so may be impossible if the number of points to embed or the original dimension of the data is large. Also note that the indices of the lower bound can be permuted without changing the no-failure probability, so that the maximum can be taken over permutations of the indices.
Limitations
We briefly discuss two limitations of our result that also help situate this work in the broader context of random projection. First, the lower bound on the no-failure probability relied on the assumption that all of the elements of the projection matrix were Gaussian distributed. However, sparse, non-Gaussian random projections are often used since they can be implemented efficiently and approximately preserve distances [Achlioptas, 2003, Li et al., 2006b]. The bivariate lower bound derived here on the no-failure probability does not extend to these cases where pairs of projection errors do not have the bivariate gamma distribution. Furthermore, the improvement in the lower bound on the no-failure probability found here is so small that it does not support using a Gaussian projection instead of a sparse projection.
A second limitation is that we derive a lower bound on the no-failure probability using the exact value of the failure probability instead of a bound on the failure probability. This difference complicates a comparison of our work with the work of others on the JL lemma since using bounds obtained with Markov's inequality appears more common [Dasgupta and Gupta, 2003]. However, as suggested in [Li et al., 2006a, footnote 7] and [Rojo and Nguyen, 2010], the failure probability can be evaluated numerically and so does not need to be bounded using Markov's inequality. One advantage with having an algebraic expression for the bound on the failure probability is that inverting the bound yields a closedform expression for the embedding dimension. This advantage is not helpful here though since our main goal is to bound the no-failure probability for a fixed embedding dimension, and not to determine an embedding dimension.
Conclusion
This paper considered the application of the bivariate gamma distribution to the proof of the JL lemma. It was found that modifying the standard proof of the JL lemma to incorporate this distribution led to a small improvement in the lower bound on the no-failure probability. Specifically, we followed the standard approach of using Bonferroni's inequality to bound the probability of at least one failure, where a failure is a significantly distorted distance. We adapted this approach by applying Bonferroni's inequality to pairs of failures instead of single failures. An inequality from [Jensen, 1969] that applies to the bivariate gamma distribution of two projection errors was then used to bound the probability of two successes. One surprising aspect of this result is that it holds even if all of the pairwise differences between the points to be embedded are orthogonal. Further work is needed to see if the result can be extended to higher-dimensional gamma distributions, or if the bivariate lower bound obtained here is the best lower bound that can be found with our approach.
A Appendix
This appendix provides additional details on the bivariate gamma distribution of a pair of projection errors. Section A.1 gives a derivation of the joint characteristic function of V i and V i ′ , which is used to derive their covariance. Section A.2 then obtains a Gaussian approximation of the bivariate gamma distribution with the central limit theorem. This approximation helps evaluate the bivariate gamma distribution function when the embedding dimension is large. Section A.3 gives the joint distribution of all of the random projections and the conditional distribution of two random projections given a third.
A.1 Characteristic Function of the Bivariate Gamma Distribution
We now provide a derivation of the characteristic function of the bivariate gamma distribution and then use the function to derive the covariance of the distribution. The characteristic or moment generating function is also given in [Wicksell, 1933, eqn. 20], [Kibble, 1941], [Moran, 1967], and [Kotz et al., 2004, eqn. 48.12], for example. The more general characteristic or moment generating function of the multivariate gamma distribution can be found in [Krishnamoorthy and Parthasarathy, 1951, eqn. 2.3] and [Krishnaiah and Rao, 1961, eqn. 2].
The joint characteristic function of V i and V i ′ can be derived from (10). Since Z iℓ and Z i ′ ℓ ′ are independent if ℓ ̸ = ℓ ′ and the pairs (Z iℓ , Z i ′ ℓ ) are identically distributed, the joint characteristic function satisfies
ϕ ii ′ (t 1 , t 2 ) = [θ ii ′ (t 1 , t 2 )] k ,(48)
where θ ii ′ (t 1 , t 2 ) is the joint characteristic function of Z ii ′ = (Z i1 , Z i ′ 1 ) ′ and t 1 , t 2 ∈ R. The joint characteristic function of Z ii ′ is derived using standard algebraic manipulation and the 'integrate-to-one' property of the Gaussian distribution. Letting ι = √ −1 and
D = t 1 0 0 t 2 ,(49)
it follows that
θ ii ′ (t 1 , t 2 ) = E[exp(ι(t 1 Z 2 i1 + t 2 Z 2 i ′ 1 ))](50)= E[exp(ιZ ′ ii ′ DZ ii ′ )](51)= 1 2π|Σ ii ′ | 1/2 exp − 1 2 z ′ ii ′ Σ −1 ii ′ z ii ′ + ιz ′ ii ′ Dz ii ′ dz ii ′ (52) = 1 2π|Σ ii ′ | 1/2 exp − 1 2 z ′ ii ′ Σ −1 ii ′ − 2ιD z ii ′ dz ii ′ (53) = |R ii ′ | 1/2 |Σ ii ′ | 1/2 1 2π|R ii ′ | 1/2 exp − 1 2 z ′ ii ′ R −1 ii ′ z ii ′ dz ii ′ , R −1 ii ′ = Σ −1 ii ′ − 2ιD (54) = |I 2 − 2ιDΣ ii ′ | −1/2 · 1 (55) = ((1 − 2ιt 1 )(1 − 2ιt 2 ) + 4t 1 t 2 ρ 2 ii ′ ) −1/2 ,(56)
where the one in (55) is due to a bivariate normal density function being integrated over its support. The joint characteristic function ϕ ii ′ (t 1 , t 2 ) is given in [Jensen, 1969, eqn. 6] as an expression involving a matrix determinant,
ϕ ii ′ (t 1 , t 2 ) = (1 − 2ιt 1 )I k −2ιt 1 ρ ii ′ I k −2ιt 2 ρ ii ′ I k (1 − 2ιt 2 )I k −1/2 ,(57)
where a scaling by 2 has been applied since the joint characteristic function of V i /2 and V i ′ /2 was given originally. An equivalent expression for the joint characteristic function is ϕ ii ′ (t 1 , t 2 ) = |(I 2 − 2ιDΣ ii ′ ) ⊗ I k | −1/2
= |I 2 − 2ιDΣ ii ′ | −k/2 |I k | −2/2 (59) = ((1 − 2ιt 1 )(1 − 2ιt 2 ) + 4t 1 t 2 ρ 2 ii ′ ) −k/2 ,
which is equivalent to [θ ii ′ (t 1 , t 2 )] k , as expected. The second equality is an application of an identity for the determinant of a Kronecker product. The covariance of V i and V i ′ is given in (11) and derived as follows. First, E[V i V i ′ ] is computed from the characteristic function of V i and V i ′ ,
E[V i V i ′ ] = − d dt 1 d dt 2 ϕ(t 1 , t 2 ) t1=t2=0 (61) = k 2 + 2kρ 2 ii ′ .(62)
Then, since E[V i ] = E[V i ′ ] = k, it follows that the covariance of V i and V i ′ is
Cov[V i , V i ′ ] = E[V i V i ′ ] − E[V i ] E[V i ′ ](63)= 2kρ 2 ii ′ .(64)
The result is useful since it shows how the data, through the dot product ρ ii ′ , appears in the joint distribution of two projection errors. Taking k = 1 also implies that Cov[Z 2 iℓ , Z 2 i ′ ℓ ] = 2ρ 2 ii ′ , as noted in Section 2.2.
A.2 Gaussian Approximation of the Bivariate Gamma Distribution
We now detail the Gaussian approximation to the bivariate gamma distribution that is justified using the central limit theorem. The basic idea is to approximate the asymptotic distribution of the right side of (10) using the multivariate central limit theorem. Following Theorem 1.17 in [DasGupta, 2008], the multivariate central limit theorem asserts that
√ k 1 k V i V i ′ − 1 1 d → N (0 2 , Ω ii ′ )(65)
as k → ∞, where the convergence is in distribution and the asymptotic covariance matrix is the covariance matrix of (Z 2 i1 , Z 2 i ′ 1 ) ′ ,
Ω ii ′ = 2 2ρ 2 ii ′ 2ρ 2 ii ′ 2 .(66)
Hence, if k is large, then the joint distribution of V i and V i ′ can be approximated as
V i V i ′ · ∼ N 2 k k , kΩ ii ′ .(67)
Joint success probabilities can thus be computed by replacing the actual bivariate gamma distribution, H(v i , v i ′ ), with this approximate Gaussian distribution. For the results shown in this paper, we used the Gaussian approximation if k > 300 and evaluated the joint distribution function directly using numerical integration if k ≤ 300. This cut-off value of k was justified by comparing values of the joint success probability evaluated using the Gaussian approximation and integrating (14) directly. When k = 300, the difference in the two values is less than one in ten-thousand for values of the correlation parameter across (−1, 1) and ϵ ∈ {0.01, 0.025, 0.05}.
A.3 Joint Distribution of Random Projections
The n 2 random projections are jointly Gaussian distributed,
Z 1 . . . Z ( n 2 ) ∼ N (0 ( n 2 )k , Σ ⊗ I k ),(68)
where Σ is a correlation matrix of dimension n 2 × n 2 with (i, i ′ )th element ρ ii ′ . Since ρ ii = 1, the correlation matrix Σ ii ′ defined in (9) is the 2 × 2 submatrix formed by the i and i ′ th rows and columns of Σ.
It follows from (68) that the joint distribution of three random projections, say Z 1 , Z 2 , and Z 3 , is multivariate Gaussian, Z 1 Z 2 Z 3 ∼ N 0 3k , 1 ρ 12 ρ 13 ρ 12 1 ρ 23 ρ 13 ρ 23 1
⊗ I k .(69)
This implies that the conditional distribution of Z 1 and Z 2 given Z 3 is also multivariate Gaussian,
Z 1 Z 2 |Z 3 ∼ N ρ 13 ρ 23 ⊗ Z 3 , Σ 12·3 ⊗ I k ,(70)
where Σ 12·3 is the covariance matrix of Z 1ℓ and Z 2ℓ given Z 3ℓ and given by Σ 12·3 = 1 − ρ 2 13 ρ 12 − ρ 13 ρ 23 ρ 12 − ρ 13 ρ 23 1 − ρ 2 23 .
Note that Σ 12·3 is only a correlation matrix if the variances equal one. The conditional correlation is thus ρ 12·3 = ρ 12 − ρ 13 ρ 23 (1 − ρ 2 13 ) 1/2 (1 − ρ 2 23 ) 1/2 ,
which is called a partial correlation coefficient in the statistics literature [Rencher and Schaalje, 2008, sec. 4.5]. Hence, Z 1 and Z 2 are conditionally independent given Z 3 if and only if ρ 12·3 = 0.
:
The complement of S i , a 'failure' µ = p(S i ) : Success probability, independent of i p(∩ i S i ) : No-failure probability p(S i ∩ S i ′ ) : Joint success probability h(v i , v i ′ ): Joint probability density function of
Figure 1 :
1The success probability for different embedding dimensions and tolerances.
Figure 2 :
2The bivariate gamma density function for different values of the correlation parameter, ρ 2 ii ′ , and an embedding dimension of k = 50. The marginal densities are Γ(25, 2) for all ρ 2 ii ′ .
Figure 3 :
3A bivariate gamma distribution with k = 50 and ρ 2 ii ′ = 0.5. The red shaded area indicates the integration domain for computing the joint success probability. In each coordinate, the integration limits are (k(1 − ϵ), k(1 + ϵ)), where ϵ = 0.15. The red dot is at (k, k) and indicates the marginal means.
Figure 5 :
5The smallest embedding dimensions that lead to a positive no-failure probability using the bivariate or marginal lower bound.
follows from Bonferroni's inequality, (30) follows from De Morgan's laws, and (31) follows from (20). Taking the complement of the first expression and applying De Morgan's laws again gives
AcknowledgmentsThe authors thank Geoffrey Sanders for helpful conversations on this topic.DisclaimerThis document was prepared as an account of work sponsored by an agency of the United States government. Neither the United States government nor Lawrence Livermore National Security, LLC, nor any of their employees makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States government or Lawrence Livermore National Security, LLC. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States government or Lawrence Livermore National Security, LLC, and shall not be used for advertising or product endorsement purposes.Lawrence Livermore National Laboratory is operated by Lawrence Livermore National Security, LLC, for the U.S. Department of Energy, National Nuclear Security Administration under Contract DE-AC52-07NA27344.
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| {'fraction_non_alphanumeric': 0.05879742987198659, 'fraction_numerical': 0.034936650617060784, 'mean_word_length': 4.027178851714168, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 10, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 40, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Probabilistic proofs of the Johnson-Lindenstrauss lemma imply that random projection can reduce the dimension of a data set and approximately preserve pairwise distances. If a distance being approximately preserved is called a success, and the complement of this event is called a failure, then such a random projection likely results in no failures. Assuming a Gaussian random projection, the lemma is proved by showing that the no-failure probability is positive using a combination of Bonferroni's inequality and Markov's inequality. This paper modifies this proof in two ways to obtain a greater lower bound on the no-failure probability. First, Bonferroni's inequality is applied to pairs of failures instead of individual failures. Second, since a pair of projection errors has a bivariate gamma distribution, the probability of a pair of successes is bounded using an inequality from[Jensen, 1969]. If n is the number of points to be embedded and µ is the probability of a success, then this leads to an increase in the lower bound on the no-failure probability of 1 2 n 2 (1 − µ) 2 if n 2 is even and 1 2 n 2 − 1 (1 − µ) 2 if n 2 is odd. For example, if n = 10 5 points are to be embedded in k = 10 4 dimensions with a tolerance of ϵ = 0.1, then the improvement in the lower bound is on the order of 10 −14 . We also show that further improvement is possible if the inequality in[Jensen, 1969]extends to three successes, though we do not have a proof of this result.", 'arxivid': '2305.17123', 'author': ['Jason Bernstein \nLawrence Livermore National Laboratory Livermore\n94550CA\n', 'Alec M Dunton \nLawrence Livermore National Laboratory Livermore\n94550CA\n', 'Benjamin W Priest \nLawrence Livermore National Laboratory Livermore\n94550CA\n'], 'authoraffiliation': ['Lawrence Livermore National Laboratory Livermore\n94550CA', 'Lawrence Livermore National Laboratory Livermore\n94550CA', 'Lawrence Livermore National Laboratory Livermore\n94550CA'], 'corpusid': 257470126, 'doi': '10.2172/1959476', 'github_urls': [], 'n_tokens_mistral': 18941, 'n_tokens_neox': 16069, 'n_words': 10622, 'pdfsha': '91b84309d8bd6d622de197f766672f5a4f11cf3a', 'pdfurls': ['https://export.arxiv.org/pdf/2305.17123v1.pdf'], 'title': ['An Analysis of the Johnson-Lindenstrauss Lemma with the Bivariate Gamma Distribution', 'An Analysis of the Johnson-Lindenstrauss Lemma with the Bivariate Gamma Distribution'], 'venue': []} |
arxiv |
Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smoothed particle hydrodynamics
Wenxiao Pan
Michael Daily
Nathan A Baker
Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smoothed particle hydrodynamics
10.1186/s13628-015-0021-yPan et al. BMC Biophysics (2015) 8:7 RESEARCH ARTICLE Open AccessDiffusionSmoluchowski equationSmoothed particle hydrodynamicsProtein-ligand interactionsBinding ratesAcetylcholinesterase
Background: The calculation of diffusion-controlled ligand binding rates is important for understanding enzyme mechanisms as well as designing enzyme inhibitors. Methods: We demonstrate the accuracy and effectiveness of a Lagrangian particle-based method, smoothed particle hydrodynamics (SPH), to study diffusion in biomolecular systems by numerically solving the time-dependent Smoluchowski equation for continuum diffusion. Unlike previous studies, a reactive Robin boundary condition (BC), rather than the absolute absorbing (Dirichlet) BC, is considered on the reactive boundaries. This new BC treatment allows for the analysis of enzymes with "imperfect" reaction rates. Results: The numerical method is first verified in simple systems and then applied to the calculation of ligand binding to a mouse acetylcholinesterase (mAChE) monomer. Rates for inhibitor binding to mAChE are calculated at various ionic strengths and compared with experiment and other numerical methods. We find that imposition of the Robin BC improves agreement between calculated and experimental reaction rates. Conclusions: Although this initial application focuses on a single monomer system, our new method provides a framework to explore broader applications of SPH in larger-scale biomolecular complexes by taking advantage of its Lagrangian particle-based nature.
Background
In the "perfect" enzyme [1] acetylcholinesterase (AChE), the rate-limiting step for catalysis is diffusional encounter [2,3]. Specifically, the active site lies at the bottom of a 20 Å-deep gorge, and the diffusion of substrate into it is accelerated by electrostatic steering [4,5]. Its diffusionlimited behavior, complex geometry, and strong electrostatic influence has made AChE a useful target for both experimental and computational studies of biomolecular diffusion [4][5][6][7][8][9][10][11]. Two major classes of methods have been used to estimate diffusion rates in biomolecular systems. Mesoscopic coarse-grained methods like Monte Carlo [12][13][14], Brownian dynamics (BD) [8,9,15], and Langevin dynamics [16,17] simulations trace the trajectories of individual coarse-grained particles driven by Brownian motion. Such simulations typically consider dilute ligand concentrations so that electrostatic protein-ligand interactions can be modeled by the Poisson-Boltzmann equation [18,19] with a few notable exceptions [20]. Alternatively, continuum models can be used to treat the diffusion of ligand concentration in space around a biomolecule by the Smoluchowski equation [6,[21][22][23][24][25]. In particular, an adaptive finite element approach [26] has been used to numerically solve the Smoluchowski equation, and it shows higher accuracy in predicting experimental data about the ligand binding rates than the coarse-grained BD modeling [6]. For dilute ligand concentrations, electrostatic interactions can also be modeled with the Poisson-Boltzmann equation like the mesoscale approach [6,7]. However, for more concentrated ligand solutions, continuum models can also model the electrostatic potential near the biomolecular surface using a regularized Poisson-Nernst-Planck formulation [24,25], allowing screening of the ligand-receptor interactions by its time-dependent distribution around the protein.
Here, we follow the continuum approach but solve the Smoluchowski equation using a new smoothed particle hydrodynamics (SPH) method [27,28]. Unlike Eulerian grid-based methods such as finite element method (FEM), SPH is a Lagrangian particle-based method. SPH has been used with good accuracy for numerically solving partial differential equations (PDEs) describing momentum, mass and energy conservation laws [27]. In SPH, the domain is discretized into a set of "particles" that serve as interpolation points to numerically solve the governing PDEs. The SPH discretization of PDEs is based on a meshless interpolation scheme, which allows the PDEs to be written in the form of a system of ordinary differential equations (ODEs). SPH has a straightforward discretization without the need for time-consuming FEM mesh construction around complicated geometries such as biomolecules. Due to its Lagrangian nature, SPH has many advantages for modeling physical phenomena involving moving boundaries, large deformation of materials, multiphases, and advection-dominated diffusive transport [28][29][30]. Specifically, in SPH, free surfaces and interfaces between fluids move with particles, and hence, there is no need for front tracking schemes. And the non-linear advection term is embedded in the material derivative in the Lagrangian coordinate system, and hence, SPH models advection exactly. In addition, the similarity of SPH to molecular dynamics and mesoscopic coarse-grained particle methods (e.g., dissipative particle dynamics, BD, and Langevin dynamics), allows coupling of simulations across scales to build a multiscale modeling framework. This is our primary goal with the current work: to enable the multiscale and multiphysics description of biomolecular dynamics and ligand recognition. To the best of our knowledge, SPH has not been widely used in modeling biomolecular systems. Thus, in the present work, we aim to take the first step to introduce SPH into this field through the development of a SPH model for biomolecular diffusion with AChE as a test case.
In the SPH model, the Smoluchowski equation is numerically solved and the ligand binding rates are calculated from flux across the reactive boundary as in the previous studies using FEM [6,[21][22][23][24][25]. However, in the previous FEM studies, active sites were modeled using the absolute absorbing (Dirichlet) boundary condition (BC). This BC has a simple description on the reactive boundaries but assumes infinitely fast chemical reactions between the enzyme and the ligand; i.e., a "perfect enzyme". In our model, we take into account imperfect and non-instantaneous reactivity and thus solve the equation subject to a reactive (Robin) BC.
To solve the Smoluchowski equation subject to Robin BC using SPH, we use a continuum surface reaction method [31] which we have recently adapted to solve the Navier-Stokes equations subject to slip (Robin) boundary conditions [32]. In this formulation, the Robin BC is replaced by a reflective Neumann BC and a source term added into the governing equation. The derivation of the method is based on the approximation of the sharp boundary with a diffuse interface of finite thickness by means of a color function. This method is general for any arbitrary complex geometries and thus appropriate for modeling Robin BC in biomolecular systems with complex structures.
Results and discussion
Spherical test systems
Before the numerical method was applied to a biomolecular system with complicated geometry, we verified it on simple spherical test cases. Specifically, we considered a diffusing sphere with a radius R 1 . The entire domain was confined by the outer boundary b determined as a spherical surface with the radius of R 2 = 125 Å.
For the first test case, we let R 1 = 0 and assume no external potential, for which the time-dependent analytical solution of the Smoluchowski equation can be easily derived. Figure 1 compares the SPH numerical solutions with the analytical solution at different times. SPH solutions are compared at different resolutions and their corresponding L 2 errors are calculated relative to the analytical solution. Figure 1 shows that, even at the coarsest resolution ( x = 8 Å), the SPH solution agrees well with the analytical solution with about 3% relative error. This relative error is further reduced to 1% by increasing the resolution to x = 2 Å.
Next, the spherical system is assumed to have a Coulombic form of the PMF, i.e., W (r) = q/βr with +1 e charge. We set R 1 = 50 Å and impose either a Dirichlet BC as specified in Eq. 6 or a Robin BC as in Eq. 5. In these two tests, the corresponding SPH solutions of concentration at steady-state are compared with the analytical solutions. The converged SPH solutions are shown for the Dirichlet BC ( Figure 2) and Robin BC ( Figure 3) imposed on the inner spherical boundary (r = R 1 ). The reactive coefficient for the Robin BC is α = 1 × 10 3 . In both tests, the SPH solutions show very good agreement with the analytical solution even at the resolution of x = 8 Å, which can be further improved with increasing resolution to x = 2 Å. Moreover, at x = 2Å, the calculated reaction rate is 2.83 × 10 12 M −1 min −1 for the Dirichlet BC, and is 8.24 × 10 11 M −1 min −1 for the Robin BC, both with L 2 errors less than 3% relative to the analytically evaluated ones.
Application to AChE-ligand binding rates
We applied the SPH method to study the ligand binding kinetics of a simple spherical cationic ligand to the mouse acetylcholinesterase (mAChE) under various ionic strength conditions. Specifically, we performed the time- In previous studies by Song et al. [6], a simple but realistic set of boundaries was used inspired by Tara et al. [9], encompassing the active site as well as the gorge and the peripheral anionic site (PAS) of mAChE. We constructed these spherical active boundaries ( a ) at varying distances from the active site along an axis defined by the carbonyl carbon of S203 at the origin and the gorge. Spheres 1-6 were placed at 16.6, 13.6, 10.6, 7.6, 4.6, and 1.6 Å along the this axis, respectively. The outermost spheres 1 and 2 were assigned radii of 12 and 9 Å, respectively, while all others were given radii of 6 Å. Each reactive boundary N is defined as the intersection of the (surface) union of spheres N through 6 with the mAChE structure. Figure 4A shows the discretized domain with R 2 = 128 Å. Figure 4B and 4C depict the constructed reactive boundaries 1 and 4.
In most prior studies [6,7,23], an absolute absorbing (Dirichlet) BC (Eq. 6) was assumed. However, in the present work, we demonstrated improved performance with the reactive (Robin) BC (Eq. 5) imposed on the reactive boundaries. Figure 5 shows the steady-state spatial distribution of ligand throughout the simulation domain at different ionic strengths. At zero ionic strength, there are three large ligand-attracting regions, two on either side of the active site and one on the opposite side of the protein. There is also one ligand-depleted region at the top and another one near the opening of the gorge. At nonzero ionic strengths, electrostatic screening reduces the size of the ligand-enriched and ligand-depleted regions. However, a large region around the active site remains ligand-depleted at up to 0.50 M ionic strength. Figure 6 illustrates the temporal evolution of the concentration distribution as ligand moves inward in the bulk region from the outer boundary ( b ). The distribution has clearly reached steady state by 190 ns.
We calculated the reaction rates from these solutions according to Eq. 22. In Figure 7, the left panel shows the time evolution of reaction rate k on (t) on reactive boundary 1 at different ionic strengths. For this boundary, k on (t) converges within 150 ns for all ionic strengths. The right panel shows k on (t) on reactive boundaries 1-4, respectively, at 0.15 M ionic strength.
We have quantitatively compared the reaction rates calculated by SPH with experimental results [4] and previous computational studies by FEM [6,8]. Radic et al. [4] fit their experimentally measured reaction rates as a function of ionic strength using the Debye-Hückel limiting law
k on = k 0 on − k H on 10 −1.18|Z E Z I | √ I + k H on ,( 1 )
where I is the ionic strength, k 0 on is the effective reaction rate at zero ionic strength rate, k H on is the effective limiting reaction rate at infinite ionic strength and set to the value of k on calculated at 0.67 M ionic strength, z E is the effective enzyme charge, and z I is the effective inhibitor charge with a fixed value of +1 e. In SPH calculations with the Robin BC, the reaction coefficient α was varied, as shown in Figure 8, to identify the value of 8.0 × 10 3 which optimized agreement between computational and experimental results. Figure 9 and Table 1 compare the reaction rates from SPH, FEM [6,8], BD, and experimental data by Radic et al. [4]. As noted by Song et al. We also assessed the accuracy of SPH method for describing the ligand-binding kinetics of a mAChE surface mutant. We tested the surface hexa-mutant (E84Q, E91Q, D280V, D283N, E292Q, and D372N) from Radic et al. [4], which reduces the reaction rate by about a factor of 4 across the 0 to 0.67 M ionic strengths. For the mutants, which are nearly isosteric with the wild-type protein, we used the same SPH model as the wild type, but recalculated the electrostatic potentials for the mutant charge distribution. As presented in Table 1, the Robin BC SPH model has qualitative accuracy: predicting k 0 on of 2.23
Conclusions
The Robin BC offers a new way to incorporate reactive surfaces into continuum diffusion models for rate calculations. This Robin-based model incorporates a new parameter α, which has units of Å/μs and can be related to the probability of reaction within distance x to the boundary and time interval t by P = 1 − exp(−α t x ) [33]. Thus, α = 0 corresponds to zero reactivity (reflective Neumann BC) while α = ∞ corresponds to absolute reactivity (absorbing Dirichlet BC).
There are two possible origins for the differences between the current SPH model results and past FEM calculations using the Dirichlet BC. First, the current SPH work uses a more recent mAChE structure (4B82) while the previous FEM calculations used an older structure (1MAH). Second, our SPH model uses a fixed resolution uniformly on both solution domain and boundaries, while the FEM adaptively meshes the reactive boundary with higher resolution. This work has provided an initial demonstration that the Lagrangian (particle-based) SPH method out-performs the Eulerian (grid-based) FEM [6] in accurately predicting ligand binding rates in AChE. This result is important because while both methods can be used to study molecules of the size of AChE, SPH is more scalable to larger systems such as the synapse geometry where AChE operates. Additionally, due to its Lagrangian nature, SPH can easily incorporate other physical phenomena such as fluid flow or protein flexibility.
We have demonstrated that superior performance can be achieved using a probabilistic reactive (Robin) BC rather than a simple Dirichlet BC. In fact, the Robin BC is likely more biologically relevant than the Dirichlet BC. While the AChE enzyme is considered nearly "perfect" with a diffusion-limited reaction rate, there is experimental evidence that a very small fraction of substrates entering the active site gorge do not react. Specifically, recent kinetic experiments suggest that through unknown mechanisms, the PAS limits the rate of progression of non-substrates of any size to the catalytic site [34]. In addition, molecular dynamics simulations suggest that the PAS provides a selective gating function, for example by fluctuations in the gorge width that are likely to let acetylcholine but not let larger molecule pass through [35,36].
Methods
Governing equation and boundary conditions
The time-dependent Smoluchowski equation can be written as: dp(x, t) dt
= ∇ · J(x, t), x ∈ ,( 2 )
where p(x, t) is the concentration distribution of the reactants, and the concentration flux J(x, t) is defined as:
J(x, t) = D(x)[ ∇p(x, t) + βp(x, t)∇W (x)] ,(3)
where D(x) is the diffusion coefficient; for simplicity, it is assumed to be constant. β = 1/k B T is the inverse Boltzmann energy with the Boltzmann constant k B and kinetic temperature T. W (x) is the potential mean force (PMF) for the diffusing particle due to solvent-mediated interactions with the target molecule. The equation is solved in a three-dimensional domain , subject to the following boundary conditions. First, Figure 9 Reaction rates of mAChE on reactive boundary 1 obtained from different methods. Black: from experimental data [4] (symbol) and fitted (line) to the Debye-Hückel limiting law (Eq. 1); blue: from BD [9]; red: from FEM with Dirichlet BC [6]; green: from SPH with Dirichlet BC; magenta: from SPH with Robin BC using α = 8 × 10 3 . For standardization, both computed and experimental data are fitted to the Debye-Hückel limiting law. specifying a Dirichlet BC on the outer boundary b where the concentration is equal to a bulk concentration p bulk . The outer boundary is often a spherical surface with a radius chosen to ensure that the ligand-protein potential is spherically symmetric and/or can be approximated analytically [6]. For the current study with mAChE, this outer boundary has radius R 2 ≈ 128Å as determined following a procedure similar to Song et al. and Chen et al. [6,23]. Also following Song et al. and Chen et al., p is normalized such that p bulk = 1.
p(x, t) = p bulk for x ∈ b ,( 4 )
The active site boundary a was modeled using either reactive Robin or absolute absorbing Dirichlet BC:
n(x) · J(x, t) = αp(x, t) for x ∈ a ,( 5 )
or
p(x) = 0 for x ∈ a ,( 6 )
respectively. The coefficient α is chosen to model an intrinsic reaction rate for the active site. Finally, a reflective Neumann BC is defined on the non-reactive boundary of molecule Figure 10 shows the simulation domain along with all boundaries. Given a solution to Eq. 2, the reaction rate is calculated from the integral of the flux across the reactive boundary [37]:
n(x) · J(x, t) = 0 for x ∈ m .( 7 )k on = p −1 bulk a n(x s ) · J(x s , t)dx s .( 8 )
In order to solve the Smoluchowski equation (Eq. 2) subject to the reactive Robin BC (Eq. 5), the simulation domain is extended to include a sub-domain a that is separated from by a , and we then reformulate Eq. 2 as:
dp r (x, t) dt =∇ · D(x)[ ∇p r (x, t) + βp r (x, t)∇W (x)] −αp r (x, t) a [ n(x) + n(x )] ·∇ x w(x − x , h r )dx , x ∈ ,(9)
subject to the reflective Neumann BC:
n(x) · J r (x, t) = 0 for x ∈ a .(10)
The derivation of Eq. 9 is detailed in Appendix A, which demonstrates
lim h r →0 p r (x, t) = p(x, t).(11)
In Eq. 9, the normalized kernel function, w(x), is a positive bell-shaped function with at least first continuous derivative and compact support κh r such that w(|r| > κh r ) = 0. The value of κ depends on the specific functional form of w(x), which is specified in Section 'SPH discretization of equations and boundary conditions' . In particular, w(x) satisfies the following conditions:
∪ a w(x − x , h r )dx = 1(12)
and
lim h r →0 w(x − x , h r ) = δ(x − x ).(13)
The normal unit vector n in Eq. 9 can be found in terms of a smoothed color functionφ as defined in Appendix A:
n(x) = ∇φ(x) |∇φ(x)| , x ∈ ∪ a .(14)
SPH discretization of equations and boundary conditions
In this section, we present SPH discretization of the Smoluchowski equation, using Eq. 2 if the Dirichlet BC is used and Eq. 9 if the Robin BC is assumed. To simplify notation, we omit superscript r for the variables in Eq. 9 in the subsequent derivations. The domain , and the boundaries a and b (extended as domains a and b respectively), are discretized with a set of N points with positions denoted by a vector r i (i = 1, ..., N). The points (which are commonly referred to as particles in SPH) are used to discretize and solve the governing equation. Initially, the particles are distributed uniformly (e.g., placed on a regular cubic lattice) with d i as the prescribed number density at r i . The discretization is based on a meshless interpolation scheme:
A i ≈ j A j d j w(r ij , h),(15)
where, A i = A(r i ) is a function defined at particle i, A j = A(r j ) is the function defined at particle i's neighboring particles j with distances r ij = r i − r j , and w(r ij , h) is the weighting kernel function. The interpolation scheme assumes a summation over all neighboring SPH particles but, due to the compact support of w, only particles within distance κh from r i have a non-zero contribution to the summation. Spatial derivatives of A can be calculated as
∇ i A i ≈ j A j d j ∇ i w(r ij , h).(16)
In the present work, we use a cubic spline kernel as the weighting function
w(r, h) = 1 πh 3 ⎧ ⎪ ⎨ ⎪ ⎩ 1 − 3 2 q 2 + 3 4 q 3 0 ≤ q ≤ 1 1 4 (2 − q) 3 1 < q ≤ 2 0 q > 2,(17)
where q = |r|/h. With this form of weighting function, only particles within 2h distance from particle i contribute to the summations in the SPH equations. We have chosen h = 1.3 x where x is the size of the cubic lattice. The SPH approximation of functions and their spatial derivatives allows the Smoluchowski equation subject to the Dirichlet BC (Eq. 2) to be written as a ODE governing the evolution of concentration on particle i as:
dp i dt = j∈ ∪ b ∪ a D i + D j d j (p i − p j ) 1 r ij dw(r ij , h) dr ij +β j∈ D i p i +D j p j d j (W i −W j ) 1 r ij dw(r ij , h) dr ij .(18)
The derivations of the first and second terms on the right-hand side of Equation 18 can be found in Monaghan et al. [27] where r ij is the magnitude of the vector r ij . If the reactive Robin BC on reactive boundary is imposed, Equation 9 is then solved instead and its corresponding SPH discretization form is:
dp i dt = j∈ ∪ b D i + D j d j (p i − p j ) 1 r ij dw(r ij , h) dr ij +β j∈ D i p i + D j p j d j (W i − W j ) 1 r ij dw(r ij , h) dr ij −αp i k∈ a n i + n k d k · r ik r ik dw(r ik , h r ) dr ik .(19)
To integrate the SPH Eqs. 18 and 19, an explicit Verlet scheme [38] is employed. The last term in Eq. 19 is obtained by discretizing the integral in Eq. 9 as a Riemann sum:
αp(x, t) a [ n(x) + n(x )] ·∇ x w(x − x , h r )dx = αp(x, t) k∈ a V k [ n(x) + n k ] ·∇ x w(x − r k , h r ),(20)
where V k = 1 d k is the volume of particle k and k∈ a is the summation over the reactive boundary particles within 2h r distance from particle i. The SPH expression for calculating the normal unit vector is obtained as:
n i = j∈ ∪ a 1 d j (φ j − φ i )∇ i w(r ij , h r ) j∈ ∪ a 1 d j (φ j − φ i )∇ i w(r ij , h r ) .(21)
In the simulations presented below, we set h r = h but it could be set differently in future applications.
Note that the reflective Neumann BC (Equation 7 or 10) can be simply enforced in SPH by excluding the contribution from the boundary particles in the summation. The Dirichlet BC (Equation 4 or 6) is enforced by assigning the fixed boundary value of concentration on the boundary particles. If the Robin BC is imposed, the reaction rate k on (t) can be calculated by Equation 42 in Appendix B and its corresponding SPH discretization form is:
k on = i∈ αp i ⎡ ⎣ k∈ a n i + n k d k · r ik r ik dw(r ik , h) dr ik ⎤ ⎦ .(22)
Otherwise, when the Dirichlet BC is enforced on the reactive boundary, the discretization of Eq. 41 in Appendix B is:
k on = i∈ (n i ·J i ) ⎡ ⎣ k∈ a n i + n k d k · r ik r ik dw(r ik , h) dr ik ⎤ ⎦ ,(23)
where
J i = D i j∈ ∪ b ∪ a p j − p i d j r ij r ij dw(r ij , h) dr ij + βD i p i j∈ W j − W i d j r ij r ij dw(r ij , h) dr ij .(24)
Calculation of potentials of mean force
We calculated the potential of mean force W (x) using the recently published 2.1 Å resolution structure of mAChE [39]. To prepare this structure for the calculation, we assigned titration states of ionizable residues using PROPKA [40] at pH 7, and we used PDB2PQR v1.8 [41,42] to assign atomic radii and charges. APBS v1.4 was used to perform a nonlinear Poisson-Boltzmann multi-grid calculation of the electrostatic potential over the entire simulation domain [43]. The small and large domains were set to 600 Å and 400 Å, respectively, with a fine grid spacing of 0.600 Å. For APBS calculations, we used the single Debye-Hückel boundary condition, a smoothed molecular surface, and protein and solvent dielectrics of 2 and 78.54, respectively. Atomic charges were mapped onto the grids using cubic B-spline discretization. The calculated potential was mapped onto the SPH discretization points of protein and solvent via trilinear interpolation.
Appendix A: continuum surface reaction method
In this appendix, we present a detailed derivation of the continuum surface reaction method for solving the Smoluchowski equation subject to Robin BC. We start from a two-sided problem; i.e., the concentration field p(x, t) is extended into the sub-domain a that is separated from by a such that Eq. 2 can be approximated as
dp r (x, t) dt = ∇ · D(x)[ ∇p r (x, t) + βp r (x, t)∇W (x)] − P (x, t) for x ∈ ∪ a(25)
subject to
n(x s )·[ J r (x s , t)| x s ∈ F a − J r (x s , t)| x s ∈ S a ] = 0 for x s ∈ a ,(26)
where F a and S a are the two sides of a , respectively. The boundary condition Eq. 26 emphasizes that the extended concentration field is continuous across a . Comparison of the weak formulations of Eq. 2 subject to Eq. 5 and Eq. 25 subject to Eq. 26 yields the relationship
This weak formulation is obtained by integrating the equations over their respected domains and then applying Gauss' theorem with the corresponding boundary conditions. To derive the formulation of P , we define a color function (i.e., a sharp characteristic function) as:
φ(x) = 0, x ∈ , 1, x ∈ a .(28)
and its smooth counterpart as
φ(x) = ∪ a φ(x )w(x − x , h r )dx .(29)
The gradient ofφ can then be found from Eq. 29 as
∇φ(x) = ∪ a φ(x )∇ x w(x − x , h r )dx .(30)
Using the definition of the surface delta function [44]:
δ[ n(x s )·(x s −x)] = n(x)·∇φ(x), x ∈ ∪ a , for x s ∈ a , (31) and noting that
lim h r →0φ = φ,(32)
we can rewrite the surface delta function in terms ofφ as:
δ[ n(x s )·(x s −x)]= n(x)· lim h r →0
∇φ(x), for x ∈ ∪ a , x s ∈ a .
The surface integral can then be rewritten as a volume integral through the surface delta function:
a αp(x s , t)dx s = ∪ a αp(x, t)δ[ n(x s ) · (x s − x)] dx, for x s ∈ a , = ∪ a αp r (x, t)n(x) · ∇φ(x)dx.(34)
To uniquely define P (x, t), we require it to vanish at a normal distance greater than h r from a and require that
lim h r →0 ∪ a P (x, t)dx = a αp(x s , t)dx s .(35)
Comparing Eqs. 34 and 35 yields an expression for P (x, t) as:
P (x, t) = αp r (x, t)n(x) · ∇φ(x), for x ∈ ∪ a . (36) Eq. 25 can then be rewritten by combining Eqs. 36 and 30 as:
dp r (x, t) dt = ∇ · D(x)[ ∇p r (x, t) + βp r (x, t)∇W (x)] −αp r (x, t) ∪ a n(x)·[ φ(x )∇ x w(x−x , h r )] × dx , for x ∈ ∪ a .(37)
Since p r is not uniquely defined on a , we introduce a one-sided formulation by approximating Eq. 37 as:
dp r (x, t) dt = ∇ · D(x)[ ∇p r (x, t) + βp r (x, t)∇W (x)] −αp r (x, t) ∪ a [n(x) + n(x )] ·[ φ(x )∇ x w(x − x , h r )] × dx , for x ∈ ,(38)
subject to the reflective Neumann BC (Eq. 10). Note that φ is non-zero only in a , where it is equal to 1 as defined in Eq. 28. Thus, the modified governing equation takes its final form as Eq. 9.
*
Correspondence: [email protected] 3 Computational and Statistical Analytics Division, Pacific Northwest National Laboratory, PO Box 999, MSID K7-20, 99352 Richland, WA, USA Full list of author information is available at the end of the article
Figure 1
1Comparison of SPH solutions to the analytical solutions for the Smoluchowski equation subject to the Dirichlet BC on r = R 2 at different times with the relative L 2 errors for different resolutions. Specifically, L 2 = 0.0326 for x = 8Å(green square), L 2 = 0.0180 for x = 4Å(blue circle), and L 2 = 0.0103 for x = 2Å(red triangle).
Figure 2
2Comparison of SPH solutions to the analytical solution for the Smoluchowski equation subject to the Dirichlet BC on both r = R 1 and r = R 2 at steady-state with the relative L 2 errors for different resolutions. Specifically, L 2 = 0.0666 for x = 8Å(green square), L 2 = 0.0321 for x = 4Å(blue circle), and L 2 = 0.0153 for x = 2Å(red triangle).
Figure 3
3Comparison of SPH solutions (symbol) to the analytical solution (line) for the Smoluchowski equation subject to the Robin BC on r = R 1 and Dirichlet BC on r = R 2 at steady-state with the relative L 2 errors for different resolutions. Specifically, L 2 = 0.00914 for x = 8Å(green square), L 2 = 0.00598 for x = 4Å(blue circle), and L 2 = 0.00377 for x = 2Å(red triangle). dependent calculations at ionic strengths of 0.0, 0.05, 0.10, 0.15, 0.20, 0.50 and 0.67 M until the diffusion reaches the steady-state. To achieve the highest accuracy with affordable computational cost, a resolution of x = 2 Å was used in all the following calculations.
Figure 4
4Panel A shows the discretized domain with R 2 = 128 Å and the mAChE molecule in the center with the reactive boundary shown in purple. Light blue indicates the outer boundary (R 2 ), blue the solvent, green the protein, and magenta the first (outermost) reactive boundary. Panels B and C show reactive boundaries 1 and 4, respectively in magenta spheres.
[6], BD simulations systematically overestimate the experimental k on , while the FEM produces good agreement with experimental k on at RMSD = 0.37 M −1 min −1 . With the Dirichlet BC, SPH predicts k on with RMSD of 0.57 M −1 min −1 , intermediate between FEM and BD results. However, with the Robin BC, SPH predicts k on with RMSD of 0.33 M −1 min −1 , better than the FEM and BD results.
Figure 5
5Contour of concentration distribution around mAChE (shown in dark gray) with the Robin BC (α = 8 × 10 3 ) on reactive boundary 1 at steady state with a range of ionic strengths. Reactive boundary 1 is shown in purple.
Figure 6
6Time evolution of the concentration distribution around mAChE (shown in gray) with a Robin BC (α = 8 × 10 3 ) on reactive boundary 1 at 0.15 M ionic strength. Reactive boundary 1 is shown in purple. compared to 1.80 from Radic et al. [4] with a k on RMSD of 0.88 over the entire ionic strength range studied.
Figure 7 (
7Left) k on as a function of t on reactive boundary 1 at different ionic strengths. Black square: 0.05M; red right-pointing triangle: 0.10M; blue asterisk: 0.15M; green circle: 0.20M; magenta diamond: 0.50M; cyan triangle: 0.67M. (Right) k on as a function of t on reactive boundaries 1-4, respectively, at 0.15 M ionic strength. Black square: reactive boundary 1; red circle: reactive boundary 2; blue diamond: reactive boundary 3; green triangle: reactive boundary 4.
Figure 8
8Root mean square deviation (RMSD) of computed by SPH to experimental reaction rates (over 0-0.67 M ionic strengths) vs. α for the Robin BC.
Figure 10
10Illustration of the simulation domain and all boundaries: b indicates the outer boundary, m the molecular surface, and a the reactive boundary 1; indicates the problem domain between b and a ∪ m .
P
(x, t)dx = a αp(x s , t)dx s .
Table 1
1Comparison of Debye-Hückel fits vs. ionic strength between experiment and simulations min -1 . And the error is the standard deviation of parameter fits using nonlinear least squares.Data
k 0
on
k H
on
Z E
RMSD exp
Experimental [4]
9.8 ± 0.6
1.30
2.3± 0.2
0.00
BD
9.1 ± 0.3
2.66
1.7± 0.2
1.52
SMOL FEM [7]
9.5 ± 0.1
1.92
2.8± 0.1
0.37
SPH (Dirichlet)
10.2 ± 0.1 2.19
2.9 ± 0.1
0.57
SPH (Robin, α = 8 × 10 3 )
9.6± 0.1
1.88
3.0± 0.1
0.33
Hexa-mutant (experiment)
1.8 ± 0.1
0.57
1.2± 0.2
0.00
Hexa-mutant (SPH, Robin) 2.23 ± 0.00 1.77 2.37 ± 0.02
0.88
RMSD exp of simulation results to experimental k on is calculated over the range
of ionic strengths between 0 and 0.67 M. The unit of k 0
on , k H
on and RMSD is
10 11 M -1
AcknowledgmentsThe authors gratefully acknowledge the funding support by the Applied Mathematics Program within the U.S. Department of Energy Office of Advanced Scientific Computing Research as part of the Collaboration on Mathematics for Mesoscopic Modeling of Materials (CM4). The research was performed using PNNL Institutional Computing at Pacific Northwest National Laboratory. The Pacific Northwest National Laboratory is operated for the U.S. Department of Energy by Battelle under Contract DE-AC06-76RL01830. This research was also supported by National Institutes of Health grant GM069702 to NAB.Appendix B: calculation of reaction rateSimilar to the derivation in Eq. 34, using the definition of the surface delta function and given p bulk = 1, the expression for the reaction rate can be rewritten asSubstituting Eq. 30 into the above equation and using Eq. 28, a new expression of k on can be obtained:Similar to Eq. 38, the corresponding one-sided formulation is:If the Robin BC (Eq. 5) is enforced, Eq. 41 can be reduced toCompeting interestsThe authors declare that they have no competing interests.Authors' contributionsWP implemented the new SPH methods, performed simulations, and participated in the writing of the manuscript. MD performed simulations, and participated in the writing of the manuscript. NAB conceived of the study, participated in its design and coordination, and helped to draft the manuscript. All authors read and approved the final manuscript.
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| {'fraction_non_alphanumeric': 0.061315427421507626, 'fraction_numerical': 0.04345058420090762, 'mean_word_length': 4.262589150637562, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 21, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Background: The calculation of diffusion-controlled ligand binding rates is important for understanding enzyme mechanisms as well as designing enzyme inhibitors. Methods: We demonstrate the accuracy and effectiveness of a Lagrangian particle-based method, smoothed particle hydrodynamics (SPH), to study diffusion in biomolecular systems by numerically solving the time-dependent Smoluchowski equation for continuum diffusion. Unlike previous studies, a reactive Robin boundary condition (BC), rather than the absolute absorbing (Dirichlet) BC, is considered on the reactive boundaries. This new BC treatment allows for the analysis of enzymes with "imperfect" reaction rates. Results: The numerical method is first verified in simple systems and then applied to the calculation of ligand binding to a mouse acetylcholinesterase (mAChE) monomer. Rates for inhibitor binding to mAChE are calculated at various ionic strengths and compared with experiment and other numerical methods. We find that imposition of the Robin BC improves agreement between calculated and experimental reaction rates. Conclusions: Although this initial application focuses on a single monomer system, our new method provides a framework to explore broader applications of SPH in larger-scale biomolecular complexes by taking advantage of its Lagrangian particle-based nature.', 'arxivid': '1501.04240', 'author': ['Wenxiao Pan ', 'Michael Daily ', 'Nathan A Baker '], 'authoraffiliation': [], 'corpusid': 14258757, 'doi': '10.1186/s13628-015-0021-y', 'github_urls': [], 'n_tokens_mistral': 16042, 'n_tokens_neox': 13270, 'n_words': 7789, 'pdfsha': 'cdbace53e7c88e48614962da9be0bde455ef4003', 'pdfurls': None, 'title': ['Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smoothed particle hydrodynamics', 'Numerical calculation of protein-ligand binding rates through solution of the Smoluchowski equation using smoothed particle hydrodynamics'], 'venue': []} |
arxiv |
Article no.JAMCS.38773 Foukzon and Men'kova
2018. 2018
Jaykov Foukzon
Israel Institute of Technology
HaifaIsrael
Elena Men'kova
All-Russian Research Institute for Optical and Physical Measurements
MoscowRussia
Article no.JAMCS.38773 Foukzon and Men'kova
Journal of Advances in Mathematics and Computer Science
2622018. 201810.9734/JAMCS/2018/38773Received: 4 th November 2017 Accepted: 8 th January 2018 Original Research Article Published: 30 th January 2018(Past name: British Journal of Mathematics & Computer Science, Past ISSN: 2231-0851) Authors' contributions This work was carried out in collaboration between both authors. Author JF designed the study, carried out the model analysis and wrote the first draft of the manuscript. Author EM wrote Section 3 of the manuscript and managed the literature searches. Both authors read and approved the final manuscript. Article Information/38773 Editor(s): (1) Dariusz Jacek Jakbczak, Assistant Professor, Chair of Computer Science and Management in this Department, Technical University of Koszalin, Poland. Reviewers: (1) Xiaolan Liu, Sichuan University of Science Engineering, China. (2) Derya Doan Durgun, Manisa Celal Bayar University, Turkey. Complete Peer review History: http://www.sciencedomain.org/review-history/22940Gödel encodingRussell's paradoxstandard modelHenkin semanticsinaccessible cardinal 2010 Mathematics Subject Classification: 53C2583C0557N16
In this paper we view the first order set theory ZF C under the canonical first order semantics and the second order set theory ZF C2 under the Henkin semantics. Main results are: (i) Let M ZF C st be a standard model of ZF C, then ¬Con(ZF C + ∃M ZF C st ). (ii) Let M ZF C 2 st be a standard model of ZF C2 with Henkin semantics, then ¬Con(ZF C2 + ∃M ZF C 2 st ). (iii) Let k be inaccessible cardinal then ¬Con(ZF C + ∃κ). In order to obtain the statements (i) and (ii) examples of the inconsistent countable set in a set theory ZF C + ∃M ZF C st and in a set theory ZF C2 + ∃M ZF C 2 st were derived. It is widely believed that ZF C + ∃M ZF C st and ZF C2 + ∃M ZF C 2 st are inconsistent, i.e. ZF C and ZF C2 have a standard models. Unfortunately this belief is wrong.We call ⟨U, S⟩ the Henkin model, if ⟨U, S⟩ satisfies the axioms of DED2 and truth in ⟨U, S⟩ is preserved by the rules of DED2. We call this semantics of second-order logic the Henkin semantics and second-order logic with the Henkin semantics the Henkin second-order logic. There is a special class of Henkin models, namely those ⟨U, S⟩ where S is the set of all subsets of A.We call these full models. We call this semantics of second-order logic the full semantics and secondorder logic with the full semantics the full second-order logic.Remark 1.1.3. We emphasize that the following facts are the main features of second-order logic:1.The Completeness Theorem: A sentence is provable in DED2 if and only if it holds in allHenkin models [2], [6].2.The Löwenheim-Skolem Theorem:A sentence with an infinite Henkin model has a countable Henkin model.3.The Compactness Theorem:A set of sentences, every finite subset of which has a Henkin model, has itself a Henkin model.The Incompleteness Theorem:Neither DED2 nor any other effectively given deductive calculus is complete for full models, that is, there are always sentences which are true in all full models but which are unprovable.
Introduction
Main results
Let us remind that accordingly to naive set theory, any definable collection is a set. Let R be the set of all sets that are not members of themselves. If R qualifies as a member of itself, it would contradict its own definition as a set containing all sets that are not members of themselves. On the other hand, if such a set is not a member of itself, it would qualify as a member of itself by the same definition. This contradiction is the Russell's paradox. In 1908, two ways of avoiding the paradox were proposed, the Russell's type theory and the Zermelo set theory, the first constructed axiomatic set theory. Zermelo's axioms went well beyond Frege's axioms of extensionality and unlimited set abstraction, and evolved into the now-canonical Zermelo-Fraenkel set theory ZF C. "But how do we know that ZF C is a consistent theory, free of contradictions? The short answer is that we don't; it is a matter of faith (or of skepticism)"-E. Nelson wrote in his paper [1]. However, it is deemed unlikely that even ZF C2 which is significantly stronger than ZF C harbors an unsuspected contradiction; it is widely believed that if ZF C and ZF C2 were inconsistent, that fact would have been uncovered by now. This much is certain -ZF C and ZF C2 is immune to the classic paradoxes of naive set theory: the Russell's paradox, the Burali-Forti paradox, and Cantor's paradox. Remark 1.1.1. Note that in this paper we view (i) the first order set theory ZF C under the canonical first order semantics, (ii) the second order set theory ZF C2 under the Henkin semantics [2], [3], [4], [5], [6]. Remark 1.1.2. Second-order logic essantially differs from the usual first-order predicate calculus in that it has variables and quantifiers not only for individuals but also for subsets of the universe and variables for n-ary relations as well [2], [6]. The deductive calculus DED2 of second order logic is based on rules and axioms which guarantee that the quantifiers range at least over definable subsets [6]. As to the semantics, there are two tipes of models: (i) Suppose U is an ordinary first-order structure and S is a set of subsets of the domain A of U. The main idea is that the set-variables range over S, i.e.
5.
Failure of the Compactness Theorem for full models.
6.
Failure of the Löwenheim-Skolem Theorem for full models.
7.
There is a finite second-order axiom system Z2 such that the semiring N of natural numbers is the only full model (up to isomorphism) of Z2.
8.
There is a finite second-order axiom system RCF2 such that the field R of real numbers is the only (up to isomorphism) full model of RCF2. Remark 1.1.4. For let second-order ZF C be, as usual, the theory that results obtained from ZF C when the axiom schema of replacement is replaced by its second-order universal closure, i.e.
∀X [F unc (X) =⇒ ∀u∃ν∀r [r ∈ ν ⇐⇒ ∃s (s ∈ u ∧ (s, r) ∈ X)]] , (1.1.1)
where X is a second-order variable, and where F unc (X) abbreviates " X is a functional relation", see [7]. is inconsistent, then we won't believe ZF C Hs 2 -proofs. What's slightly more subtle is that the mere consistency of ZF C2 isn't quite enough to get us to believe arithmetical theorems of ZF C Hs 2 ; we must also believe that these arithmetical theorems are asserting something about the standard naturals. It is "conceivable" that ZF C Hs
} . (2.1.2) (iv) For any Ψ (X) ∈ Γ Hs X let [Ψ (X)] Hs { Φ (X) ∈ Γ Hs X |Ψ (X) ∼ Φ (X)
} denotes the equivalence class to which Ψ (X) belongs. All elements of Γ Hs X equivalent to each other are also elements of the same equivalence class. Definition 2.1.2. [9]. Let T h be any theory in the recursive language L Th ⊃ LPA, where LPA is a language of Peano arithmetic.
We say that a number-theoretic relation R (x1, ..., xn) of n arguments is expressible in T h if and only if there is a wff R (x1, ..., xn) of T h with the free variables x1, ..., xn such that, for any natural numbers k1, ..., kn, the following hold:
(i) If R (k1, ..., kn) is true, then ⊢ T h R ( k1, ..., kn ) ; (ii) If R (k1, ..., kn) is false, then ⊢ T h ¬ R ( k1, ..., kn ) .(y, v) ⇐⇒ ∃!Ψ (X) [( g ZF C Hs 2 (Ψ (X)) = y ) ∧ ( g ZF C Hs 2 (X) = ν )] , (2.1.3)
where Ψ (X) is a unique wff of ZF C Hs 2 which contains free occurrences of the variable X with Gödel number v. We denote a unique wff Ψ (X) defined by using equivalence (1.2.3) by symbol Ψy,ν (X) , i.e.
Fr
Hs
2 (y, v) ⇐⇒ ∃!Ψy,ν (X) [( g ZF C Hs 2 (Ψy,ν (X)) = y ) ∧ ( g ZF C Hs 2 (X) = ν )] , (2.1.4) (v) Let ℘ Hs 2 (y, v, ν1
) be a Gödel number of the following wff:
∃!X [Ψ (X) ∧ Y = X] , where g ZF C Hs 2 (Ψ (X)) = y, g ZF C Hs 2 (X) = ν, g ZF C Hs 2 (Y ) = ν1.∀Y { Y ∈ ℑ Hs 2 ⇐⇒ ∃Ψ (X) [( [Ψ (X)] Hs ∈ Γ * Hs X / ∼X ) ∧ (Y = X) ]} , (2.1.11)
where the countable collection Γ * Hs
X / ∼X is defined by ∀Ψ (X) { [Ψ (X)] ∈ Γ * Hs X / ∼X ⇐⇒ [( [Ψ (X)] ∈ Γ Hs X / ∼X ) ∧ ∃!XΨ (X) ]} (2.1.12) Definition 2.1.8. Let ℜ Hs 2
be the countable collection of the all sets such that
∀X ( X ∈ ℑ Hs 2 ) [ X ∈ ℜ Hs 2 ⇐⇒ X / ∈ X ] .(Ψ ( Z, ℑ Hs 2 ) ∀X ( X ∈ ℑ Hs 2 ) [X ∈ Z ⇐⇒ X / ∈ X] .
From ( for the reason that the countable collection ℑ Hs 2 is not a set in the sense of the set theory ZF C Hs 2 .
In order to obtain a contradiction inside ZF C
ν / ∼ν ) ∧ ( g ZF C Hs 2 (X) = ν ) ∧ Y = X ]} . (2.1.17)
Note that from the axiom schema of replacement (1.1.1) it follows directly that ℑ * Hs 2 is a set in the sense of the set theory ZF C Hs 2 . Definition 2.1.11. We define now the countable set ℜ * Hs
2 by formula ∀X ( X ∈ ℑ * Hs 2 ) [ X ∈ ℜ * Hs 2 ⇐⇒ X / ∈ X ] . (2.1.18)
Note that from the axiom schema of separation it follows directly that ℜ * Hs 2 is a set in the sense of the set theory ZF C Hs 2 . Remark 2.1.5. Note that ℜ * Hs 2 ∈ ℑ * Hs 2 since ℜ * Hs 2 is definable by the following formula (ii) Let Frst(y, v) be the relation : y is the Gödel number of a wff of the set theoryZF Cst that contains free occurrences of the variable X with Gödel number v [9].
Ψ * (Z) ∀X ( X ∈ ℑ * Hs 2 ) [X ∈ Z ⇐⇒ X / ∈ X] .(
(iii) Note that the relation Frst(y, v) is expressible in ZF Cst by a wff Frst(y, v).
(iv) Note that for any y, v ∈ N by definition of the relation Frst(y, v) follows that
Frst(y, v) ⇐⇒ ∃!Ψ (X) [(gZF C st (Ψ (X)) = y) ∧ (gZF C st (X) = ν)] , (2.2.1)
where Ψ (X) is a unique wff of ZF Cst which contains free occurrences of the variable X with Gödel number v. We denote a unique wff Ψ (X) defined by using equivalence (2.2.1) by symbol Ψy,ν (X) , i.e.
Frst(y, v) ⇐⇒ ∃!Ψy,ν (X) [(gZF Cst (Ψy,ν (X)) = y) ∧ (gZF Cst (X) = ν)] , (2.2.2) (v) Let ℘st (y, v, ν1) be a Gödel number of the following wff: ∃!X [Ψ (X) ∧ Y = X] , where gZF Cst (Ψ (X)) = y, gZF Cst (X) = ν, gZF Cst (Y ) = ν1
(2.6) in section 2, see Remark 2.2 and Designation 2.3, (see also [8]- [9]). or in the following equivalent form:
∀y (y ∈ N) [ y ∈ Γ st ν ⇐⇒ (y ∈ N) ∧ Frst(y, v)
] .
Remark 2.2.1. Note that from the axiom of separation it follows directly that Γ st ν is a set in the sense of the set theory ZF Cst.
Definition 2.2.3. (i) We define now the equivalence relation (· ∼X ·) ⊂ Γ st X × Γ st X by Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ (∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)]) (2.2.4) (ii) A subcollection Λ st X of Γ st X such that Ψ1 (X) ∼X Ψ2 (X) holds for all Ψ1 (X) and Ψ2 (X) in Λ st X , and never for Ψ1 (X) in Λ st X and Ψ2 (X) outside Λ st X , is an equivalence class of Γ st X . (iii) For any Ψ (X) ∈ Γ st X let [Ψ (X)] st { Φ (X) ∈ Γ st X |Ψ (X) ∼X Φ (X)
} denote the equivalence class to which Ψ (X) belongs. All elements of Γ st X equivalent to each other are also elements of the same equivalence class.
(iv) The collection of all possible equivalence classes of Γ st
X by˜X , denoted Γ st X / ∼X Γ st X / ∼X { [Ψ (X)] st |Ψ (X) ∈ Γ st X } . (2.2.5) Definition 2.2.4. (i) We define now the equivalence relation (· ∼ν ·) ⊂ Γ st ν × Γ st ν in the sense of the set theory ZF Cst by y1 ∼ν y2 ⇐⇒ (∀X [Ψy 1 ,ν (X) ⇐⇒ Ψy 2 ,ν (X)]) (2.2.6)
Note that from the axiom of separation it follows directly that the equivalence relation (· ∼ν ·) is a relation in the sense of the set theory ZF Cst.
(ii) A subset Λ st ν of Γ st ν such that y1 ∼ν∀Y { Y ∈ ℑst ⇐⇒ ∃Ψ (X) [( [Ψ (X)] st ∈ Γ st X / ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} .(∀Y { Y ∈ ℑst ⇐⇒ ∃Ψ (X) [( [Ψ (X)] st ∈ Γ * st X / ∼X ) ∧ (Y = X) ]} , (2.2.9)
where the countable collection Γ * st ] .
X / ∼X is defined by ∀Ψ (X) { [Ψ (X)] st ∈ Γ * st X / ∼X ⇐⇒ [( [Ψ (X)] st ∈ Γ st X / ∼X ) ∧ ∃!XΨ (X) ]}([( [y] st ∈ Γ * st ν / ∼ν ) ∧ (gZF Cst (X) = ν) ∧ Y = X ]} .(
Derivation of the inconsistent definable set in ZF C N st
(ii) Let ZF CNst be the theory
ZF CNst = ZF C + M ZF C N st [ P A ] .
Designation 2.3.2. (i) Let gZF C N st (u) be a Gödel number of given an expression u of the set theory ZF CNst ZF C
+ ∃M ZF C N st [ P A ] .
(ii) Let FrNst(y, v) be the relation : y is the Gödel number of a wff of the set theory ZF CNst that contains free occurrences of the variable X with Gödel number v [9].
(iii) Note that the relation FrNst(y, v) is expressible in ZF CNst by a wff FrNst(y, v).
(iv) Note that for any y, v ∈ N by definition of the relation FrNst(y, v) follows that
FrNst(y, v) ⇐⇒ ∃!Ψ (X) [(gZF C N st (Ψ (X)) = y) ∧ (gZF C N st (X) = ν)] , (2.3.1)
where Ψ (X) is a unique wff of ZF Cst which contains free occurrences of the variable X with Gödel number v. We denote a unique wff Ψ (X) defined by using equivalence (2.3.1) by symbol Ψy,ν (X) ,
i.e. ℘Nst (y, v, ν1) be a Gödel number of the following wff: (2.3.3) or in the following equivalent form
FrNst(y, v) ⇐⇒ ∃!Ψy,ν (X) [(gZF C N st (Ψy,ν (X)) = y) ∧ (gZF C N st (X) = ν)] . (2.3.2) (v) Let∃!X [Ψ (X) ∧ Y = X] , where gZF C N st (Ψ (X)) = y, gZF C N st (X) = ν, gZF C N st (Y ) = ν1.Γ N st ν = {y ∈ N| ⟨y, ν⟩ ∈ Fr N st (y, v)} ,∀y (y ∈ N) [ y ∈ Γ N st ν ⇐⇒ (y ∈ N) ∧ Fr N st (y, v)
] .
Remark 2.3.1.
Note that from the axiom of separation it follows directly that Γ st ν is a set in the sense of the set theory ZF CNst.
Definition 2.3.4. (i) We define now the equivalence relation (· ∼X ·) ⊂ Γ N st
X × Γ N st X by Ψ1 (X) ∼X Ψ2 (X) ⇐⇒ (∀X [Ψ1 (X) ⇐⇒ Ψ2 (X)]) (2.3.4) (ii) A subcollection Λ st X of Γ st X such that Ψ1 (X) ∼X Ψ2 (X){ Φ (X) ∈ Γ N st X |Ψ (X) ∼X Φ (X)
} denote the equivalence class to which Ψ (X) belongs. All elements of Γ st X equivalent to each other are also elements of the same equivalence class.
∀Y { Y ∈ ℑNst ⇐⇒ ∃Ψ (X) [( [Ψ (X)] N st ∈ Γ N st X / ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} . (2.3.8)
Definition 2.3.7.
We rewrite now (2.3.8) in the following equivalent form
∀Y { Y ∈ ℑNst ⇐⇒ ∃Ψ (X) [( [Ψ (X)] N st ∈ Γ * N st X / ∼X ) ∧ (Y = X) ]} , (2.3.9)
where the countable collection Γ * N st
X / ∼X is defined by formula ∀Ψ (X) { [Ψ (X)] N st ∈ Γ * N st X / ∼X ⇐⇒ [( [Ψ (X)] N st ∈ Γ N st X / ∼X ) ∧ ∃!XΨ (X) ]} .(2.[( [y] N st ∈ Γ * N st ν / ∼ν ) ∧ (gZF C N st (X) = ν) ∧ Y = X ]} . (2.3.15)
Note that from the axiom schema of replacement it follows directly that ℑ * st is a set in the sense of the set theory ZF CNst. Definition 2.3.11. We define now the countable set ℜ * N st by formula
∀X (X ∈ ℑ * N st ) [X ∈ ℜ * N st ⇐⇒ X / ∈ X] . (2.3.16)
Note that from the axiom schema of separation it follows directly that ℜ * N st is a set in the sense of the set theory ZF CNst.
Remark 2.3.5. Note that ℜ * N st ∈ ℑ * N st since ℜ * N st is definable by the following formula Ψ * (Z) ∀X (X ∈ ℑ * N st ) [X ∈ Z ⇐⇒ X / ∈ X] .(2.(ℜ * N st ∈ ℜ * N st ) ∧ (ℜ * N st / ∈ ℜ * N st ) . (2.3.18)
Avoiding the Contradictions from Set Theory ZF C
Hs 2
and Set Theory ZF C st Using Quinean Approach
In order to avoid difficulties mentioned above we use well known Quinean approach [13].
Quinean set theory N F
Remind that the primitive predicates of Russellian unramified typed set theory (TST), a streamlined version of the theory of types, are equality = and membership ∈ . TST has a linear hierarchy of types: type 0 consists of individuals otherwise undescribed. For each (meta-) natural number n, type n + 1 objects are sets of type n objects; sets of type n have members of type n − 1. Objects connected by identity must have the same type. The following two atomic formulas succinctly describe the typing rules: x n = y n and x n ∈ y n+1 .
The axioms of TST are:
Extensionality: sets of the same (positive) type with the same members are equal.
Axiom schema of comprehension:
If Φ(x n ) is a formula, then the set {x n | Φ(x n )} n+1 exists i.e., given any formula Φ(x n ), the formula
∃A n+1 ∀x n [x n ∈ A n+1 ↔ Φ(x n )] (3.1.1)
is an axiom where A n+1 represents the set {x n | Φ(x n )} n+1 and is not free in Φ(x n ).
Quinean set theory [13] (New Foundations) seeks to eliminate the need for such superscripts.
New Foundations has a universal set, so it is a non-well founded set theory. That is to say, it is a logical theory that allows infinite descending chains of membership such as . . . xn ∈ xn−1 ∈ . . . x3 ∈ x2 ∈ x1. It avoids Russell's paradox by only allowing stratifiable formulae in the axiom of comprehension. For instance x ∈ y is a stratifiable formula, but x ∈ x is not (for details of how this works see below). Quinean set theory.
Axioms and stratification are:
the well-formed formulas of New Foundations (N F ) are the same as the well-formed formulas of TST, but with the type annotations erased. The axioms of N F are.
Extensionality: two objects with the same elements are the same object.
A comprehension schema: all instances of TST Comprehension but with type indices dropped (and without introducing new identifications between variables).
By convention, NF's Comprehension schema is stated using the concept of stratified formula and making no direct reference to types. Comprehension then becomes.
Axiom schema of comprehension:
{x | Φ s } exists for each stratified formula Φ s .
Even the indirect reference to types implicit in the notion of stratification can be eliminated. Theodore Hailperin showed in 1944 that Comprehension is equivalent to a finite conjunction of its instances, [14] so that N F can be finitely axiomatized without any reference to the notion of type. Comprehension may seem to run afoul of problems similar to those in naive set theory, but this is not the case. For example, the existence of the impossible Russell class {x | x / ∈ x} is not an axiom of N F, because x / ∈ x cannot be stratified.
Set theory ZF C
Hs 2 , ZF C st and set theory ZF C N st with stratified axiom schema of replacement
The stratified axiom schema of replacement asserts that the image of a set under any function definable by stratified formula of the theory ZF Cst will also fall inside a set.
Stratified Axiom schema of replacement.
Let Φ s (x, y, w1, w2, . . . , wn) be any stratified formula in the language of ZF Cst whose free variables are among x, y, A, w1, w2 Remind that urlogic has the following characteristics [6].
1.
Sentences of urlogic are finite strings of symbols. That a string of symbols is a sentence of 1.5) or in the following equivalent form:
[( g ZF C U l 2 ( Ψ U l y,ν (X) ) = y ) ∧ ( g ZF C U l 2 (X) = ν )] .Γ U l ν = { y ∈ N| ⟨y, ν⟩ ∈ Fr U l 2 (y, v) } ,(4.∀y (y ∈ N) [ y ∈ Γ U l ν ⇐⇒ (y ∈ N) ∧ Fr U l 2 (y, v)
] . where the countable collection Γ * U l X / ∼X is defined by for the reason that the countable collection ℑ U l 2 is not a set in the sense of the set theory ZF C U l 2 . In order to obtain a contradiction inside ZF C U l 2 we introduce the following definitions. Definition 4.1.7. We define now the countable set Γ * U l ν / ∼ν by
∀Ψ (X) { [Ψ (X)] U l ∈ Γ * U l X / ∼X ⇐⇒ [( [Ψ (X)] U l ∈ Γ U l X / ∼X ) ∧ ∃!XΨ (X) ]} .
Definition 2 .3. 1 .
21Let P A be a first order theory which contain usual postulates of Peano arithmetic[9] and recursive defining equations for every primitive recursive function as desired. So for any (n + 1)-place function f defined by primitive recursion over any n-place base function g and (n + 2)place iteration function h there would be the defining equations:(i) f (0, y1, ..., yn) = g (y1, ..., yn) ,(ii) f (x + 1, y1, ..., yn) = h (x,f (x, y1, ..., yn) , y1, ..., yn).Designation 2.3.1. (i) Let M ZF C N st be a nonstandard model of ZF C and let M P A st be a standard model of P A. We assume now that M P A st ⊂ M ZF C N st and denote such nonstandard model of the set theory ZF C by M ZF C N st [ P A
Definition 2 .3. 2 .
22Let Γ N st X be the countable collection of the all 1-place open wff's of the set theory ZF CNst that contains free occurrences of the variable X. Definition 2.3.3. Let gZF C N st (X) = ν.Let Γ N st ν be a set of the all Gödel numbers of the 1-place open wff's of the set theory ZF CNst that contains free occurrences of the variable X with Gödel number v, i.e.
holds for all Ψ1 (X) and Ψ2 (X) in Λ st X , and never for Ψ1 (X) in Λ N st X and Ψ2 (X) outside Λ N st X , is an equivalence class of Γ N st X . (iii) For any Ψ (X) ∈ Γ N st X let [Ψ (X)] N st
( iv ). 2 .
iv2The collection of all possible equivalence classes of Γ N st X by˜X , denoted Γ N st X / ∼X Γ N st X / ∼X { [Ψ (X)] N st |Ψ (X) ∈ Γ N st X } . (2.3.5) Definition 2.3.5. (i) We define now the equivalence relation (· ∼ν ·) ⊂ Γ N st ν × Γ N st ν in the sense of the set theory ZF CNst byy1 ∼ν y2 ⇐⇒ (∀X [Ψy 1 ,ν (X) ⇐⇒ Ψy 2 ,ν (X)]) (2.3.6)Note that from the axiom of separation it follows directly that the equivalence relation (· ∼ν ·) is a relation in the sense of the set theory ZF CNst.(ii) A subset Λ N st ν of Γ Nst ν such that y1 ∼ν y2 holds for all y1 and y1 in Λ N st ν , and never for y1 in Λ N st ν and y2 outside Λ N st ν , is an equivalence class of Γ N st ν . (iii) For any y ∈ Γ N st ν let [y] N st { z ∈ Γ N st ν |y ∼ν z } denote the equivalence class to which y belongs. All elements of Γ N st ν equivalent to each other are also elements of the same equivalence class. (iv) The collection of all possible equivalence classes of Γ N st ν by˜ν , denoted Γ Note that from the axiom of separation it follows directly that Γ N st ν / ∼ν is a set in the sense of the set theory ZF CNst. Definition 2.3.6. Let ℑNst be the countable collection of the all sets definable by 1-place open wff of the set theory ZF CNst, i.e.
Definition 3.1. 1 .
1In New Foundations (N F ) and related set theories, a formula Φ in the language of first-order logic with equality and membership is said to be stratified if and only if there is a function f (x ) which sends each variable appearing in Φ [considered as an item of syntax] to a natural number (this works equally well if all integers are used) in such a way that any atomic formula x ∈ y appearing in Φ satisfies f (x ) + 1 = f (y) and any atomic formula x = y appearing in Φ satisfies f (x ) = f (y).
. 2 .
2Let g ZF C U l 2 (X) = ν. Let Γ U l ν be a set of the all Gödel numbers of the 1-place open wff's of the set theory ZF C U l 2 that contains free occurrences of the variable X with Gödel number v, i.e.
. 2 .. 3 .
23Note that from the axiom of separation it follows directly that Γ U l ν is a set in the sense of the set theory ZF C U l 2 . Definition 4.1.3. (i) We define now the equivalence relation(· ∼ν ·) ⊂ Γ U l ν × Γ sense of the set theory ZF C U l 2 by y1 ∼ν y2 ⇐⇒ ( ∀X [ Ψ U l y 1 ,ν (X) ⇐⇒ Ψ U l y 2 ,ν (X) y1 ∈ Γ U l v let [y1] U l { y ∈ Γ U l X |y1 ∼ν y2} denote the equivalence class to which y1 belongs. The collection of all possible equivalence classes of Γ U l ν by˜ν ,denoted Γ U l ν / ∼ν Γ U l 2 / ∼ν {[y] U l |y ∈ Γ Note that from the axiom of separation it follows directly that Γ Hs ν / ∼ν is a set in the sense of the set theory ZF C U l 2 .Definition 4.1.4. Let ℑ U l 2 be the countable collection of all sets definable by 1-place open wff of the set theory ZF C U l 2, i.e. ∀Y { Y ∈ ℑ U l 2 ⇐⇒ ∃Ψ (X) [( [Ψ (X)] U l ∈ Γ U l X / ∼X ) ∧ [∃!X [Ψ (X) ∧ Y = X]] ]} .(4.1.10) Definition 4.1.5. We rewrite now (4.1.10) in the following equivalent form ∀Y { Y ∈ ℑ U l 2 ⇐⇒ ∃Ψ (X) [( [Ψ (X)] U l ∈ Γ * U l X / ∼X ) ∧ (Y = X) ]} , (4.1.11)
. 6 .. 4 . 2 )
642Let ℜ U l 2 be the countable collection of all sets such that∀X ( X ∈ ℑ Note that ℜ U l 2 ∈ ℑ U l 2 since ℜ U l 2 is a collection definable by 1-place open wff Ψ ( Z, ℑ U l ∀X ( X ∈ ℑ U l 2 ) [X ∈ Z ⇐⇒ X / ∈ X] . (2.1.16) it is not a contradiction inside ZF C U l 2
.1. 5 .
5∀y { [y] U l ∈ Γ * U l ν / ∼ν ⇐⇒ ( [y] U l ∈ Γ U l ν / ∼νNote that from the axiom of separation it follows directly that Γ * U l ν / ∼ν is a set in the sense of the set theory ZF C U l 2 . Definition 4.1.8.We define now the countable set ℑ * U from the axiom schema of replacement (1.1.1) it follows directly that ℑ * Hs 2 is a set in the sense of the set theory ZF C U l 2 . Definition 4.1.9. We define now the countable set ℜ * U l 2 by formula ∀X
Axiom ∃M ZF C .[8]. There is a set M ZF C and a binary relation ϵ ⊆ M ZF C × M ZF C which makes M ZF C a model for ZF C. Remark 1.1.5. (i) We emphasize that it is well known that axiom ∃M ZF C a single statement in ZF C see[8], Ch.II, section 7. We denote this statement throught all this paper by symbolCon ( ZF C;M ZF C ). The completness theorem says that ∃M ZF C ⇐⇒ Con (ZF C) .is a model for ZF C under the relation R.st is called a standard model since the relation ∈ used is merely the standard ∈-relation. Remark 1.1.6.[8]. Note that axiom ∃M ZF C doesn't imply axiom ∃M ZF C st . Remark 1.1.7. Note that in order to deduce: ZF C) from Con(ZF C), by using Gödel encoding, one needs something more than the consistency of ZF C Hs 2 , e.g., that ZF C Hs st i.e., a model in which the integers are the standard integers. To put it another way, why should we believe a statement just because there's a ZF C Hs 2 -proof of it? It's clear that if ZF C HsDesignation 1.1.1. We will denote
(i) by ZF C Hs
2
set theory ZF C2 with the Henkin semantics,
(ii) by ZF C
Hs
2
set theory ZF C Hs
2
+ ∃M
ZF C Hs
2
st
, and
(iii) by ZF Cst set theory
ZF C + ∃M ZF C
st
, where M T h
st is a standard model of the theory T h.
(ii) Obviously there exists a single statement in ZF C Hs
2
such that ∃M ZF C Hs
2
⇐⇒ Con
(
ZF C Hs
2
)
.
We denote this statement throught all this paper by symbol Con
(
ZF C Hs
2 ;M ZF C Hs
2
)
and there
exists a single statement ∃M Z Hs
2
in Z Hs
2 . We denote this statement throught all this paper by
symbol Con
(
Z Hs
2 ;M Z Hs
2
)
.
Axiom ∃M ZF C
st
. [[8]]. There is a set M ZF C
st
such that if R is
{
⟨x, y⟩ |x ∈ y ∧ x ∈ M ZF C
st
∧ y ∈ M ZF C
st
}
,
then M ZF C
st
Definition 1.1.1. [8].The model M ZF C
st
and M
Z Hs
2
(i)˜Con(ZF C Hs
2 ) from Con(ZF C Hs
2 ), and
(ii)˜Con(2
has an omega-model M
ZF C Hs
2
ω
or an standard model
M
ZF C Hs
2
2
that contains free occurrences of the variable X.be a set of the all Gödel numbers of the 1-place open wff's of the set theoryZF C that contains free occurrences of the variable X with Gödel number v, i.e.Definition 2.1.3. Let Γ Hs
X be the countable collection of the all 1-place open wff's of the set
theoryZF C
Hs
2
Definition 2.1.4. Let g ZF C Hs
2
(X) = ν.
Let Γ Hs
ν
Hs
2
Γ Hs
ν
=
{
y ∈ N| ⟨y, ν⟩ ∈ Fr Hs
2 (y, v)
}
(2.1.5)
or in the following equivalent form:
∀y (y ∈ N)
[
y ∈ Γν ⇐⇒ (y ∈ N) ∧ Fr
Hs
2 (y, v)
]
.
(2.1.6)
Remark 2.1.1. Note that from the axiom of separation it follows directly that Γ Hs
ν
is a set in the
sense of the set theory ZF C
Hs
2 .
Definition 2.1.5. (i) We define now the equivalence relation
(· ∼ν ·) ⊂ Γ Hs
ν × Γ Hs
ν
(2.1.7)
in the sense of the set theory ZF C
Hs
2
∀Y
{
Y ∈ ℑ Hs
2
⇐⇒ ∃Ψ (X)
[(
[Ψ (X)] Hs ∈ Γ Hs
X / ∼X
)
∧ [∃!X [Ψ (X) ∧ Y = X]]
]}
.
(2.1.10)
Definition 2.1.7. We rewrite now (2.1.10) in the following equivalent form
Hs 2
Hswe introduce the following definitions. Definition 2.1.9.We define now the countable set Γ * Hs Definition 2.1.10. We define now the countable set ℑ * Hsν
/ ∼ν by
∀y
{
[y] Hs ∈ Γ * Hs
ν
/ ∼ν ⇐⇒
(
[y] Hs ∈ Γ Hs
ν / ∼ν
)
∧ Fr
Hs
2 (y, v) ∧ [∃!XΨy,ν (X)]
}
.
(2.1.16)
Remark 2.1.4. Note that from the axiom of separation it follows directly that Γ *
ν / is a set in the
sense of the set theory ZF C
Hs
2 .
2
by formula
∀Y
{
Y ∈ ℑ * Hs
2
⇐⇒ ∃y
[ (
[y] ∈ Γ * Hs
Definition 2.2.1. Let Γ st X be the countable collection of the all 1-place open wff's of the set theory ZF Cst that contains free occurrences of the variable X. Definition 2.2.2. Let gZF C st (X) = ν. Let Γ st ν be a set of the all Gödel numbers of the 1-place open wff's of the set theory ZF Cst that contains free occurrences of the variable X with Gödel number v, i.e.Γ st
ν = {y ∈ N| ⟨y, ν⟩ ∈ Frst(y, v)} ,
(2.2.3)
y2 holds for all y1 and y1 in Λ st ν ,and never for y1 in Λ st Definition 2.2.5. Let ℑst be the countable collection of all sets definable by 1-place open wff of the set theory ZF Cst, i.e.ν
and y2 outside Λ st
ν , is an equivalence class of Γ st
ν .
(iii) For any y ∈ Γ st
ν let [y] st
{
z ∈ Γ st
ν |y ∼ν z
}
denote the equivalence class to which y belongs.
All elements of Γ st
ν equivalent to each other are also elements of the same equivalence class.
(iv) The collection of all possible equivalence classes of Γ st
ν by˜ν , denoted Γ st
ν / ∼ν
Γ st
ν / ∼ν
{
[y] st |y ∈ Γ st
ν
}
.
(2.2.7)
Remark 2.2.2. Note that from the axiom of separation it follows directly that Γ st
ν / ∼ν is a set in
the sense of the set theory ZF Cst.
ℑst is not a set in the sense of the set theory ZF Cst. Note that from the axiom of separation it follows directly that Γ * st ν / ∼ν is a set in the sense of the set theory ZF Cst.2.2.10)
Definition 2.2.7. Let ℜst be the countable collection of the all sets such that
∀X (X ∈ ℑst) [X ∈ ℜst ⇐⇒ X /
∈ X] .
(2.2.11)
Remark 2.2.3. Note that ℜst ∈ ℑst since ℜst is a collection definable by 1-place open wff is
definable by formula
Ψ (Z, ℑst) ∀X (X ∈ ℑst) [X ∈ Z ⇐⇒ X /
∈ X] .
From (2.2.11) and Remark 2.2.3 one obtains directly
ℜst ∈ ℜst ⇐⇒ ℜst /
∈ ℜst.
(2.2.12)
But (2.2.12) immediately gives a contradiction
(ℜst ∈ ℜst) ∧ (ℜst /
∈ ℜst) .
(2.2.13)
However contradiction (2.2.13) it is not a true contradiction inside ZF Cst for the reason that the
countable collection In order to obtain a true contradiction inside ZF Cst we introduce the following definitions.
Definition 2.2.8. We define now the countable set Γ * st
ν / ∼ν by formula
∀y
{
[y] st ∈ Γ * st
ν / ∼ν ⇐⇒
(
[y] st ∈ Γ st
ν / ∼ν
)
∧ Frst(y, v) ∧ [∃!XΨy,ν (X)]
}
.
(2.2.14)
Remark 2.2.4. Definition 2.2.9. We define now the countable set ℑ *
st by formula
∀Y
{
Y ∈ ℑ *
st ⇐⇒ ∃y
, . . . , wn, so that in particular B is not free in Φ s . Then ∀A∀w1∀w2...∀wn [∀x (x ∈ A =⇒ ∃!yΦ s (x, y, w1, w2, . . . , wn)) =⇒ =⇒ ∃B∀x (x ∈ A =⇒ ∃y (y ∈ B ∧ Φ s (x, y, w1, w2, . . . , wn)))] , (3.2.1) i.e., if the relation Φ s (x, y, ...) represents a definable function f, A represents its domain, and f (x) is a set for every x ∈ A, then the range of f is a subset of some set B. Axiom schema of separation. Let Φ s (x, w1, w2, . . . , wn) be any stratified formula in the language of ZF Cst whose free variables are among x, A, w1, w2, . . . , wn, so that in particular B is not free in Φ s . Then ∀w1∀w2...∀wn∀A∃B∀x [x ∈ B ⇐⇒ (x ∈ A ∧ Φ s (x, w1, w2, . . . , wn))] , (3.2.2) Remark 3.2.1. Notice that the stratified axiom schema of separation follows from the stratified axiom schema of replacement together with the axiom of empty set. Remark 3.2.2. Notice that the stratified axiom schema of replacement (separation) obviously violeted any contradictions (2.1.20), (2.2.18) and (2.3.18) mentioned above. The existence of the countable Russell sets ℜ * Hs , ℜ * st and ℜ * N st impossible, because x / ∈ x cannot be stratified. Second-order Set Theory ZF C 2 with the Full Secondorder Semantics 4.1 Second order set theory ZF C 2 with urlogicStratified 2
4
, i.e.
(y, v) ⇐⇒ ∃!Ψ U l y,ν (X)
AcknowledgementThe reviewers provided important clarifications.urlogic, is a non-mathematical judgement.2. Some sentences are accepted as axioms. That a sentence is an axiom, is a non-mathematical judgement.3.Derivations are made from axioms. The derivations obey certain rules of proof. That a derivation obeys the rules of proof, is a non-mathematical judgement.4.Derived sentences can be asserted as facts. We define now the equivalence} denote the equivalence class to which Ψ (X) belongs. All elements of Γ U l X equivalent to each other are also elements of the same equivalence class. The collection of all possible equivalence classes ofLet Fr U l 2 (y, v) be the relation : y is the Gödel number of a wff of the set theoryZF C U l 2 that contains free occurrences of the variable X with Gödel number v[9].Note that the relation Fr U l 2 (y, v) is expressible in ZF C U l 2 by a wff Fr U l 2 (y, v). Note that for any y, v ∈ N by definition of the relation Fr U l 2 (y, v) follows thatwhere Ψ (X) is a unique wff of ZF C U l 2 which contains free occurrences of the variable X with Gödel number v. We denote a unique wff Ψ (X) defined by using equivalence (4.1.3) by symbol Ψ U l y,ν (X) , i.e.Note that from the axiom schema of separation it follows directly that ℜ * U l 2 is a set in the sense of the set theory ZF C U l 2 ..(4.1.21)Second-order set theory ZF C 2 with the full second-order semanticsRemind that the canonical approach of second order logic with full second-order semantics to the foundations of mathematics is that mathematical propositions have the formwhere U is a mathematical structure, such as integers, reals etc., and is a mathematical statement written in second order logic. If A is one of the structures, such as (N, +, ×, <) or (R, +, ×, <), for which there is a second order sentence Ξ U such that be second order set theory ZF C2 with the full second-order semantics.(1) There is no completeness theorem for second-order logic.(2) Nor do the axioms of second-order ZF C f ss 2 imply a reflection principle which ensures that if a sentence of second-order set theory is true, then it is true in some standard model. (i) that are true but unsatisfiable, or (ii) sentences that are valid, but false.Remark 4.2.3. For example let Z be the conjunction of all the axioms of second-order ZF C f ss2 . Z is surely true. But the existence of a model for Z requires the existence of strongly inaccessible cardinals. The axioms of ZF C f ss 2 don't entail the existence of strongly inaccessible cardinals, and hence the satisfiability of Z is independent of ZF C f ss 2 . Thus, Z is true but its unsatisfiability is consistent with ZF C f ss 2 . Note that for any y, v ∈ N by definition of the relation Fr ♯f ss 2 (y, v) follows thatwhich contains free occurrences of the variable X with Gödel number v. We denote a unique wff Ψ (X) defined by using equivalence (4.2.6) by symbol Ψ ♯f ss y,ν (X) , i.e.Competing InterestsAuthors have declared that no competing interests exist.
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Foundations without Foundationalism: A Case for Second-order Logic. S Shapiro, 0-19-825029-0Oxford University PressShapiro S. Foundations without Foundationalism: A Case for Second-order Logic. Oxford University Press; 1991. ISBN 0-19-825029-0
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Inconsistent countable set in second order zfc and nonexistence of the strongly inaccessible cardinals. J Foukzon, British Journal of Mathematics & Computer Science. 95Foukzon J. Inconsistent countable set in second order zfc and nonexistence of the strongly inaccessible cardinals. British Journal of Mathematics & Computer Science. 2015;9(5). ISSN: 2231-0851 Available: http://www.sciencedomain.org/abstract/9622
Generalized lob's theorem. Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology. J Foukzon, 12Foukzon J. (). Generalized lob's theorem. Strong Reflection Principles and Large Cardinal Axioms. Consistency Results in Topology. V. 2017;12. Available: http://arxiv.org/abs/1301.5340v12
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Inconsistent countable set in second order ZFC and unexistence of the strongly inaccessible cardinals. Logic Colloquium16. J Foukzon, 23240Leeds, UKAbstract of contributed talks. The Bulletin of Symbolic LogicFoukzon J. Inconsistent countable set in second order ZFC and unexistence of the strongly inaccessible cardinals. Logic Colloquium16, Leeds, UK, July 31-August 6, 2016. Abstract of contributed talks. The Bulletin of Symbolic Logic. 2017;23(2):240.
| {'fraction_non_alphanumeric': 0.0895986726122576, 'fraction_numerical': 0.03256248055584362, 'mean_word_length': 3.0913237165888843, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 2, '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': 'In this paper we view the first order set theory ZF C under the canonical first order semantics and the second order set theory ZF C2 under the Henkin semantics. Main results are: (i) Let M ZF C st be a standard model of ZF C, then ¬Con(ZF C + ∃M ZF C st ). (ii) Let M ZF C 2 st be a standard model of ZF C2 with Henkin semantics, then ¬Con(ZF C2 + ∃M ZF C 2 st ). (iii) Let k be inaccessible cardinal then ¬Con(ZF C + ∃κ). In order to obtain the statements (i) and (ii) examples of the inconsistent countable set in a set theory ZF C + ∃M ZF C st and in a set theory ZF C2 + ∃M ZF C 2 st were derived. It is widely believed that ZF C + ∃M ZF C st and ZF C2 + ∃M ZF C 2 st are inconsistent, i.e. ZF C and ZF C2 have a standard models. Unfortunately this belief is wrong.We call ⟨U, S⟩ the Henkin model, if ⟨U, S⟩ satisfies the axioms of DED2 and truth in ⟨U, S⟩ is preserved by the rules of DED2. We call this semantics of second-order logic the Henkin semantics and second-order logic with the Henkin semantics the Henkin second-order logic. There is a special class of Henkin models, namely those ⟨U, S⟩ where S is the set of all subsets of A.We call these full models. We call this semantics of second-order logic the full semantics and secondorder logic with the full semantics the full second-order logic.Remark 1.1.3. We emphasize that the following facts are the main features of second-order logic:1.The Completeness Theorem: A sentence is provable in DED2 if and only if it holds in allHenkin models [2], [6].2.The Löwenheim-Skolem Theorem:A sentence with an infinite Henkin model has a countable Henkin model.3.The Compactness Theorem:A set of sentences, every finite subset of which has a Henkin model, has itself a Henkin model.The Incompleteness Theorem:Neither DED2 nor any other effectively given deductive calculus is complete for full models, that is, there are always sentences which are true in all full models but which are unprovable.', 'arxivid': '1301.5340', 'author': ['Jaykov Foukzon \nIsrael Institute of Technology\nHaifaIsrael\n', 'Elena Men'kova \nAll-Russian Research Institute for Optical and Physical Measurements\nMoscowRussia\n', 'Jaykov Foukzon \nIsrael Institute of Technology\nHaifaIsrael\n', 'Elena Men'kova \nAll-Russian Research Institute for Optical and Physical Measurements\nMoscowRussia\n'], 'authoraffiliation': ['Israel Institute of Technology\nHaifaIsrael', 'All-Russian Research Institute for Optical and Physical Measurements\nMoscowRussia', 'Israel Institute of Technology\nHaifaIsrael', 'All-Russian Research Institute for Optical and Physical Measurements\nMoscowRussia'], 'corpusid': 203103432, 'doi': '10.4236/apm.2019.99034 10.4236/apm.2019.99034 10.4236/apm.2019.99034', 'github_urls': [], 'n_tokens_mistral': 15554, 'n_tokens_neox': 13906, 'n_words': 7366, 'pdfsha': 'c568b9269fdab88908748dc6b185f8193b5bfe66', 'pdfurls': None, 'title': ["Article no.JAMCS.38773 Foukzon and Men'kova", "Article no.JAMCS.38773 Foukzon and Men'kova", "Article no.JAMCS.38773 Foukzon and Men'kova", "Article no.JAMCS.38773 Foukzon and Men'kova"], 'venue': ['Journal of Advances in Mathematics and Computer Science', 'Journal of Advances in Mathematics and Computer Science']} |
arxiv |
Rationalizing right-handed neutrinos
Apr 2003
Graham D Kribs [email protected]
Department of Physics
University of Wisconsin
53706MadisonWI
Rationalizing right-handed neutrinos
Apr 2003arXiv:hep-ph/0304256v1 27
A simple argument based on an SU(3) gauged horizontal symmetry is presented that connects the explanation for three generations of matter with the existence of a triplet of right-handed neutrinos. This rationale for right-handed neutrinos is analogous to, but completely independent of, grand unification or extra universal dimensions. A brief discussion of the supersymmetrized SU(3) model is also given, pointing out that certain problems in ordinary supersymmetric models such as fast proton decay via dimension-5 Planck-suppressed operators can be naturally solved.
One of the most remarkable features of the Standard Model (SM) is that matter fermions are chiral and yet all gauge [1,2] and gravitational [3] anomalies vanish for each generation. A known but not often emphasized fact about the matter content is that, given one generation with unfixed hypercharges, anomaly cancellation determines the relative hypercharge assignment to be precisely what has been established by experiment [4]. In other words, electric charge quantization is essentially automatic without grand unification. This fact, taken at face value, is circumstantial evidence against the existence of right-handed neutrinos. By definition a candidate for a right-handed neutrino is any fermion that is uncharged under all of the SM gauge symmetries. Yet, gauge symmetries are precisely the reason that each type of matter (Q, u, d, L, e) is tied with the other matter fields together in a self-consistent, exclusive fashion. In addition, non-chiral matter allows a new mass scale unconnected to electroweak symmetry breaking that only further complicates our understanding of mass generation and mass hierarchies. Extensions of the SM with non-chiral matter, such as adding right-handed neutrinos, therefore appear to be contrary to all of the guiding wisdom gleaned from experiment, at least until recently. (Those who are still in doubt need only observe the agony that the µ problem causes avatars of supersymmetry.) Neutrino experiments [5,6,7], however, have firmly established that the neutrinos oscillate between each generation and thus they have mass. The largest mass of any one neutrino is constrained to be less than about 2 eV [8], and more likely their mass is one to a few orders of magnitude below this, depending on the generation. The mechanism of mass generation for neutrinos remains a mystery. If neutrinos acquire mass analogously to the SM matter fermions, preserving lepton number, then the particle content must be extended with at least two right-handed neutrinos ν 1,2 . Ordinary Yukawa terms L = λ ν LHν c with tiny couplings λ ν < ∼ 10 −12 suffice to explain the two undisputed mass differences found in neutrino oscillation experiments.
But, the global symmetry behind lepton number conservation is not expected to be exact. At dimension-5, the operator HLHL/M violates lepton number by two units and leads to a tiny Majorana mass v 2 /M for left-handed neutrinos. This transmutes the neutrino mass hierarchy problem from explaining λ ν < ∼ 10 −12 to instead explaining v/M < ∼ 10 −12 . To embrace the dimension-5 neutrino mass explanation means the SM effective theory breaks down at M < ∼ 10 14 GeV. This is somewhat disconcerting since there are dimension-6 operators that violate lepton and baryon number, leading to a proton decay rate that is excluded by experiment unless M > ∼ 10 16 GeV. Hence, while lepton number must be be violated at M to explain neutrino masses, baryon number must be preserved to keep the proton stable. The simplest phenomenological explanation for lepton number violation without baryon number violation at the cutoff scale M is to add righthanded neutrinos to the SM with ordinary Yukawa couplings, plus a heavy Majorana mass term L = M ν c ν for the right-handed neutrinos. The resulting combination of a Dirac mass and a heavy Majorana mass leads to the famous "see-saw" neutrino mass matrix [9]
0 λ ν v λ ν v M .(1)
Diagonalizing this mass matrix, or equivalently integrating out the right-handed neutrino gives back the SM plus the dimension-5 operator HLHL/M with a welldefined coefficient, λ 2 ν , and thus predicted neutrino mass, λ 2 ν v 2 /M . A right-handed neutrino with Majorana mass M therefore provides an ultraviolet completion of the effective theory beyond the cutoff M , explaining why only lepton number was violated at M . The difficulty with such neutrino mass generation mechanisms is that they do not really solve the neutrino mass hierarchy problem, and worse still, require precisely those odd-ball fields -right-handed neutrinos -that are unconnected to SM matter through gauge anomalies. Furthermore, the see-saw explanation requires a new Majorana mass scale unconnected with electroweak symmetry breaking. These facts would ordinarily be highly distressing except for a remarkable coincidence: the scale M ∼ 10 14 GeV is tantalizingly close to where the SM gauge couplings come to an approximate intersection. Such an intersection is predicted by grand unified theories (GUTs), providing justification for the new scale. Furthermore, in an SO(10) GUT each right-handed neutrino is elegantly fused with each generation of SM matter into a single 16 representation [10]. This is really just an artifact of unifying into a GUT group with rank greater than that of the SM, since candidates for righthanded neutrinos in GUTs are those fields uncharged under the SM symmetries but charged under some additional gauge symmetry [for SO (10) this is the extra U(1) under the decomposition SO(10) → SU(5) × U(1)]. Rank > 4 GUTs therefore provide a rationale for n = 0 mod 3 right-handed neutrinos whenever each generation is unified into a single representation of the group.
Unfortunately, grand unification has many well-known problems of implementation. Non-supersymmetric grand unification proposals suffer from the hierarchy problem as well as a rather inexact unification of gauge couplings. Both non-supersymmetric and supersymmetric unification models predict proton decay at a rate that has been experimentally ruled out in the simplest models. Also, several theoretical problems pervade unification ranging from understanding how the Higgs is embedded into a GUT representation (the doublet-triplet splitting problem), how (or if) Yukawa couplings are unified, etc. Such experimental and theoretical problems ought to induce us to reconsider GUTs as the origin of right-handed neutrinos and the Majorana mass scale. Is there any rationale independent of unification that predicts righthanded neutrinos as well as the Majorana mass scale?
Suppose the explanation for the number of generations is that each field's three generations (Q 1,2,3 , u 1,2,3 , etc.) correspond to three components of a multiplet of a "horizontal" flavor symmetry. There are only two continuous symmetries that are suitable for this purpose possessing a 3 representation: SU(3) [11] and SU(2) ∼ SO(3) [12,13]. SU(2) can be summarily dismissed if right-handed neutrinos are required to be in a chiral representation of the new symmetry. It has already been emphasized that nonchiral fermions, and right-handed neutrinos in particular, seem to have no (aesthetic) place in the SM if anomaly cancellation is to connect all matter together. There is no hope with SU(2) since it is anomaly-free. SU(2) also does not predict the number of generations since representations of any dimension are possible [14]. Instead, SU(3) admits chiral fermions with only certain dimensionality -there can be three but not two, four, five, seven, etc. generations. Moreover, SU(3) provides two additional key ingredients: (1) there is an additional anomaly cancellation condition on the matter content if SU(3) is at least weakly gauged, and (2) all fermion masses, including right-handed neutrino masses, arise from spontaneous symmetry breaking. Before proceeding, note that the connection between SU(3) anomaly cancellation and the existence of right-handed neutrinos was made some time ago in [11]. In this paper the argument in presented in detail, contrasting with grand unification and universal extra dimensions, and then implications for a supersym-metrized version are briefly discussed.
Gauging a new symmetry in which SM fermions transform is non-trivial and requires the cancellation of all gauge anomalies associated with the new symmetry. There are potentially eight new gauge anomalies as-
sociated with SU(3) f : [SU(3) f ] 3 ; SU(3) f × [SU(3) c ] 2 ; [SU(3) f ] 2 × SU(3) c ; SU(3) f × [SU(2) L ] 2 ; [SU(3) f ] 2 × SU(2) L ; SU(3) f × [U(1) Y ] 2 ; [SU(3) f ] 2 × U(1) Y ; and SU(3) f × [grav] 2 .
Six of these are trivially zero since tr[t a ] = 0 for SU(N) gauge groups with N > 1. This leaves the mixed flavor symmetry/hypercharge anomaly 3 anomaly. The mixed anomaly leads to a condition on the sum of the hypercharges of the SM fermions that is equivalent to
[SU(3) f ] 2 × U(1) Y , and the [SU(3) f ]the mixed [grav] 2 × U(1) Y anomaly [SU (3) f ] 2 × U (1) Y : (6Y Q + 3Y u + 3Y d + 2Y L + Y e ) = 0,
and so automatically cancels. The [SU(3) f ] 3 anomaly, however, does not cancel with just the SM fermion content [11,12]. This is straightforward to see: Five types of matter (Q, u, d, L, e) can be assigned to either 3 or 3 representations. Two of the five fields contribute an even number of 3's or 3's to the anomaly (Q : ±6; L : ±2) while the remaining three fields contribute an odd number of 3's or 3's (u : ±3, d : ±3, e : ±1). The sum of two even numbers and three odd numbers is an odd number, and so [SU(3) f ] 3 anomaly a ∝ n 3 − n 3 cannot be canceled no matter how SM matter is assigned to SU(3) f .
The simplest assignment of matter in 3 or 3 representations allows ordinary Yukawa couplings. Here the Higgs scalar doublet is assumed to be a singlet under SU(3) f , since there is no need for (and various reasons that disfavor) more than one Higgs doublet in the SM. Gauge invariance of the three Yukawa couplings of the SM implies three relations among the anomaly coefficients of SM matter
QHu c ⇒ a(Q) + a(u c ) = 0 (2) QH * d c ⇒ a(Q) + a(d c ) = 0 (3) LH * e c ⇒ a(L) + a(e c ) = 0 .(4)
Without loss of generality Q can be chosen to be a 3,
a = 6 − 3 − 3 ± (2 − 1) = ±1 .(5)
Notice that the anomaly associated with colored fermions self-cancels, but with the leptons it does not cancel regardless of assigning (L, e c ) into a (3,3) or ( It is important to emphasize that this flavor symmetry rationale for right-handed neutrinos is completely independent of grand unification. In fact, the simplest assignment that allows Yukawa couplings to be gauge invariant under SU(3) f does not commute with the usual matter embeddings in unified representations of GUTs. For example, SU(5) [as well as SO(10) and E 6 ] unifies Q and u into a single representation; this is inconsistent with the SU(3) f assignment given above. However, Yukawa couplings are notoriously over-constrained in GUTs as well as flavor symmetry models. SU(5) predicts the down and lepton Yukawas of each generation should unify, and SO(10) predicts up, down, and lepton Yukawas to unify. These predictions are badly broken at low energies, and not much better at the GUT scale for all but perhaps λ b and λ τ . Analogously, the simplest SU(3) f assignment allows Yukawa couplings for all generations, but no generational differences. This must come from additional structure related to the flavor symmetry breaking that has not been specified here. Nevertheless, the matter (and Higgs) assignments under SU(3) f can be suitably modified to commute with grand unification. This was done in several early works on gauged SU(3) f × SU(5) [15]. There they found that many more triplets (or perhaps larger representations) of right-handed neutrinos were needed to cancel the [SU(3) f ] 3 anomaly. For the purposes of this paper, it is enough to observe that there must be at least one triplet of right-handed neutrinos to gauge SU(3) f .
The absence of a signal for new physics in flavorchanging neutral current processes places an important constraint on the scale of SU(3) f symmetry breaking. The constraint arises from the tree-level exchange of flavor gauge bosons that lead to transitions between samecharge, different-generation quarks or leptons. Integrating out heavy flavor gauge bosons results in a low-energy effective theory with new contributions to four-fermion, flavor-violating operators
g 4 f M 2 f (f i γ µ f i )(f j γ µ f j ) ,(6)
where g f is the SU(3) f gauge coupling and M f is the symmetry breaking scale. If the couplings are CP-conserving, one of the strongest constraints comes from the ∆s = 2 process that contributes to the K 0 − K 0 mass difference. Estimates of the bound on the four-quark operator suggest M f > ∼ g 2 f × 1600 TeV [16]. The bound is significantly stronger if the couplings maximally violate CP. In any case, for a flavor gauge coupling that is of order the SM gauge couplings, the bound on the symmetry breaking scale is at least hundreds of TeV. This is reminiscent of the constraints on extended technicolor [17].
The benefit of right-handed neutrinos transforming under a chiral representation of the flavor symmetry is that the Majorana mass scale is no longer arbitrary. The right-handed neutrino Majorana mass is generated through flavor symmetry breaking, analogous to SM fermion masses generated through electroweak symmetry breaking. The scale M f is not predicted, but obviously there is no conflict between the lower bound M f > ∼ 1000 TeV from flavor-changing constraints and the upper bound M f < ∼ 10 14 GeV needed for a successful see-saw explanation of neutrino masses. If M f were near the lower bound, future experiments could search for deviations from (or as-yet unobserved) flavor-changing neutral current processes as a signal for SU(3) f . This would require neutrino Yukawa couplings λ ν ∼ 10 −4 nearer in value to their lepton cousins.
How are right-handed neutrino Majorana masses generated from flavor symmetry breaking? Consider a pair of complex scalar fields in the fundamental representation Σ 1,2 (3) that acquire unaligned vacuum expectation values. This is sufficient to break SU(3) f → nothing. A right-handed neutrino mass arises from the dimension-4 operator ǫ ijk ν c i ν j Σ * k replacing Σ by its vev. Curiously, this two-field breaking model gives mass to just two offdiagonal components of the 3 × 3 Majorana mass matrix in flavor space
Σ 1 Σ 2 Σ 1 Σ 2 (7)
due to the anti-symmetric contraction of SU(3) f indices. This may be a useful starting point for generating an interesting neutrino mass texture. Also, the flavor symmetry could be broken in stages, such as SU(3) → SU(2) → nothing, that may be similarly useful for quark, lepton, or neutrino mass textures. SU(3) f is not the only rationale for three generations and three right-handed neutrinos. In a recent proposal called "universal extra dimensions" (UED) [18], all matter, Higgs, and gauge bosons are promoted to six dimensional fields, and the more complicated gauge and gravitational anomaly structure of six dimensional theories is used to constrain the matter content [19]. Ref. [19] found that cancellation of the global gauge anomaly [20] required the number of generations to be n g = 0 mod 3, and cancellation of the pure gravitational anomaly required n = n g fermionic fields uncharged under the SM gauge group. This is intriguingly similar to the SU(3) f symmetry argument, since the matter content is similarly restricted by anomaly cancellation of a larger symmetry structure. Other similarities are remarkable: [19] required that all matter was chiral in 6-D, analogous to requiring all matter to be in chiral representations of SU(3) f . This led to two possible chirality assignments in UED that are precisely analogous to the 3 versus 3 "chirality" possibilities for the SM fermions under SU(3) f . Specifically, the quark doublet (Q) must have the opposite chirality to the quark singlets (u, d), and the lepton doublet (L) must have the opposite chirality to the lepton singlets (e, ν). In UED the lepton doublet could have the same or the opposite chirality of the quark doublet, just as here the lepton doublet could be assigned to the same (3) or opposite (3) representation of the quark doublet. Finally, the UED rationale for three generations and three right-handed neutrinos does not depend on the compactification scale, just as the SU(3) f argument does not depend on the flavor symmetry breaking scale.
There are a few important differences between the six dimensional UED model and the SU(3) f model. The higher dimensional nature of UED implies there is an effective theory cutoff scale that is only an order of magnitude above the compactification scale; in the SU(3) f model, there is no such restriction. Several gauge anomalies, such as [SU(2) L ] 2 U(1) Y that are automatically canceled in the SU(3) f model, are canceled in UED only via the Green-Schwarz mechanism with additional matter. Finally, the prediction of three generations is not easily extended to a supersymmetric six-dimensional "universal" model for a variety of reasons [19], whereas the SU(3) f model can be quite simply supersymmetrized as will be briefly sketched below.
Everything that has been said for the SM with a gauged SU(3) f flavor symmetry also applies to a straightforward extension of the minimal supersymmetric standard model (MSSM). This means promoting matter supermultiplets to (anti-)fundamental representations of SU(3) f while the Higgs supermultiplets remain singlets, in exact analogy with the non-supersymmetric case. (In the following discussion the same notation is used for the MSSM chiral superfields as for the SM fermion fields.) There are, however, new restrictions on the allowed operators in the supersymmetrized SU(3) f model. The most interesting, model-independent restriction is that the dimension-5 operators leading to proton decay
QQQL M Pl , u c u c d c e c M Pl(8)
Below the SU(3) f symmetry breaking scale, these dimension-6 operators map onto the dimension-5 operators above with tiny coefficients of order Σ /M Pl . This is sufficient to cure the fast proton decay problem that results from the ordinarily unsuppressed dimension-5 operators.
A supersymmetrized version of the SU(3) f model has even more interesting constraints. All dimension ≤ 4 lepton number violating superpotential terms QLd c , LLe c , and LH u are forbidden by SU(3) f . Again, higher dimension operators with SU(3) f breaking fields will reintroduce these terms, but (for the first two) this leads to significant suppression. If the flavor symmetry were promoted to U(3) f , the dimension-4 baryon number violating term u c d c d c would also be forbidden. An exact flavor symmetry could serve in precisely the same role as matter parity on superfields (R-parity on fields). Of course the flavor symmetry is broken, and this reintroduces these so-called R-parity violating operators. It would be interesting to see if R-parity could be discarded in favor of a spontaneously broken U(3) f flavor symmetry without sacrificing a long-lived proton.
In summary, an extension of the Standard Model with an SU(3) f gauged flavor symmetry is presented that explains why there are three generations of matter and predicts the existence of three right-handed neutrinos. This argument is independent of grand unification or extra universal dimensions. The right-handed Majorana mass scale results from spontaneous SU(3) f symmetry breaking. If the breaking scale is "low", less than of order 1000 TeV, deviations in flavor changing neutral current processes are expected due to tree-level flavor gauge boson exchange. It should be emphasized that such a Majorana mass scale is completely consistent with the see-saw explanation for neutrino mass generation so long as the Dirac masses of the neutrinos are less than but of order the muon mass. This is a perfectly reasonable possibility given that SU(3) f has freed us from thinking only in terms of grand unification. The supersymmetric extension including a gauged SU(3) f is straightforward. The fast proton decay problem from dimension-5 Planck-suppressed operators is automatically cured, and certain R-parity violating couplings are naturally suppressed. Combining the SU(3) f gauged flavor symmetry with models that attempt to explain the structure of the quark, lepton, or neutrino mass matrices is an extremely interesting direction left for future work.
I have benefited from discussions with B. Balantekin, V. Barger, A. Nelson, Y. Nir, G. Shiu, and L.-T. Wang. I
then u c and d c must both be 3's. There are two choices for the leptons: [L(3), e c (3)] or [L(3), e c (3)]. In either case, the [SU(3) f ] 3 anomaly coefficient becomes
3, 3). Intriguingly, the [SU(3) f ] 3 anomaly is canceled by adding a single new field that transforms as a 3 [for L(3), e c (3)] or 3 [for L(3), e c (3)] under SU(3) f . To avoid spoiling the SM anomaly cancellation conditions this field must be neutral under SM gauge symmetries. Hence, this anomaly-cancellation field has precisely the quantum numbers of a right-handed neutrino. Also, a Yukawa interaction connecting the left-handed with the right-handed neutrino, L = LHν c , is automatically allowed by SU(3) f gauge invariance regardless of the initial choice of (3, 3, 3) versus (3, 3, 3) for (L, e c , ν c ). This is a remarkable result. Let me restate the assumptions and the implication: Assuming (1) the explanation for the number of generations is a gauged SU(3) f flavor symmetry, (2) all matter is assigned to chiral representations (3 or 3) of SU(3) f , and (3) ordinary Yukawa couplings are SU(3) f gauge invariant, then there must exist one set of right-handed neutrinos ν 1,2,3 transforming as a triplet of SU(3) f .
are forbidden by SU(3) f . Technically the second operator in Eq. (8) could be made gauge-invariant if u c were assigned the conjugate representation to that of d c and e c , but this does not happen for the SU(3) f model norfor the embeddings of matter into SU(5) or SO(10) repre-
sentations. These operators can be made gauge-invariant
by adding a pair of SU(3) f breaking superfields Σ(3) and
Σ(3), whereby Eq. (8) becomes
QQQLΣ
M 2
Pl
,
u c u c d c e c Σ
M 2
Pl
.
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with only odd dimensional representations, but SO(3) with just 3's is physically indistinguishable from SU(2) with just 3's, and so there is no unambiguous explanation for three generations. SO(3) is naively better than SU. SO(3) is naively better than SU(2), with only odd dimen- sional representations, but SO(3) with just 3's is physi- cally indistinguishable from SU(2) with just 3's, and so there is no unambiguous explanation for three genera- tions.
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arxiv |
ON CONNECTIVE K-THEORY OF ELEMENTARY ABELIAN 2-GROUPS AND LOCAL DUALITY
29 Dec 2011
Geoffrey M L Powell
ON CONNECTIVE K-THEORY OF ELEMENTARY ABELIAN 2-GROUPS AND LOCAL DUALITY
29 Dec 2011arXiv:1112.6327v1 [math.AT]
The connective ku-(co)homology of elementary abelian 2-groups is determined as a functor of the elementary abelian 2-group. The argument requires only the calculation of the rank one case and the Atiyah-Segal theorem for KU -cohomology together with an analysis of the functorial structure of the integral group ring. The methods can also be applied to the odd primary case.These results are used to analyse the local cohomology spectral sequence calculating ku-homology, via a functorial version of local duality for Koszul complexes. This gives a conceptual explanation of results of Bruner and Greenlees.2000 Mathematics Subject Classification. 19L41; 20J06. Key words and phrases. connective K-theory; elementary abelian group; group cohomology; group homology; local cohomology.This work was partly financed by the project ANR BLAN08-2 338236, HGRT.1 2 GEOFFREY POWELL explicit the functorial nature of the duality between ku * (BV + ) and ku * (BV + ), via Pontrjagin duality. The second part of the paper applies these results to give an analysis of the local cohomology spectral sequence relating ku * (BV + ) to ku * (BV + ) (see Theorem 8.26); this sheds light upon the description given by Bruner and Greenlees [BG03]: local duality appears as an explicit functor defined in the functorial context. The key observation which explains the origin of the differentials in the local cohomology spectral sequence comes from the analysis of ku * (BV + ), which shows how the vtorsion tors v ku * (BV + ) and the v-cotorsion of ku * (BV + ) are related.The functorial description of V → ku * (BV + ) identifies the mod-p cohomology of the spaces of the Ω-spectrum for ku (up to nilpotent unstable modules), via Lannes' theory [Lan92]. This gives a conceptual framework for understanding the results of[Sto63,Sin68], and can be related to the description of the mod-p homology in terms of Hopf rings [Har91] (which does not a priori retain information on the action of the Steenrod algebra). This will be explained elsewhere.ContentsThe categories F A , F are tensor abelian, with structure induced from A b. (For basic properties of F , see [Kuh94a, Kuh94b, Kuh95] or [FFSS99].) There is an exact Pontrjagin duality functor which generalizes the duality for F introduced in [Kuh94a]:Recall that the socle of an object is its largest semi-simple subobject and the head its largest semi-simple quotient.Example 2.3. The symmetric powers, divided powers and exterior powers are fundamental examples of (polynomial) functors in F . For n ∈ N, the nth symmetric power functor S n is defined by S n (V ) := (V ⊗n )/S n , the nth divided power functor by Γ n (V ) := (V ⊗n ) Sn and the nth exterior power functor identifies as Λ n (V ) ∼ = (V ⊗n ⊗ sign) Sn , where sign is the sign representation of S n . By convention, these functors are zero for negative integers n. There is a duality relation S n ∼ = DΓ n ,
Introduction
The calculation of the ku-(co)homology of finite groups is an interesting and highly non-trivial problem. The case of elementary abelian p-groups illustrates important features; these groups were first calculated by Ossa [Oss89] and were studied further by Bruner and Greenlees [BG03], exhibiting a form of duality via local cohomology. Neither of these references exploit the full naturality of the functors V → ku * (BV + ) and V → ku * (BV + ).
This paper shows how studying these as functors of the elementary abelian pgroup V gives a new and conceptual approach. The methods apply to any prime p; the case p = 2 is privileged here since this requires an additional filtration argument when studying the local cohomology. Moreover, this is the case of interest when extending the methods to the study of ko-(co)homology (cf. [BG10]); this will be developed elsewhere. The main results of the first part of the paper give complete descriptions of the functors V → ku * (BV + ) (Theorem 5.19) and V → ku * (BV + ) (Theorem 5.22).
For ku-cohomology, the only input which is required is the graded abelian group structure of ku * (BZ/2 + ) and the identification of the functor V → KU 0 (BV + ) for periodic complex K-theory, which is provided by the Atiyah-Segal completion theorem. In particular, the method gives a conceptual proof of an algebraic form of Ossa's theorem [Oss89], which gives a reduction to the rank one case. Both Ossa [Oss89] and Bruner and Greenlees [BG03] use Ossa's theorem as a starting point for their calculations.
To give a full functorial description, the integral group ring functor V → Z[V ] is studied, stressing the functorial viewpoint and extending results of Passi and others [PV77]. Of independent interest is the observation that the quotients which arise from studying the filtration of Z[V ] by powers of the augmentation ideal are self-dual under Pontrjagin duality (see Theorems 3.14 and 3.12).
Similarly, for ku-homology, only knowledge of ku * (BZ/2 + ) is required, together with an understanding of the functor V → KU 1 (BV + ). The arguments make 1. Introduction 1 2. Background 2 3. The integral group ring functor 4 4. Milnor derivations 8 5. Connective complex K-cohomology and homology 10 6. Local duality 18 7. Filtering symmetric powers and Koszul complexes 21 8. The local cohomology spectral sequence 24 References 31 2. Background 2.1. Definitions and notation. Fix a prime p and let F denote the prime field F p ; V f denotes the full subcategory of finite-dimensional spaces in the category V of F-vector spaces and vector space duality is denoted by (−) ♯ : V op → V .
Notation 2.1. The category of functors from V f to abelian groups is denoted F A and the full subcategory of functors with values in V is denoted F .
whereas the functor Λ n is self-dual (there is a canonical isomorphism DΛ n ∼ = Λ n ). For p = 2, the functor Λ n is the head of S n and the socle of Γ n . These functors are examples of graded exponential functors; for example, the exponential structure induces natural coproducts S n ∆ → S i ⊗ S j and products S i ⊗ S j µ → S n , for integers i + j = n, and, upon evaluation on V ∈ ObV f , these correspond to the primitively-generated Hopf algebra structure on a polynomial algebra.
Example 2.4. Yoneda's lemma provides the standard injective and projective objects of F (the case of F A is considered in Section 3). The projective functor P F is the functor V → F[V ], which corepresents evaluation at F; the injective functor I F is given by V → F V ♯ (the vector space of set maps) and represents the dual evaluation functor F → DF (F). Duality provides the relation I F ∼ = DP F , which relates the canonical decompositions I F ∼ = F ⊕ I F and P F ∼ = F ⊕ P F , where F is the constant functor and I F (respectively P F ) is the complementary constant-free summand.
The functor I F is ungraded exponential and has associated diagonal ∆ : I F → I F ⊗ I F and multiplication µ : I F ⊗ I F → I F ; these morphisms induce I F → I F ⊗ I F and I F ⊗ I F → I F respectively and, in both cases, these are the unique non-trivial morphisms of the given form. Dually, there is a product P F ⊗ P F → P F and coproduct P F → P F ⊗ P F . Notation 2.5. For n > 0 an integer:
(1) let P n F be the image of the iterated product µ (n−1) : P ⊗n F → P F and q n−1 P F denote its cokernel, so that there is a short exact sequence 0 → P n F → P F → q n−1 P F → 0;
(2) dually, let p n−1 I F denote the kernel of the iterated diagonal I F ∆ (n) → I ⊗n F . Lemma 2.6. [Kuh94b] Suppose that p = 2. Then
(1) I F is the injective envelope of Λ 1 , is uniserial, I F ∼ = colim → p n I F , and there are non-split short exact sequences 0 → p n I F → p n+1 I F → Λ n+1 → 0;
(2) P F is the projective cover of Λ 1 , is uniserial, P F ∼ = lim ← q n P F , and there are short exact sequences 0 → Λ n+1 → q n+1 P F → q n P F → 0.
The functor q n P F is the dual of p n I F .
The fact that p n I F has a simple socle (for n > 0) implies that it is easy to detect non-triviality of a subobject:
Lemma 2.7. Suppose that p = 2 and let n > 0 be an integer.
(1) If G ⊂ p n I F , then G = 0 if and only if G(F) = 0.
(2) If H ⊂ q n P F , then H = q n P F if and only if H(F) = 0.
Proof. The two statements are equivalent by duality, hence it suffices to prove the first. The socle of p n I F is Λ 1 ; if G = 0 then Λ 1 ⊂ G, hence G(F) = 0, since Λ 1 (F) = F. The converse is obvious.
Remark 2.8. There are analogous results for odd primes, taking into account the weight splitting of the category F provided by the action of the units F × p (cf. [Kuh94a]).
The integral group ring functor
This section provides a functorial analysis of the structure of the integral group ring functor; throughout, the prime is taken to be 2 (there are analogous results for odd primes). These results are necessary to complete the full functorial description of the ku-(co)homology of elementary abelian 2-groups but are not required for the proof of the algebraic version of Ossa's theorem.
3.1. The functors P Z , P Z2 .
Notation 3.1. Let (1) P Z denote the integral group ring functor V → Z[V ] and P Z the augmentation ideal, so that there is a direct sum decomposition P Z ∼ = Z ⊕ P Z in F A ; (2) P Z2 denote the functor Z 2 ⊗ P Z and P Z2 the functor Z 2 ⊗ P Z , where Z 2 denotes the 2-adic integers.
Yoneda's lemma implies:
Lemma 3.2. The functor P Z is projective in F A and corepresents evaluation on F.
The ring structure of Z[V ] gives a morphism µ : P Z ⊗ P Z → P Z , which induces µ : P Z ⊗P Z → P Z . There is a reduced diagonal ∆ : P Z → P Z ⊗P Z ; the composition with the canonical projection P Z ։ P Z is determined by the element
([1] − [0]) ⊗ ([1] − [0]) ∈ (P Z ⊗ P Z )(F), by Yoneda. Definition 3.3. For n ∈ N, let P n Z denote the image of the iterated product µ (n−1) : P ⊗n Z → P Z (respectively P n Z2 ⊂ P Z2 ).
The structure of P Z (V ) and the filtration
. . . ⊂ P n+1 Z (V ) ⊂ P n Z (V ) ⊂ . . . ⊂ P 1 Z (V ) = P Z (V )
for a fixed V has received much attention (see [PV77,BV00], for instance). These references do not exploit functoriality.
Lemma 3.4.
(1) There is a natural isomorphism (P Z2 )/2 ∼ = P F .
(2) For n ∈ N, the canonical surjection P Z2 ։ P F induces a commutative diagram in F A :
P n Z2 1 / / P Z2 P n F 1 / / P F .
(3) The head of P Z2 is the functor Λ 1 .
Proof. The commutative diagram follows from the fact that Z → F induces a morphism of group rings
Z[V ] → F[V ]
; the remaining statements are clear.
Lemma 3.5. The composite P Z ∆ → P Z ⊗ P Z µ → P Z is the morphism P Z −2 → P Z . Hence, for n ∈ N, there are inclusions of subobjects of P Z : 2P n Z ⊂ P n+1 Z and 2 n P Z ⊂ P n+1 Z .
Proof. The first statement is straightforward (cf. [BV00, Lemma 3.2]); this gives rise to the natural inclusion 2P n Z ⊂ P n+1 Z . The final statement follows by induction.
Lemma 3.6. For n ∈ N, there is a unique non-trivial morphism P n Z → I F in F A and this induces a surjection
P n Z /P n+1 Z ։ p n I F .
Proof. A morphism P n Z → I F factors naturally across (P n Z )/2. It is straightforward to see that (P n Z )/2 (F) = F and (P n Z )/2 (0) = 0; this implies that there is a unique non-trivial morphism (P n Z )/2 → I F , by Yoneda. The composite P ⊗n Z ։ P n Z → I F factorizes across the projection P ⊗n Z ։ P ⊗n F . Again, there is a unique non-trivial morphism from P ⊗n F to I F ; for n = 1, this is the composite P F ։ Λ 1 ֒→ I F and, for n > 1,
P ⊗n F → I ⊗n F µ (n−1) −→ I F ,
where the first morphism is the iterated tensor product of the morphism P F → I F ; it follows easily that the composite has image p n I F . Hence the morphism P n Z → I F has image p n I F , by Lemma 2.6.
Finally, the fact that P 2 F is the kernel of P F ։ Λ 1 implies that the composite P n+1 Z ֒→ P n Z → I F is trivial, giving the stated factorization.
A non-functorial version of the following result (expressed using different notation) is proved in [PV77]. Since the functorial version is required here, a direct proof is given.
Proposition 3.7. For n > 0 an integer, the following statements hold:
(1) there is a short exact sequence:
0 → 2P n Z → P n+1 Z → P n+1 F → 0;
(2) the canonical morphism P n Z /P n+1 Z → p n I F is an isomorphism.
Proof. The statements are proved in parallel by induction upon n. For n = 1, consider the commutative diagram
2P Z 1 / / P 2 Z / / / / P 2 Z /2P Z 2P Z 1 / / P Z / / / / P F .
The image of P 2 Z in P F is P 2 F , by Lemma 3.4, which proves the first statement. The second statement follows by studying the cokernels of the monomorphisms.
For the inductive step, suppose the result true for n < N . Lemma 3.5 provides a natural inclusion 2 i P Z ⊂ P i+1 Z , for all natural numbers i; for current purposes, it is sufficient to work with the inclusion 2 i+1 P Z ⊂ P i+1 Z . The proof proceeds by providing upper and lower bounds for P N +1 Z /2 N +1 P Z . It is sufficient to do this in the Grothendieck group of F A , since the functors considered below only have finitely many composition factors of a given isomorphism type, which allows comparison arguments. (In fact, evaluation on finite dimensional vector spaces allows the arguments to be reduced to objects which are finite.)
The element of the Grothendieck group associated to an object F is denoted [F ]; for the functors considered here, this lies in the submonoid λ N indexed by the isomorphism classes of simple objects S λ of F A ; for two objects
M = ΣM λ [S λ ] and N = ΣN λ [S λ ], write M ≤ N if M λ ≤ N λ , for all λ.
The inductive hypothesis implies that:
P N Z /2 N P Z = N i=1 P i F in the Grothendieck group. The inclusion 2P N Z ֒→ P N +1 Z induces a monomorphism (P N Z /2 N P Z ) ∼ = 2P N Z /2 N +1 P Z ֒→ P N +1 Z /2 N +1 P Z ;
the composite of this morphism with the projection P
N +1 Z /2 N +1 P Z ։ P N +1 F
is clearly trivial, hence this gives a lower bound for P
N +1 Z /2 N +1 P Z : N +1 i=1 P i F ≤ P N +1 Z /2 N +1 P Z ,
with equality if and only if the first statement holds.
Consider the inclusion P N +1 Z ֒→ P N Z , which induces the monomorphism P N +1 Z /2 N +1 P Z ֒→ P N Z /2 N +1 P Z with cokernel which surjects to p N I F , by Lemma 3.6; this gives the inequality:
P N Z /2 N +1 P Z ≥ P N +1 Z /2 N +1 P Z + p N I F ,(1)
with equality if and only if the second statement holds.
The short exact sequence
0 → P F ∼ = (2 N P Z /2 N +1 P Z ) → P N Z /2 N +1 P Z → P N
Z /2 N P Z → 0 and the inductive hypothesis give:
P N Z /2 N +1 P Z = [P F ] + N i=1 P i F . Now P F = P N +1 F + p N I F , hence (1) implies that N +1 i=1 P i F ≥ P N +1 Z /2 N +1 P Z ,
with equality if and only if the second statement holds. Hence, both inequalities are equalities and the inductive step is established.
Corollary 3.8. For V ∈ ObV f , the topologies on the abelian group P Z (V ) induced by the 2-adic filtration 2 i P Z and by the filtration P i Z are equivalent.
Proof. By Lemma 3.5, 2 n P Z ⊂ P n+1 Z ; conversely, it is straightforward to show, using Proposition 3.7, that P n+k Z
(F n ) ⊂ 2 k P Z (F n ).
3.2. The structure of the quotients P Z /P n+1 Z . Notation 3.9. For n ∈ N, let R n Z denote the quotient P Z /P n+1 Z . Lemma 3.10. For n > 0 an integer:
(1) 2 n+1 R n Z = 0 and R n Z (F) ∼ = Z/2 n ; (2) there are short exact sequences:
0 → p n I F → R n Z → R n−1 Z → 0 0 → R n−1 Z → R n Z → q n P F → 0;
(3) the largest subfunctor 2 R n Z of R n Z annihilated by 2 is isomorphic to p n I F .
Proof. The first statement is clear and the first short exact sequence is provided by Proposition 3.7. The second short exact sequence is induced by the inclusion 2P Z ֒→ P Z , since Proposition 3.7 implies that
2P Z ∩ P n+1 Z is isomorphic to P n Z under the isomorphism P Z ∼ = 2P Z .
The proof that the largest subfunctor of R n Z annihilated by 2 is p n I F is by induction on n; for n = 1, R 1 Z ∼ = Λ 1 and the result is immediate. For the inductive step, the short exact sequence
0 → p n I F → R n Z → R n−1 Z → 0,
implies that there is an exact sequence 0 → p n I F → 2 R n Z → p n−1 I F , where the right hand term is given by the inductive hypothesis. To complete the result, it suffices to show that the image of 2 R n Z in p n−1 I F is trivial; hence, by Lemma 2.7, it suffices to show this after evaluation on F. This follows from the fact that R n Z (F) ∼ = Z/2 n . Lemma 3.11. For n > 0 an integer, the functor R n Z has simple head Λ 1 and simple socle Λ 1 .
Proof. The functor R n Z is a non-trivial quotient of P Z2 , which has simple head Λ 1 by Lemma 3.4, hence R n Z has simple head Λ 1 . The proof that the socle is Λ 1 is by induction on n, starting from the case n = 1, which is clear, since R 1
Z ∼ = Λ 1 is simple. The exact sequence 0 → p n I F → R n Z → R n−1 Z → 0 shows that the socle of R n Z is either Λ 1 or Λ 1 ⊕ Λ 1 .
The latter possibility is excluded since R n Z (F) = Z/2 n , by Lemma 3.10. Theorem 3.12. Let n > 0 be an integer. The functor R n Z is self-dual; more precisely, any surjection P Z2 ։ DR n Z factors canonically across an isomorphism R n Z ∼ = → DR n Z . Proof. The proof is by induction upon n, starting from the case n = 1, when R n Z is the simple functor Λ 1 , which is self-dual. The factorization statement follows from the fact that P Z2 has simple head.
For the inductive step, R n Z has simple socle Λ 1 , hence DR n Z has simple head Λ 1 and there exists a surjection P Z2 ։ DR n Z (by projectivity of P Z ). This gives rise to a morphism of short exact sequences
0 / / P Z2 2 / / P Z2 / / P F / / 0 0 / / DR n−1 Z / / DR n Z / / q n P F / / 0,
where the lower exact sequence is the dual of the first exact sequence of Lemma 3.10, the commutativity of the right hand square follows from the fact that there is a unique non-trivial morphism P Z2 → q n P F and the surjectivity of the left hand vertical morphism is seen by evaluating on F, since DR n−1 Z has simple head. By the inductive hypothesis, the left hand vertical morphism factorizes across an isomorphism R n−1 Z ∼ = → DR n−1 Z . In particular, this implies that P n+1 Z2 lies in the kernel of P Z2 ։ DR n Z , hence this induces a surjection R n Z ։ DR n Z which is an isomorphism, since the objects have finite composition series with isomorphic associated graded functors.
Definition 3.13. Let R ∞ Z be the direct limit of the diagram R 1 Z ֒→ R 2 Z ֒→ R 3 Z ֒→ .
. . of monomorphisms provided by Lemma 3.10.
Theorem 3.14. There are Pontrjagin duality isomorphisms:
R ∞ Z ∼ = DP Z2 P Z2 ∼ = DR ∞ Z . Proof.
Observe that the functor R ∞ Z takes values in torsion 2-groups. By construction and Corollary 3.8, the functor P Z2 is isomorphic to the inverse limit of the natural system of quotients
. . . ։ R n Z ։ R n−1 Z ։ . . . ։ R 1 Z ∼ = Λ 1 .
Applying the Pontrjagin duality functor and using the fact that each R n Z is self dual, gives a direct system which is isomorphic to that defining R ∞ Z (the latter fact follows from the proof of Theorem 3.12). The result follows from Pontrjagin duality for abelian groups.
Milnor derivations
This section establishes the fundamental ingredient, Proposition 4.10, to the proof of an algebraic version of Ossa's theorem; the prime p is taken to be 2.
Milnor derivations on symmetric powers.
Definition 4.1. For i ∈ N, let Q i : S 1 → S 2 i+1 denote the iterated Frobenius x → x 2 i+1 and also its extension S n Qi → S n+2 i+1 −1 to a derivation of S * , defined by the composite S n ∆ → S n−1 ⊗ S 1 1⊗Qi → S n−1 ⊗ S 2 i+1 µ → S n+2 i+1 −1 .
Lemma 4.2. For i, j ∈ N, Q i • Q i = 0 and the derivations Q i , Q j commute. Hence the graded algebra S * is a module in F over the exterior algebra Λ(Q i |i ≥ 0); in particular, (S * , Q i ) has the structure of a commutative differential graded algebra in F .
Proof. Straightforward.
Notation 4.3. For j > 0 an integer, let S * / x 2 j denote the truncated symmetric power functor, so that S n / x 2 j is the cokernel of the composite
S n−2 j ⊗ S 1 1⊗Qj−1 −→ S n−2 j ⊗ S 2 j µ → S n .
In the following statement, the degree corresponds to the grading inherited from that of S * .
Proposition 4.4. For i ∈ N, the homology H(S * , Q i ) is the truncated symmetric power functor S * / x 2 i , concentrated in even degrees. Explicitly:
H(S * , Q i ) k ∼ = 0 k ≡ 1 mod 2 S d / x 2 i k = 2d.
Proof. The result follows from the calculation of the homology of the differential graded algebra (F[x], dx = x 2 i+1 ), which is the truncated polynomial algebra F[y]/y 2 i , where y = x 2 . The homology of the tensor product of such algebras is calculated by using the Künneth theorem. It remains to show that this corresponds to the stated functorial isomorphism.
The above establishes that the homology is concentrated in even degrees; moreover, in degree k = 2d, the Frobenius S d ֒→ S 2d maps to the cycles in degree k and induces a surjection onto the homology.
It is straightforward to check that, for integers i, d ≥ 1, there is a commutative diagram in F :
S d−2 i ⊗ S 1 Φ⊗1 / / 1⊗Qi−1 S 2d−2 i+1 ⊗ S 1 µ / / S 2d−2 i+1 +1 Qi S d−2 i ⊗ S 2 i µ / / S d Φ / / S 2d ,
where Φ is the Frobenius. It follows that the morphism S d → H(S * , Q i ) 2d factorizes across the canonical projection S d ։ S d / x 2 i . This completes the proof.
Remark 4.5. For i = 0, the homology is F concentrated in degree zero; in particular, the Q 0 : S n → S n+1 induce an exact complex
0 → S 1 → S 2 → S 3 → . . . .
For i = 1, there is non-trivial homology in even degrees; the homology of the complex S 2d−3 → S 2d → S 2d+3 is Λ d . Moreover, the proof of the Proposition gives an exact complex
S d ⊕ S 2d−3 (Φ,Q1) → S 2d Q1 → S 2d+3 . 4.2. The Q 0 -kernel complex.
Notation 4.6. For n ∈ N, let K n denote the kernel of Q 0 : S n → S n+1 .
By construction, there are short exact sequences
0 → K n → S n → K n+1 → 0, for n ≥ 0. Initial values of K n are K 0 = F = S 0 , K 1 = 0, K 2 = Λ 1 = S 1 ֒→ S 2 , K 3 = Λ 2 .
Lemma 4.7. The derivation Q 1 on S * induces a differential Q 1 : K n → K n+3 and there is a short exact sequence of complexes
0 → (K i+3• , Q 1 ) → (S i+3• , Q 1 ) → (K i+1+3• , Q 1 ) → 0, where 0 ≤ i < 3 and • ≥ 0.
Proof. A consequence of the exactness of the Q 0 -complex in positive dimensions and the commutation of Q 0 , Q 1 (Lemma 4.2).
Notation 4.8. For n ∈ N, let L n denote the image of Q 1 : K n−3 → K n andL n the kernel of Q 1 :
K n → K n+3 .
The following is clear:
Proposition 4.9. The graded functors K * ,L * have unique graded algebra structures such that the canonical inclusionsL * ֒→ K * ֒→ S * are morphisms of commutative graded algebras in F .
K i+3• , Q 1 ) is determined bỹ L n /L n ∼ = F n = 0 0 n ≡ 1 mod 2 p d I F n = 2d > 0.
Proof. The proof is by induction upon n; the case n = 0 is clear. It is straightforward to show that the odd degree homology is trivial (independently of the calculation of the even degree homology).
For the inductive step, consider the commutative diagram arising from the short exact sequence of complexes given by Lemma 4.7:
S n−6 / / K n−5 K n−3 / / S n−3 / / K n−2 K n / / S n / / K n+1 K n+3 / / S n+3 / / K n+4 ;
the homology H at the middle of the left hand column is calculated in terms of the known homologies in the other two columns, via the long exact sequence in homology.
For n = 2d > 0, there is a short exact sequence
0 → p d−1 I F → H → Λ d → 0,
by the inductive hypothesis for the left hand term and Proposition 4.4 for the exterior power, using the vanishing of homology in odd degrees. It suffices to show that p d I F is a subquotient of H. The Frobenius Φ : S d ֒→ S 2d maps to K 2d and the image lies in the kernelL 2d of K 2d → K 2d+3 . There is a unique non-trivial morphism S d → I F and this has image p d I F ; since I F is injective, this extends to give a commutative diagram
K 2d−3 S d 1 / / L 2d p d I F 1 / / I F . The composite morphism K 2d−3 → I F is trivial, since K 2d−3 (F) = 0; it follows that p d I F is a subquotient of the homology H, as required.
Example 4.11. It is straightforward to verify that L n = 0 for n ≤ 5, hence the initial values ofL i are given bỹ
L n = 0 n ∈ {1, 3, 5} Λ 1 n = 2 S 2 n = 4.
Connective complex K-cohomology and homology
A complete functorial description of both V → ku * (BV + ) and V → ku * (BV + ) is given, using the results of Section 3 and Section 4. 5.1. Recollections. The Postnikov towers of ku and KU provide morphisms ku → HZ and ku → KU of commutative ring spectra relating connective (resp. periodic) complex K-theory ku (resp. KU ) and the integral Eilenberg-MacLane spectrum HZ.
There are cofibre sequences
Σ 2 ku v → ku → HZ, HZ 2 → HZ ρ → HF, where v is multiplication by the Bott element (ku * ∼ = Z[v]
, with |v| = 2). These give rise to (generalized) Bocksteins; in particular:
Notation 5.1. Let Q denote the first k-invariant of ku, given by the composite HZ → Σ 3 ku → Σ 3 HZ.
Recall that the Milnor derivation Q 1 ∈ HF 3 HF is the commutator [Sq 2 , Sq 1 ] (the Milnor derivation Q 0 is the Bockstein β = Sq 1 ).
Lemma 5.2. There is a commutative diagram:
HZ Q / / ρ Σ 3 HZ ρ HF Q1 / / Σ 3 HF.
Proof. The morphism Q is the image under the integral Bockstein HF 2 HZ → HZ 3 HZ of the class of Sq 2 (recall that HF * HZ ∼ = A/(Sq 1 )) [Ada74, proof of III.16.6]. Hence there is a commutative diagram
HZ Q * * ρ / / Σ 3 ku / / Σ 3 HZ ρ HF Sq 2 / / Sq 3 4 4 Σ 2 HF Sq 1 / / 9 9
Since the composite Sq 1 • ρ is trivial, the result follows.
Notation 5.3. The morphisms in (co)homology induced by Q (respectively Q 1 ) will be denoted simply Q (resp. Q 1 ).
Lemma 5.4. For Y a spectrum, the following conditions are equivalent:
(1) HZ * Y ρ → HF * Y is a monomorphism; (2) the Bockstein complex (HF * Y, β) is exact; (3) HZ * Y ρ → HF * Y is a monomorphism; (4) the Bockstein complex (HF * Y, β) is exact.
When these conditions are satisfied, the respective morphisms Q are determined by the commutative diagrams:
HZ * Y Q / / ρ HZ * +3 Y ρ HZ * Y Q / / ρ HZ * −3 Y ρ HF * Y Q1 / / HF * +3 Y HF * Y Q1 / / HF * −3 Y.
Proof. The equivalence of the conditions is standard and the commutative diagrams follow from Lemma 5.2; these determine Q, since the vertical morphisms are injective, by hypothesis.
Example 5.5. The hypotheses of Lemma 5.4 are satisfied for Y = Σ ∞ BV , where V is an elementary abelian 2-group. In particular, the action of Q on HZ * (BV ) can be understood in terms of the action of Λ(Q 0 , Q 1 ) on HF * (BV ).
5.2.
On v-torsion and cotorsion. There is a hereditary torsion theory (T , F ) on the category of Z[v]-modules where T is the category of v-torsion modules (every element is annihilated by some power of v) and F the category of v-cotorsion
modules (those Z[v]-modules M which embed in M [ 1 v ])
. This torsion theory has associated torsion functor tors v and cotorsion functor cotors v so that, for a Z[v]module, there is a natural short exact sequence of Z[v]-modules:
0 → tors v M → M → cotors v M → 0.
The submodule of M annihilated by v is written ann v M .
Lemma 5.6. For a Z[v]-module M , there is an isomorphism tors v M ∩ vM ∼ = vtors v M , hence tors v M ֒→ M induces a monomorphism (tors v M )/v ֒→ M/v.
Moreover, there is a short exact sequence of Z[v]-modules:
0 → cotors v M v → cotors v M → (M/v)/ (tors v M )/v → 0. Proof. Straightforward. For a Z[v]-module M , there is a composite ann v M ֒→ tors v M ։ (tors v M )/v.
The following algebraic result is required in the proof of Proposition 5.12.
Lemma 5.7. For M a graded Z[v]-module, the following conditions are equivalent: (2) and (3). Suppose condition (2) holds and consider x ∈ tors v M , so that there exists t ∈ N such that v t x = 0 and v t+1 x = 0. Hence v t x ∈ ann v M ; the hypothesis implies that v t x is not in vM , hence t = 0 and x ∈ ann v M , as required. Now suppose that condition (3) holds, under the additional hypothesis. Consider x ∈ tors v M ; by surjectivity of ann v M ։ (tors v M )/v, x = a + vy, for a ∈ ann v M and y ∈ tors v M . An induction shows that, for 0 < n ∈ N, x = a + v n y n , for some y n ∈ tors v M ; it suffices to show that v n y n = 0 for n ≫ 0. By construction v n y n ∈ (v n M ∩ tors v M ) |x| , hence the hypothesis implies that the element is zero for n ≥ N (|x|).
(1) ann v M = tors v M ; (2) ann v M → (tors v M )/v is injective. If, for each degree d, there exists N (d) ∈ N such that (v N (d) M ∩ tors v M ) d = 0 then these conditions are equivalent to (3) ann v M → (tors v M )/v is surjective. Proof. If ann v M = tors v M , then tors v M ∼ = (tors v M )/v and hence condition (1) implies both
Remark 5.8. The example Z[v]/v ∞ shows that the conditions (1) and (3) are not equivalent without the additional hypothesis.
The following Lemma applies when considering ku-cohomology of a spectrum and allows the cotorsion to be related to periodic K-theory. . For the applications, Z will be the suspension spectrum of a space, hence the connectivity hypothesis is not restrictive.
Lemma 5.10. For X a spectrum:
(1) there is a natural exact sequence of Z[v]-modules:
0 → tors v ku * (X) → ku * (X) → cotors v ku * (X) → 0
and a natural isomorphism
cotors v ku * (X) ∼ = image{ku * (X) → KU * (X)};
(2) there is a natural exact sequence of Z[v]-modules:
0 → tors v ku * (X) → ku * (X) → cotors v ku * (X) → 0
and, if X is connective, a natural isomorphism
cotors v ku * (X) ∼ = image{ku * (X) → KU * (X)}.
Lemma 5.11. For X a spectrum, there are natural short exact sequences
0 → ku * (X)/v → HZ * (X) → ann v ku * −3 (X) → 0; 0 → ku * (X)/v → HZ * (X) → ann v ku * +3 (X) → 0.
Moreover, there are natural inclusions
ImQ ⊂ tors v ku * (X) /v ⊂ ku * (X)/v ⊂ KerQ; ImQ ⊂ tors v ku * (X) /v ⊂ ku * (X)/v ⊂ KerQ.
Proof. The proofs for homology and cohomology are formally the same, hence consider ku-homology. The short exact sequence is induced by the cofibre sequence Σ 2 ku → ku → HZ. The inclusion (tors v ku * (X))/v ⊂ ku * (X)/v is provided by Lemma 5.6; the outer inclusions are clear, from the definition of Q.
Proposition 5.12. Let Z be a connective spectrum.
(1) The following conditions are equivalent:
0 / / tors v ku * (Z) / / ∼ = ku * (Z) / /
cotors v ku * (Z)
/ / 0 0 / / ImQ / / KerQ / / KerQ/ImQ / / 0.
(2) Suppose that, for each degree d, there exists an integer N (d) such that
(v N (d) ku * (Z) ∩ tors v ku * (Z)) d = 0,0 / / tors v ku * (Z) / / ∼ = ku * (Z) / /
cotors v ku * (Z)
/ / 0 0 / / ImQ / / KerQ / / KerQ/ImQ / / 0.
Proof. The hypotheses imply that, for M ∈ {ku * (Z), ku * (Z)}, the three conditions of Lemma 5.7 are equivalent. The equivalence of the conditions (a), (b), (c) follows from an analysis of the short exact sequences of Lemma 5.11; consider ku-homology (the argument for ku-cohomology is similar), so that there is a commutative diagram:
KerQ 0 / / ku * (Z)/v / / A Ù 6 6 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ HZ * (Z) / / ann v ku * −3 (Z) / / w w w w ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ 0 (tors v ku * (Z))/v c 1 O O ImQ, c o o
in which the middle row and column are both short exact.
By the five lemma, ImQ ∼ = ann v ku * −3 (Z) if and only if KerQ ∼ = ku * (Z)/v. Moreover, Lemma 5.7 implies that the following three conditions are equivalent: (1) Under the hypotheses of the Proposition, ku * (Z) (respectively ku * (Z)) is determined by (HZ * (Z), Q) (resp. (HZ * (Z), Q)), up to the analysis of the v-adic filtration of cotors v ku * (Z) (resp. cotors v ku * (Z)).
(1) ann v ku * (Z) = tors v ku * (Z); (2) ImQ ∼ = (tors v ku * (Z))/v; (3) ImQ ∼ = ann v ku * −3 (Z).
(2) If HZ * (Z) → HF * (Z) is a monomorphism (so that the hypotheses of Lemma 5.4 are satisfied), then the data is provided by HF * (Z), considered as a Λ(Q 0 , Q 1 )-module and there is a form of duality between ku * (Z) and ku * (Z).
(3) Related considerations for connected Morava K-theories occur in [Lel82].
The nilpotency hypothesis of Proposition 5.12 is supplied by the following result when considering the ku-cohomology of spaces. When considering the unreduced ku-cohomology of a space, under the cohomological hypothesis of Proposition 5.12, the following result is clear.
Proposition 5.16. (Cf. [BG03].) Let Y be a space such that tors v ku * (Y + ) ∼ = ann v ku * (Y + ), then the algebra structure of ku * (Y + ) is determined by the induced algebra structures of ku * (Y + )/v and cotors v ku * (Y + ). In particular, there is a monomorphism of algebras
ku * (Y + ) → KU * (Y + ) HZ * (Y + )
where cotors v ku * (Y + ) is considered as a subalgebra of KU * (Y + ) and ku * (Y + )/v ∼ = KerQ, as a subalgebra of HZ * (Y + ).
5.3.
The ku-cohomology of BV + . The above discussion applies in considering the group ku-cohomology ku * (BG + ). The periodic K-theory KU * (BG + ) of the finite group G is known by the Atiyah-Segal completion theorem to be trivial in odd degrees and isomorphic in even degrees to the completion R(G) ∧ I , where I is the augmentation ideal of the complex representation ring R(G).
Here we restrict to the case of elementary abelian 2-groups, and V → ku * (BV + ) is considered as a contravariant functor of V ∈ ObV f . The result is proved by applying Proposition 5.12, for which an understanding of HZ * (BV + ) and the action of Q is required.
Proposition 5.17. There are natural isomorphisms:
HZ n (BV + ) ∼ = Z n = 0 K n (V ♯ ) n > 0 (ImQ) n ∼ = L n (V ♯ ) (KerQ) n ∼ = Z n = 0 L n (V ♯ ) n > 0.
Proof. The algebra HF * (BV + ) is naturally isomorphic to S * (V ♯ ) and integral reduced cohomology HZ * (BV ) embeds in HF * (BV ) as the kernel of the Bockstein operator. Hence (paying attention to the behaviour in unreduced cohomology), the result follows from Lemma 5.4, using the definition of the functors K n , L n andL n from Section 4.
Lemma 5.18. There are identities tors v ku * (B0 + ) = 0 = tors v ku * (BZ/2 + ).
Proof. This follows from the identification of ku * (BZ/2 + ) using the Gysin sequence (cf. [BG03, Section 2.2]).
Theorem 5.19. For V ∈ ObV f , there are natural isomorphisms tors v ku n (BV + ) ∼ = ann v ku n (BV + ) ∼ = (ImQ) n ku n (BV + )/v ∼ = (KerQ) n and tors v ku n (BV + ) → ku n (BV + )/v is an isomorphism for n odd and, for n = 2d > 0, there is a natural short exact sequence:
0 → tors v ku 2d (BV ) → ku 2d (BV )/v → p d I F (V ♯ ) → 0.
The surjection cotors v ku * (BV + ) ։ cotors v ku * (BV + )/v induces a pullback diagram of short exact sequences
0 / / tors v ku * (BV + ) / / ku * (BV + ) / /
cotors v ku * (BV + )
/ / 0 0 / / ImQ / / KerQ / / (cotors v ku * (BV + ))/v / / 0.
There are natural isomorphisms cotors v ku * (BV + ) ∼ = Z[v] ⊕ cotors v ku * (BV ) and
cotors v ku 2d (BV ) ∼ = P d Z2 (V ♯ )
inducing an isomorphism of short exact sequences (for d > 0):
0 / / cotors v ku 2d+2 (BV ) ∼ = v / / cotors v ku 2d (BV ) ∼ = / / p d I F (V ♯ ) / / 0 0 / / P d+1 Z2 (V ♯ ) / / P d Z2 (V ♯ ) / / p d I F (V ♯ ) / / 0.
Proof. The first part of the Theorem follows from Proposition 5.12, since Lemma 2.7 implies that the cohomological hypothesis is satisfied. To apply the Proposition, it is sufficient to show that ImQ ∼ = tors v ku * (BV + ) /v. By Proposition 5.17 and Proposition 4.10,
(KerQ/ImQ) n ∼ = Z n = 0 0 n odd p d I F (V ♯ ) n = 2d, d > 0.
In odd degrees ImQ = KerQ, and the result follows from the inclusions given in Lemma 5.10.
It remains to show that the inclusion (ImQ) 2d ֒→ tors v ku 2d (BV + )/v is an isomorphism for d ∈ N. For d = 0, both terms are zero; for d > 0, the cokernel is a subfunctor of V → p d I F (V ♯ ) by the inclusions given in Lemma 5.10 and the above identification of KerQ/ImQ, hence it suffices to show that the cokernel is trivial when evaluated on V = F, by Lemma 2.7. This follows from the fact that tors v ku * (BZ/2 + ) = 0, by Lemma 5.18.
Finally, consider cotors v ku * (BV + ). For d = 0, the result is the Atiyah-Segal completion theorem; the structure in higher degree follows by induction on d from the results of Section 3.1, in particular Proposition 3.7, using the identification of the functors (cotors v ku * (BV + ))/v ∼ = KerQ/ImQ given above.
Remark 5.20. Proposition 5.16 applies to ku * (BV + ) to give a description of the algebra structure (cf. [BG03]).
5.4.
The ku-homology of elementary abelian 2-groups. The ku-homology of elementary abelian 2-groups can be determined as for ku-cohomology, by applying Proposition 5.12, for which an understanding of HZ * (BV + ) and the action of Q is required.
Proposition 5.21. There are natural isomorphisms:
HZ n (BV + ) ∼ = Z n = 0 DK n+1 (V ) n > 0 (ImQ) n ∼ = DL n+4 (KerQ) n ∼ = Z n = 0 D(K n+1 /L n+1 )(V ) n > 0.
Moreover:
(KerQ/ImQ) n ∼ = Z n = 0 0 n ≡ 0 mod 2 q d P F (V ) n = 2d − 1 > 0.
Proof. There is a natural isomorphism HF n (BV + ) ∼ = Γ n (V ) and HZ ρ → HF induces a monomorphism in reduced homology, so Lemma 5.4 applies. It remains to identify the functors upon dualizing, with the attendant shift in indexing.
There is a natural isomorphism
HZ n (BV + ) ∼ = DK n+1 (V ),
for n > 0 and the natural transformation Q is induced by DQ 1 .
Consider the Q 1 complex K n−3 → K n → K n+3 (for n > 3); this gives rise to short exact sequences
0 → L n →L n → H n → 0 0 →L n → K n → L n+3 → 0 0 → H n → K n /L n → L n+3 → 0,
where H n denotes the homology.
In the dual complex, DK n+3 DQ1 → DK n DQ1 → DK n−3 :
ImDQ 1 ∼ = DL n+3 KerDQ 1 ∼ = D(K n /L n ),
the inclusion ImDQ 1 ֒→ KerDQ 1 is dual to K n /L n ։ L n+3 and the cokernel is DH n . This proves the first statement (taking into account the unreduced homology in degree zero and the degree shift); the calculations of Example 4.11 show that the expressions are correct in low degrees. The final statement follows from Proposition 4.10, by dualizing.
Theorem 5.22. For V ∈ ObV f , there are natural isomorphisms:
tors v ku n (BV + ) ∼ = ann v ku n (BV + ) ∼ = (ImQ) n ku n (BV + )/v ∼ = (KerQ) n
and the inclusion tors v ku * (BV + ) ֒→ ku * (BV + ) induces a natural short exact sequence
0 → tors v ku * (BV + ) → ku * (BV + )/v → cotors v ku * (BV + )/v → 0, where cotors v ku n (BV + )/v ∼ = Z n = 0 0 0 < n ≡ 0 mod 2 q d P F (V ) n = 2d − 1 > 0.
There is a pullback diagram of short exact sequences
0 / / tors v ku * (BV + ) / / ∼ = ku * (BV + ) / / cotors v ku * (BV + ) / / 0 0 / / ImQ / / KerQ / / (cotors v ku * (BV + ))/v / / 0.
For d > 0 an integer, there is a natural isomorphism
cotors v ku 2d−1 (BV ) ∼ = R d Z (V ) inducing an isomorphism of short exact sequences 0 / / cotors v ku 2d−1 (BV ) ∼ = ×v / / cotors v ku 2d+1 (BV ) ∼ = / / q d+1 P F (V ) / / 0 0 / / R d Z (V ) / / R d+1 Z (V ) / / q d+1 P F (V ) / / 0, where R d Z ֒→ R d+1 Z is the dual of the natural projection R d+1 Z ։ R d Z .
Proof. There are natural monomorphisms
ImQ(V ) ֒→ ku * (BV + )/v ֒→ KerQ(V );
by Proposition 5.21, in positive even degree, these are isomorphisms; in degree
n = 2d − 1 > 0, the quotient (KerQ/ImQ)(V ) is q d P F (V ), hence there is a natural inclusion (ku * (BV + )/v)/ImQ(V ) 2d−1 ֒→ q d P F (V ).
To prove the result, by Proposition 5.12, if suffices to show that this is an isomorphism; hence, by Lemma 2.7, it suffices to show that the left hand side is non-trivial when evaluated on F, for all d > 0. It is straightforward to verify that ImQ odd (F) = 0, thus it suffices to show that ku * (BZ/2 + )/v is non-trivial in all odd degrees, which follows from the structure of ku * (BZ/2 + ) as a graded abelian group (see [BG03,Section 3.4], for example). Finally, the identification of cotors v ku * (BV + ) follows from the results of Section 3.2, in particular the short exact sequences of Lemma 3.10, the self-duality of the functors R n Z (Theorem 3.12) and the Pontrjagin duality between R ∞ Z and P Z2 (Theorem 3.14). (Compare the proof of Theorem 5.19.)
Remark 5.23. By [BG03, Proposition 3.2.1], the universal coefficient spectral sequence calculating ku * (BV ) from ku * (BV ) collapses to the short exact sequence
0 → Ext 2 ku * (Σ 2 tors v ku * (BV ), ku * ) → ku * (BV ) → Ext 1 ku * (Σ 1 cotors v ku * (BV ), ku * ) → 0. This is isomorphic to 0 → tors v ku * (BV ) → ku * (BV ) → cotors v ku * (BV ) → 0
and explains the duality between ku-homology and ku-cohomology of BV . The analysis of [BG03, Section 4.12] can be made functorial to give the identification of cotors v ku * (BV + ).
Local duality
An equivariant version of local duality (with respect to the action of the general linear groups Aut(V )) is given as it arises in the current context; this gives a refinement of the results of [BG03, Section 4.7].
6.1. Categories of S • -modules. Throughout this section, the prime p is arbitrary. The fact that the functor S • takes values in graded vector spaces of finite type will be used without further comment.
Definition 6.1.
(1) Let S • −mod F denote the category of graded right S • -modules in F and S • -module morphisms. Remark 6.2. The choice to work with right modules is dictated by the notation adopted for Koszul complexes.
An object of S • − mod F is a graded functor M • , equipped with a structure morphism M • ⊗ S • → M • which is unital and associative; this can be expressed in terms of the components M b ⊗ S a → M a+b . A similar description holds for S • (V )-modules. (1) the forgetful functor S • −mod F → F is exact and admits an exact left adjoint − ⊗ S • : F → S • −mod F which is monoidal;
(2) the forgetful functor S • (V ) − mod Aut(V ) → Aut(V ) − mod is exact and admits an exact left adjoint −⊗S • (V ) : Aut(V )−mod → S • (V )−mod Aut(V ) which is monoidal; (3) evaluation at V , F → Aut(V ) − mod, induces an exact tensor functor S • −mod F → S • (V )−mod Aut(V ) .
Proof. Clear.
Definition 6.4. For N ∈ ObS • −mod F , let Hom V S • (−, N ) be the functor (S • − mod F ) op → S • (V )−mod Aut(V ) defined by Hom V S • (M, N ) := Hom S • (V ) (M (V ), N (V )
) where the right hand side is equipped with the usual grading and Aut(V ) acts via conjugation.
Remark 6.5. The above definition can be refined to give a coefficient system for the general linear groups over F associated to the pair M, N of graded S • -modules in F .
Lemma 6.6. For F ∈ ObF and V ∈ ObV f , there is a natural isomorphism:
Hom V S • (F ⊗ S • , S • ) ∼ = F (V ) ♯ ⊗ S • (V ) in Aut(V )−mod, where F (V ) ♯ is equipped with the contragredient action.
Proof. Straightforward.
The following is clear:
Lemma 6.7. Let F, G ∈ ObF and α : F ⊗ S • → G ⊗ S • be a morphism of S • −mod F , induced byα : F → G ⊗ S • in F . Then Hom V S • (α, S • ) identifies with the morphism G(V ) ♯ ⊗ S • (V ) → F (V ) ♯ ⊗ S • (V ) of S • (V )−mod Aut(V ) induced by γ : G(V ) ♯ → F (V ) ♯ ⊗ S • (V ) in Aut(V )− mod, where γ is adjoint (in the category Aut(V )−mod) to the morphism G(V ) ♯ ⊗ S • (V ) ♯ → F (V ) ♯ dual to the evaluation ofα on V .
6.2. The dualizing functor. Recall that the exterior power functors are self-dual, so that:
Lemma 6.8. For n ∈ N, there is a natural isomorphism of contravariant functors of V : Λ n (V ♯ ) ∼ = Λ n (V ) ♯ .
This is combined with the following duality result, when restricting to the consideration of the Aut(V )-action: Lemma 6.9. Let 0 ≤ j ≤ r be integers and V ∈ ObV f have rank r. The composite
Λ r (V ) ⊗ Λ j (V ) ♯ ∆⊗1 → Λ r−j (V ) ⊗ Λ j (V ) ⊗ Λ j (V ) ♯ → Λ r−j (V ),
where the second morphism is induced by evaluation
Λ j (V ) ⊗ Λ j (V ) ♯ → F, induces an isomorphism Λ r (V ) ⊗ Λ j (V ) ♯ ∼ = Λ r−j (V ) in Aut(V )−mod, where Λ j (V ) ♯ is equipped with the contragredient Aut(V )-module structure.
Proof. The result follows from the fact that the product Λ r−j (V ) ⊗ Λ j (V ) → Λ r (V ) ∼ = F defines a perfect pairing and the equivariance of the evaluation map.
Lemma 6.10. Let 1 ≤ j ≤ r be integers, V ∈ ObV f have rank r and write µ : Λ j−1 (V ) ♯ ⊗ Λ 1 (V ) ♯ → Λ j (V ) ♯ for the product morphism (dual to the evaluation of the coproduct Λ j → Λ j−1 ⊗ Λ 1 ).
Then, under the isomorphism of Lemma 6.9, Λ r (V ) ⊗ µ is Aut(V )-equivariantly isomorphic to the morphism
Λ r−j+1 (V ) ⊗ Λ 1 (V ) ♯ → Λ r−j (V )
which is adjoint to the evaluation of the coproduct Λ r−j+1 → Λ r−j ⊗ Λ 1 on V .
Proof. A consequence of the coassociativity of the comultiplication on the exterior power functors and the fact that the multiplication µ is dual to the coproduct Λ j → Λ j−1 ⊗ Λ 1 .
Notation 6.11. For i ∈ N and V ∈ ObV f of rank r, let
(1) τ i : Λ 1 → S p i denote the composite of the isomorphism Λ 1 ∼ = S 1 with the iterated Frobenius S 1 ֒→ S p i and also the induced Koszul differential
τ i : Λ j ⊗ S • → Λ j−1 ⊗ S •+p i
in the category S • −mod F , induced by the composite morphism:
Λ j ∆ → Λ j−1 ⊗ Λ 1 1⊗τi → Λ j−1 ⊗ S p i ;
(2) Kz i denote the Koszul complex in S • −mod F :
. . . → Λ j ⊗ S • τi → Λ j−1 ⊗ S •+p i → . . . → S •+jp i → 0;
(3) Kz i (V ) denote the Koszul complex in S • (V )−mod Aut(V ) :
0 → Λ r (V ) ⊗ S • (V ) → . . . → Λ j (V ) ⊗ S •+(r−j)p i (V ) τi → Λ j−1 (V ) ⊗ S •+(r−j+1)p i (V ) → . . . → S •+rp i (V ) → 0.
Proposition 6.12. For integers 1 ≤ j ≤ r, V ∈ ObV f of rank r and i ∈ N, the morphism
Hom V S • (τ i , S • ) : Hom V S • (Λ j−1 ⊗ S • , S • ) → Hom V S • (Λ j ⊗ S • , S • )
is induced by the morphism γ :
Λ j−1 (V ) ♯ → Λ j (V ) ♯ ⊗ S p i (V ) such that, under the isomorphism of Lemma 6.9, Λ r (V ) ⊗ γ : Λ r−j+1 (V ) → Λ r−j (V ) ⊗ S p i (V )
is Aut(V )-equivariantly isomorphic to the evaluation on V of the Koszul differential.
Proof. Combine Lemma 6.7 with Lemma 6.10.
Corollary 6.13. For V ∈ ObV f of rank r and i ∈ N, there is a natural isomorphism of complexes:
Hom V S • (Kz i , Λ r ⊗ S • ) ∼ = Kz i (V ) in S • (V )−mod Aut(V ) .
Proof. After addition of the twisting functor Λ r , the result is an immediate consequence of Proposition 6.12.
Remark 6.14. Taking i = 0, so that Kz 0 is the usual Koszul complex, which has homology F concentrated in homological degree zero, this shows that Λ r ⊗S • plays the role of the dualizing object, corresponding to the fact that S • is graded Gorenstein (cf. [BH93, Section 3.7]). 6.3. Local cohomology in S • (V )−mod Aut(V ) . In this section, local cohomology is considered with respect to the augmentation ideal I of S • (V ), where V ∈ ObV f has rank r; from the results of the previous section, it follows that the local duality isomorphism (cf. [BH93]) should be interpreted as stating that the local cohomology of the S • (V )-free object of S • (V )−mod Aut(V ) , F (V ) ⊗ S • (V ), for F ∈ ObF , is concentrated in cohomological degree r, where it is isomorphic to
Hom V f Hom V S • (F ⊗ S • , Λ r ⊗ S • ), F .0 → F r ⊗ S • → F r−1 ⊗ S • → . . . → F 0 ⊗ S • → G → 0 which induces an exact sequence in S • (V )−mod Aut(V ) , after evaluation on V . Then the local cohomology of G(V ) ∈ ObS • (V )−mod Aut(V ) is H i I (G(V )) ∼ = H r−i Hom V f (Hom V S • (F j ⊗ S • , Λ r ⊗ S • )
, F) , up to shift in grading.
Filtering symmetric powers and Koszul complexes
To calculate the local cohomology of tors v ku * (BV + ) at the prime two by using a form of local duality, it is necessary to filter and study the associated Koszul complexes (for odd primes, this filtration step is unnecessary). The results of this section refine those of Section 4.
f 0 S • = ΦS • ⊂ f 1 S • ⊂ f 2 S • ⊂ . . . ⊂ f t S • ⊂ . . . ⊂ S • in ΦS • -modules;
(2) there is an isomorphism of S • -modules:
f t S • /f t−1 S • ∼ = Λ t ⊗ S • ,
where S • acts on the left hand side by restriction along S • Φ ∼ = → ΦS • and by multiplication on the right hand factor of Λ t ⊗ S • ; for V ∈ ObV f , this restricts to an isomorphism of S • (V )-modules:
t≥0 f t S • /f t−1 S • (V ) ∼ = S • (V ),
where S • (V ) acts on the right hand side via Φ;
(3) for V ∈ ObV f , the inclusion f t S • (V ) ֒→ S • (V ) is an isomorphism for t ≥ dim V .
Proof. Straightforward; to prove that S • (V ) is isomorphic to t≥0 f t S • /f t−1 S • (V ) as S • (V )-modules, it is sufficient to consider monomial bases. (Note that this statement is not true Aut(V )-equivariantly.)
For notational clarity, shifts in gradings are omitted from the following statement.
Proposition 7.3. For i, t ∈ N, the Milnor derivation Q i : S • → S • is a morphism of ΦS • -modules and restricts to f t S • ftQi → f t−1 S • ; the induced morphism on the filtration quotients
Λ t ⊗ S • ∼ = f t S • /f t−1 S • → f t−1 S • /f t−2 S • ∼ = Λ t−1 ⊗ S • is the Koszul differential τ i . Proof. Straightforward.
The differentials τ 0 and τ 1 define a bicomplex structure on Λ • ⊗ S • , which can be displayed as:
F Λ 1 / / S 1 Λ 2 / / Λ 1 ⊗ S 1 / / S 2 Λ 3 / / Λ 2 ⊗ S 1 / / Λ 1 ⊗ S 2 / / S 3 Λ 4 / / Λ 3 ⊗ S 1 / / Λ 2 ⊗ S 2 / / Λ 1 ⊗ S 3 / / S 4
The portion of the bicomplex displayed indicates the essential features:
(1) the rows (respectively columns) are Koszul complexes, with differential τ 0 (resp. τ 1 ); (2) the bicomplex is concentrated in a single quadrant and there are vanishing lines of slope 1/2 and 1.
7.2. Filtering the functors K n . This section establishes the filtered version of Proposition 4.10; the starting point is the functorial homology of the Koszul complexes, which is a standard calculation, related to Proposition 4.4:
Proposition 7.4. For i ∈ N, the homology of (Λ • ⊗S • , τ i ) is S • / x 2 i , concentrated in homological degree zero.
The following are analogous to the functors K n introduced in Notation 4.6:
Notation 7.5. For an integer a ≥ 1 and b ∈ N, let K a,b denote the image of τ 0 : Λ a ⊗ S b → Λ a−1 ⊗ S b+1 and, by convention:
K 0,b := F b = 0 0 b > 0.
Remark 7.6.
(1) The functor K 1,b identifies with the symmetric power S b+1 .
(2) Taking the image of τ 0 rather than the kernel, is best suited for the current application, where the homology of the Koszul complex intervenes.
Lemma 7.7. For 0 < n ∈ Z, the filtration f * S • induces a filtration of K n with associated graded:
grK n ∼ = a+2b+1=n K a,b .
Lemma 7.12. For n > 0 and CokerQ 1 as above, the filtration f * S • induces a finite filtration of CokerQ 1 with associated graded
grCokerQ 1 ∼ = a+2b+1=n,a≥1 L a,b ,
where, for 2(b+1) = n, the subobject L 1,b is isomorphic toL 2(b+1) /L 2(b+1) ∼ = p b+1 I F and there is an induced isomorphism:
grL n+3 ∼ = a+2b+1=n,a≥2 L a,b .
Proof. The result follows as for the proof of Lemma 7.7.
Remark 7.13. For the calculations of local cohomology, it is important that the isomorphisms of Lemma 7.7 and of Lemma 7.12 upon evaluation on V ∈ ObV f correspond to isomorphisms in the category of S • (V )-modules (as in Lemma 7.2).
The local cohomology spectral sequence
The local cohomology theorem for ku implies that there is a spectral sequence:
E 2 := H * , * I (ku * (BV + )) ⇒ ku * (BV + ),
where the E 2 -term is the local cohomology with respect to the augmentation ideal (see [BG03] for generalities on the spectral sequence, for arbitrary finite groups).
Here V is taken to be a fixed elementary abelian 2-group of rank r. The aim of this section is to indicate how the spectral sequence can be understood conceptually, by using the functorial calculations introduced in Section 4.
There are two key ingredients: the functorial description of local duality and of local cohomology given in Section 6 and an explanation of the relationship between the local cohomology of tors v ku * (BV + ) and that of cotors v ku * (BV + ).
The local cohomology spectral sequence can be made Aut(V )-equivariant but it is clearly not functorial as it stands with respect to arbitrary vector space morphisms; the techniques of this section do however show that the behaviour of the spectral sequence is largely determined by functorial structure.
Remark 8.1. Throughout, the grading shifts resulting from working with graded modules are suppressed. The gradings are not essential for the presentation of the arguments; the reader is encouraged to supply them.
8.1. The case of integral cohomology. The local cohomology spectral sequence for HZ * (BV + ) already illustrates some of the salient features of the local cohomology spectral sequence. It can also be used in the analysis of the local cohomology spectral sequence for ku * (BV + ) via the morphism induced by ku → HZ.
Let V ∈ ObV f have rank r and consider the short exact sequence
0 → HZ * (BV ) → HZ * (BV + ) → Z → 0
relating reduced and unreduced cohomology of BV , in the category of HZ * (BV + )modules, so that HZ * (BV ) corresponds to the augmentation ideal I. There is an induced exact sequence of local cohomology groups:
0 → H 0 I (HZ * (BV )) → H 0 I (HZ * (BV + )) → (2)
→ Z → H 1 I (HZ * (BV )) → H 1 I (HZ * (BV + )) → 0 and, for j > 1, a natural isomorphism H j I (HZ * (BV + )) ∼ = H j I (HZ * (BV )). Hence, up to calculating the connecting morphism Z → H 1 I (HZ * (BV )), the local cohomology of HZ * (BV + ) is determined by that of HZ * (BV ). Moreover, it is clear that H 1 I (HZ * (BV )) is annihilated by 2, hence it suffices to consider behaviour after reducing mod 2.
Notation 8.2. For a ∈ N, let σ ≥a Kz 0 denote the brutal truncation to the right of the Koszul complex:
. . . → Λ a+1 ⊗ S •−1 τ0
→ Λ a ⊗ S • and let σ ≤a Kz 0 denote the brutal truncation to the left:
Λ a ⊗ S • τ0 → Λ a−1 ⊗ S •+1 τ0 → . . . → S •+a . Lemma 8.3. For a ∈ N,
(1) σ ≥a Kz 0 is an S • -free resolution of K a,• ;
(2) for a ≥ 1, the complex σ ≤a Kz 0 has homology F in homological degree 0 and K a+1,• in homological degree a.
Moreover, there are morphisms of complexes
Kz 0 ։ σ ≥1 Kz 0 ։ σ ≥2 Kz 0 ։ . . . σ ≤0 Kz 0 ∼ = S • ֒→ σ ≤1 Kz 0 ֒→ σ ≤2 Kz 0 ֒→ . . . ⊂ Kz 0 .
Proof. Clear.
Proposition 8.4. For V ∈ ObV f of rank r and integers 0 ≤ a ≤ b ≤ r, there is a natural isomorphism of complexes: Aut(V ) and, with respect to these isomorphisms, the surjection σ ≥a Kz 0 ։ σ ≥b Kz 0 induces the inclusion σ ≤r−b Kz 0 (V ) ֒→ σ ≤r−a Kz 0 (V ).
Hom V S • (σ ≥a Kz 0 , S • ⊗ Λ r ) ∼ = σ ≤r−a Kz 0 (V ) in S • (V ) − mod
In particular, the surjection Kz 0 ։ σ ≥b Kz 0 induces the inclusion:
σ ≤r−b Kz 0 (V ) ֒→ Kz 0 (V ),
which induces an isomorphism in degree zero homology if b < r.
Proof. The result follows from Corollary 6.13.
Remark 8.5. It is useful to think of the surjection σ ≥a Kz 0 ։ σ ≥b Kz 0 as a morphism in an appropriate derived category
K a,• [a] → K b,• [b],
where [a], [b] correspond to the shift in homological degree. In particular, for a = 0, this corresponds to
F → K b,• [b].
The morphism σ ≥a Kz 0 ։ σ ≥b Kz 0 induces a morphism between local cohomology groups (a generalized connecting morphism), via the identification of local cohomology given in Proposition 6.16 and Proposition 8.4.
Using the above observation, one deduces: Lemma 8.6. For V ∈ ObV f of rank r > 1, the connecting morphism H 0 I (Z/2) ∼ = Z/2 → H 1 I (HZ * (BV )) induced by the short exact sequence of HZ * (BV + )-modules 0 → HZ * (BV ) → HZ * (BV + )/2 → Z/2 → 0 is non-trivial.
Remark 8.7. The connecting morphism in the long exact sequence for local cohomology, (2), is therefore non-trivial. For a conceptual presentation of the results, it is useful to define an associated E 1 -page, so that this connecting morphism appears as the d 1 differential.
The local cohomology (r > 1) is as follows, using Lemma 8.3:
H j I (HZ * (BV + )) ∼ = Z j = 0 0 j = 1 F 2 ≤ j ≤ r − 1 H r I (HZ * (BV )) j = r.
Moreover, H r I (HZ * (BV )) has a finite filtration such that grH r I (HZ * (BV )) ∼ = F ⊕ grHZ * (BV ), up to shift in degree, where the filtration on homology HZ * (BV ) is induced by the filtration f t S • .
The analysis of the local cohomology spectral sequence is straightforward; the local cohomology corresponds to the E 2 -page of the spectral sequence. The permanent cycles in the zero column are given by the subgroup 2 r−1 Z; the differentials d i , for 2 ≤ i ≤ r are all non-trivial, starting from the zero column, and serve to eliminate the extraneous factors of F which occur above.
Remark 8.8. Heuristically it is useful to consider that the differential d i is induced by the surjection of complexes Kz 0 ։ σ ≥i Kz 0 for i ≥ 1, using Remark 8.7 to interpret the connecting morphism as d 1 .
8.2. Bicomplexes of S • -modules. The purpose of this section is to explain the calculation of the local cohomology of tors v ku * (BV + ); a fundamental point is that the method also calculates the local cohomology of cotors v ku * (BV )/v and explains all the differentials in the local cohomology spectral sequence. This relies on the following result, in which grading shifts have been suppressed and, for variance reasons, the cohomology of V ♯ is considered.
tors v ku * (BV ♯ + ) ∼ = r i=2 L i, * (V ).
Proof. The result follows from Lemma 7.12 and Theorem 5.19.
For the consideration of local duality, it is necessary to consider the S • -action on the Λ a ⊗ S b -bicomplex introduced in Section 7.1. This gives a half plane bicomplex with differentials of the form
/ / Λ j+2 ⊗ S •−3 τ0 / / τ1 Λ j+1 ⊗ S •−2 τ1 / / / / Λ j+1 ⊗ S •−1 τ0 / / Λ j ⊗ S • / / s o o t O O
in the (s, t)-plane. Consider the following brutal truncations, which are analogues of the truncated Koszul complexes of Notation 8.2. (1) B(i) be the bicomplex in S • −mod F : B(i) s,t := 0 t < i or s < 0; Λ s+t ⊗ S • t ≥ i, considered as a quotient bicomplex, where the term of lowest total degree is Λ i ⊗ S • , in bidegree (0, i).
(2) D(i) be the bicomplex in S • −mod F : D(i) s,t := 0 t > i or s > 0; Λ s+t ⊗ S • t ≤ i, considered as a sub-bicomplex, where the term of greatest total degree is Λ i ⊗ S • , in bidegree (0, i).
Remark 8.11. When evaluated on V ∈ ObV f of rank r, the only non-trivial terms are those with s + t ≤ r and t ≥ i. In particular, B(i)(V ) is trivial if i > r.
Example 8.12. Taking r = 5 and i = 2, B(2)(F 5 ) is given by evaluating the following bicomplex on F 5 :
Λ 5 ⊗ S * −6 τ1 Λ 5 ⊗ S * −5 τ1 τ0 / / Λ 4 ⊗ S * −4 τ1 Λ 5 ⊗ S * −4 τ1 τ0 / / Λ 4 ⊗ S * −3 τ1 τ0 / / Λ 3 ⊗ S * −2 τ1 Λ 5 ⊗ S * −3 τ0 / / Λ 4 ⊗ S * −2 τ0 / / Λ 3 ⊗ S * −1 τ0 / / Λ 2 ⊗ S * s o o t O O
where Λ 2 ⊗ S * is in (s, t)-degree (0, 2).
Similarly, D(2)(F 5 ) is obtained by evaluating the following on F 5 :
Λ 2 ⊗ S * τ0 / / τ1 Λ 1 ⊗ S * +1 τ0 / / τ1 S * +2 Λ 1 ⊗ S * +2 τ0 / / τ1 Λ 1 ⊗ S * +3 S * +4 .
Remark 8.13.
(1) The grading on B(i) used in [BG03, Chapter 4] (respectively on D(i)) can be recovered by considering the grading of the 'generators' in the lowest (resp. greatest) total degree, since the morphism τ 0 raises the degree by 2 and τ 1 raises the degree by 4 (the grading is calculated relative to the S • -grading, so that the usual gradings of the odd degree generators do not contribute).
(2) The homology of the bicomplexes is calculated pointwise, by first evaluating on V ∈ ObV f ; for any i and V , the bicomplex B(i)(V ) has only finitely many non-zero terms.
The following is clear from the definition.
Lemma 8.14.
(1) There are surjections of bicomplexes in S • −mod F : The following result follows from the definition of the objects K a,b and L a,b given in Section 4.
B
Lemma 8.15. For 0 < a ∈ Z, K a,• and L a,• are objects of S • −mod F such that the surjections Λ a ⊗ S • ։ K a,• ։ L a,• are morphisms of S • −mod F .
To establish the behaviour of the indices, consider the following commutative diagram (for a ≥ 2):
Λ a+1 ⊗ S •−2 τ1 / / / / K a+1,•−2 Λ a ⊗ S • / / / / τ1 K a,• L a,• Λ a−1 ⊗ S •+2 / / / / τ0 7 7 K a−1,•+2 1 / / Λ a−2 ⊗ S •+3 .
In particular, L a,b is a quotient of Λ a ⊗ S b and a subobject of Λ a−2 ⊗ S b+3 . In the following Proposition, recall the indexing convention of B(i), which means that the term of lowest total degree is in (s, t)-degree (0, i), hence has total degree i.
Proposition 8.16. For 0 < i ∈ Z, the homology of Tot(B(i)) is concentrated in degree i and H i (Tot(B(i))) ∼ = L i, * .
Proof. The result follows by filtering the bicomplex B(i) using Lemma 8.14 and Proposition 7.9 (cf. [BG03, Proposition 4.6.3]).
Remark 8.17. Although the bicomplex B(0) is defined, it does not give a resolution of F, since the homology is not concentrated in a single homological degree -which is a consequence of the homology of (Λ • ⊗ S • , τ 0 ). (3) A notational sleight of hand has been used; the local cohomology introduces a change of variance with respect to V ; L i, * on the left hand side should be considered as being contravariant in V , by precomposition with the duality functor ♯.
8.3. The local cohomology of cotors v ku * (BV + ). Throughout this section, V ∈ ObV f has rank r > 1. To simplify notation, Q will be written for the (contravariant) functor V → cotors v ku * (BV + ). Multiplication by v induces a short exact sequence 0 → Q v → Q → Q/v → 0 (omitting the suspension associated with the grading). Moreover, the augmentation Q → Z[v] induces a short exact sequence
0 → Q/v → Q/v → Z → 0.
Theorem 5.19 implies that there is a natural isomorphism Q/v ∼ = L 1, * (V ♯ ). Hence, by using the change of rings isomorphism associated to ku * (BV + ) ։ HZ * (BV + ), Corollary 8.20 gives the local cohomology of Q/v ; in particular, the ideal I refers here to the augmentation ideal of ku * (BV + ).
(3) The distinction between the cases i ≤ r − 3 and i = r − 2 is simply to underline the isomorphism with H i+2
Proposition 4 . 10 .
410The homology of the complexes (
Lemma 5.9. [BG03, Chapter 1] If Z is a connective spectrum, there is a natural isomorphism KU * (Z) ∼ = ku * (Z)[ 1 v ]
(a) ann v ku * (Z) = tors v ku * (Z);(b) tors v ku * (Z) /v ∼ = ImQ; (c) ku * (Z)/v ∼ = KerQ.If these conditions are satisfied, then (ku * (Z)/v)/ (tors v ku * (Z))/v ∼ = KerQ/ImQ and there is a short exact sequence of Z[v]-modules:0 → cotors v ku * −2 (Z) v → cotors v ku * (Z) → KerQ/ImQ → 0and a pullback:
then the following conditions are equivalent:(a) ann v ku * (Z) = tors v ku * (Z); (b) tors v ku * (Z) /v ∼ = ImQ; (c) ku * (Z)/v ∼ = KerQ.If these conditions are satisfied, then (ku * (Z)/v)/ (tors v ku * (Z))/v ∼ = KerQ/ImQ and there is a short exact sequence of Z[v]-modules:0 → cotors v ku * +2 (Z) v → cotors v ku * (Z) → KerQ/ImQ → 0and a pullback:
This shows that conditions (a), (b), (c) are equivalent. The consequences follow, using Lemma 5.6 to provide the short exact sequence which calculates cotors v ku * (Z)/v.Remark 5.13.
Lemma 5 .
514. [BG03, Lemma 1.5.8] Let Y be a space such that ku * (Y + ) is a Noetherian Z[v]-algebra. Then there exists a natural number N such that v N tors v ku * (Y + ) = 0. Example 5.15. For G a finite group, ku * (BG + ) is a Noetherian Z[v]-algebra (see [BG03, Section 1.1] for example).
( 2 )
2For V ∈ ObV f , let S • (V )−mod Aut(V ) denotethe category of graded right S • (V )-modules in Aut(V )-modules and S • (V )-module morphisms.
Lemma 6 . 3 .
63For V ∈ ObV f , the categories S • −mod F and S • (V )−mod Aut(V ) are tensor abelian. Moreover:
Remark 6.15. In the cases of interest here, F takes finite-dimensional values, so thatHom V S • (F ⊗ S • , Λ r ⊗ S • ) isa graded vector space of finite type. Proposition 6.16. (Cf. [BG03, Lemma 4.7.1].) Let V ∈ ObV f have rank r and G ∈ ObS • −mod F . Suppose that there exists a complex
7. 1 .
1Filtering the symmetric powers. Let ΦS • denote the image of the Frobenius, so that ΦS • takes values in graded commutative algebras and the canonical morphisms S • ∼ = → ΦS • ֒→ S • are morphisms of algebras. Definition 7.1. For t ∈ N, let f t S • ⊂ S • denote the image of multiplication S t ⊗ ΦS • µ → S • . Lemma 7.2. For t ∈ N, (1) the functors f t S • ⊂ S • define an increasing filtration of S • :
Proposition 8.9. (Cf. [BG03, Section 4.6].) For V ∈ ObV f of rank r, ku * (BV ♯ + ) admits a finite natural filtration with associated graded grtors v ku * (BV ♯ + ) ∼ = r i=2 L i, * (V ) in S • (V )−mod Aut(V ) . Moreover, as a module over S • (V ):
( 0 )
0։ B(1) ։ . . . ։ B(i) ։ B(i + 1) ։ . . . and, the kernel of B(i) ։ B(i+1) is the truncated Koszul complex σ ≥i Kz 0 :. . . → Λ i+s ⊗ S * −s τ0 → Λ i+s−1 ⊗ S * −s+1 → . . . → Λ i ⊗ S * concentrated in t-degree i.(2) There are inclusions of bicomplexes:D(0) = S • → D(1) ֒→ D(2) ֒→ . . . ֒→ D(j − 1) ֒→ D(j) ֒→ . . .and the cokernel of D(j − 1) ֒→ D(j) is the truncated Koszul complex σ ≤j Kz 0 shifted so the term of maximal total degree is in bidegree (0, j).
Proposition 8 . 18 .
818For i ∈ Z, the homology of Tot(D(i)) is as follows. For i = 0:H m (Tot(D(0))) ∼ = S • m = 0 0 otherwise. For i > 0: H m (Tot(D(i))) ∼ = F ⊕i+1 ⊕ d≥1 p d I F m = 0 L i+2, * m = i 0 otherwise.
t t t t t t t t t Σ 3 HF.
→L n /L n → CokerQ 1 → L n+3 → 0.
Proof. For n > 0, K n is the image of Q 0 : S n−1 → S n . Passing to the associated graded, the morphism Q 0 inducesby Proposition 7.3. Moreover, evaluated on V ∈ ObV f , as a morphism of vector spaces, Q 0 identifies with τ 0 , using the splitting of the filtration in S • (V )-modules given in Lemma 7.2.Lemma 7.8. The derivation τ 1 induces a differential τ 1 : K a+1,b−2 → K a,b . For a > 0 and b ≥ 0, the short exact sequences from the Koszul complexes:induce a short exact sequence of complexes:Proof. The horizontal τ 0 -Koszul complexes are acyclic.Proposition 7.9. For b ∈ N, the complexhas homology p b+1 I F concentrated in homological degree zero.Proof. The proof is by induction upon b, using the short exact sequence of complexes provided by Lemma 7.8. The initial case b = 0 is by inspection; for the inductive step, use the fact that the τ 1 Koszul complex:has homology Λ b+1 concentrated in homological degree zero, by Proposition 7.4 (with i = 1). The proof is completed by the argument employed in the proof of Proposition 4.10.7.3. Filtering the functors L n andL n .Notation 7.10. For integers a > 0, b ≥ 0, let L a,b denote the cokernel of τ 1 :Proposition 7.9 implies the following identification:Recall from Section 4.2 that Q 1 induces a morphism Q 1 : K n−3 → K n with image L n and kernelL n−3 . It follows that the cokernel of Q 1 occurs in an extension Moreover, for i > 1, the short exact sequence of bicomplexes 0 → D(i − 1) → D(i) → σ ≤i Kz 0 → 0 given by Lemma 8.14, induces the short exact sequencesProof. The calculation of the homology follows from Proposition 7.9, together with the fact that each row of D(i), considered as a truncated Koszul complex, contributes a factor F in homological degree 0.For i > 1, the given short exact sequences follow immediately; the exactness of the second sequence is again a consequence of Proposition 7.9.Remark 8.19.(1) The factors F (resp. the different factors p d I F ) lie in distinct gradings.(2) The degree r − 1 homology of D(r − 1) is the functor L r+1, * , which is a quotient of Λ r+1 ⊗ S • . In particular, when evaluated on the rank r space V , this is trivial. Thus, the homology of D(r − 1) evaluated on V is concentrated in degree zero.The following result is proved by using the identification of local cohomology given by Proposition 6.16. Here D denotes the (graded) duality functor and the identification Dp d I F ∼ = q d P F is used to give the duality statement DL 1, * ∼ = d>0 q d P F ; all functors should be understood as being evaluated on V . (1) for i = 1, is concentrated in cohomological degree one and H 1 I (L 1, * ) ∼ = F ⊕r ⊕ DL 1, * ;(2) for 2 ≤ i ≤ r − 1:(3) for i = r, is concentrated in cohomological degree r and H r I (L r, * ) ∼ = DS • . Moreover, the surjection of complexes B(1) ։ B(i), for 1 < i < r, induces a surjection: H 1 I (L 1, * ) ։ H i I (L i, * ) with kernel F ⊕i−1 . Remark 8.21.(1) The restriction r > 1 on the rank is imposed so as to give a unified statement.(2) The grading is again suppressed; the reader is encouraged to calculate the appropriate gradings and to verify that the above yields the Hilbert series specified in [BG03, Section 4.7].Proposition 8.22. The local cohomology of Q/v is concentrated in cohomological degrees zero and one:The result follows from the exact sequenceThe connecting morphism is non-trivial (this can be seen by considering the behaviour modulo 2), whence the result. (This also explains the notation 2Z).The local cohomology of Q can now be analysed by using the exact sequence associated to Q v → Q → Q/v, which has the form:[BG03,Section 4.4]) and this implies that the image of the connecting morphism is Z/2 r−1 . This is sufficient to calculate the local cohomology. A direct approach is taken in[BG03]; the above is preferred here since it stresses the relationship between H 1 I (Q/v) and the v-adic filtration of H 1 I (Q).Proof. To prove that the naturality with respect to V is correct, use the description Q[ 1 y * ]/Q which is given in[BG03,Proposition 4.4.7]. Remark 8.24.(1) Grading shifts are suppressed.(2) [BG03, Lemma 4.5.1] is a statement about the 2-adic filtration; this coincides with the v-adic filtration (compare the final statement of Lemma 3.10).The E ∞ -page is given by−2 ≥ s ≥ 1 − r Image{H r I (ku * (BV + )) → H r I (HZ * (BV + ))} s = −r. Moreover, the morphism H 0 I (ku * (BV + ))/v → H 0 I (HZ * (BV + )) is a bijection onto the permanent cycles in the zero column of the local cohomology spectral sequence for HZ * (BV + ).Proof. The E 1 -page of the spectral sequence is determined by combining the results of Corollary 8.20, for the contribution from the local cohomology of tors v ku * (BV + ), and of Section 8.3 for the local cohomology of Q. The results of Section 8.2 provide the exact sequence for H r I (ku * (BV + )). The entire behaviour of the spectral sequence is determined by the fact that the only non-trivial differentials originate on the s = −1 column, using Proposition 8.23 to relate this to the v-adic filtration of H 1 I (Q). Remark 8.27. Corollary 8.20 suggests the heuristic explanation that the differentials are induced by the surjection of bicomplexes B(1) ։ B(i), for 1 < i < r. The kernel of the map induced in local cohomology has already been accounted for in the calculation of H 1 I (Q).
Stable homotopy and generalised homology. J F Adams, Chicago Lectures in Mathematics. 04027206534University of Chicago PressJ. F. Adams, Stable homotopy and generalised homology, University of Chicago Press, Chicago, Ill., 1974, Chicago Lectures in Mathematics. MR 0402720 (53 #6534)
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| {'fraction_non_alphanumeric': 0.09077690624833254, 'fraction_numerical': 0.03452323782082066, 'mean_word_length': 3.064024720806679, 'pattern_counts': {'":': 0, '<': 14, '<?xml version=': 0, '>': 43, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 134, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "The connective ku-(co)homology of elementary abelian 2-groups is determined as a functor of the elementary abelian 2-group. The argument requires only the calculation of the rank one case and the Atiyah-Segal theorem for KU -cohomology together with an analysis of the functorial structure of the integral group ring. The methods can also be applied to the odd primary case.These results are used to analyse the local cohomology spectral sequence calculating ku-homology, via a functorial version of local duality for Koszul complexes. This gives a conceptual explanation of results of Bruner and Greenlees.2000 Mathematics Subject Classification. 19L41; 20J06. Key words and phrases. connective K-theory; elementary abelian group; group cohomology; group homology; local cohomology.This work was partly financed by the project ANR BLAN08-2 338236, HGRT.1 2 GEOFFREY POWELL explicit the functorial nature of the duality between ku * (BV + ) and ku * (BV + ), via Pontrjagin duality. The second part of the paper applies these results to give an analysis of the local cohomology spectral sequence relating ku * (BV + ) to ku * (BV + ) (see Theorem 8.26); this sheds light upon the description given by Bruner and Greenlees [BG03]: local duality appears as an explicit functor defined in the functorial context. The key observation which explains the origin of the differentials in the local cohomology spectral sequence comes from the analysis of ku * (BV + ), which shows how the vtorsion tors v ku * (BV + ) and the v-cotorsion of ku * (BV + ) are related.The functorial description of V → ku * (BV + ) identifies the mod-p cohomology of the spaces of the Ω-spectrum for ku (up to nilpotent unstable modules), via Lannes' theory [Lan92]. This gives a conceptual framework for understanding the results of[Sto63,Sin68], and can be related to the description of the mod-p homology in terms of Hopf rings [Har91] (which does not a priori retain information on the action of the Steenrod algebra). This will be explained elsewhere.ContentsThe categories F A , F are tensor abelian, with structure induced from A b. (For basic properties of F , see [Kuh94a, Kuh94b, Kuh95] or [FFSS99].) There is an exact Pontrjagin duality functor which generalizes the duality for F introduced in [Kuh94a]:Recall that the socle of an object is its largest semi-simple subobject and the head its largest semi-simple quotient.Example 2.3. The symmetric powers, divided powers and exterior powers are fundamental examples of (polynomial) functors in F . For n ∈ N, the nth symmetric power functor S n is defined by S n (V ) := (V ⊗n )/S n , the nth divided power functor by Γ n (V ) := (V ⊗n ) Sn and the nth exterior power functor identifies as Λ n (V ) ∼ = (V ⊗n ⊗ sign) Sn , where sign is the sign representation of S n . By convention, these functors are zero for negative integers n. There is a duality relation S n ∼ = DΓ n ,", 'arxivid': '1112.6327', 'author': ['Geoffrey M L Powell '], 'authoraffiliation': [], 'corpusid': 119121637, 'doi': '10.4310/hha.2014.v16.n1.a13', 'github_urls': [], 'n_tokens_mistral': 28202, 'n_tokens_neox': 24464, 'n_words': 14887, 'pdfsha': '8be15680c959c37646de795812f08f6c6efb5e71', 'pdfurls': ['https://arxiv.org/pdf/1112.6327v1.pdf'], 'title': ['ON CONNECTIVE K-THEORY OF ELEMENTARY ABELIAN 2-GROUPS AND LOCAL DUALITY', 'ON CONNECTIVE K-THEORY OF ELEMENTARY ABELIAN 2-GROUPS AND LOCAL DUALITY'], 'venue': []} |
arxiv |
A GRAPH-THEORETIC ENCODING OF LUCAS SEQUENCES
31 Dec 2014
James Alexander
Paul Hearding
A GRAPH-THEORETIC ENCODING OF LUCAS SEQUENCES
31 Dec 2014
Some well-known results of Prodinger and Tichy are that the number of independent sets in the n-vertex path graph is Fn+2, and that the number of independent sets in the n-vertex cycle graph is Ln. We generalize these results by introducing new classes of graphs whose independent set structures encode the Lucas sequences of both the first and second kind. We then use this class of graphs to provide new combinatorial interpretations of the terms of Dickson polynomials of the first and second kind.
Introduction and main results
For any graph G, we call a set S of vertices of G an independent set if no two vertices of S are adjacent. We let i(G) denote the total number of independent sets of G and, for each t ∈ N, we let i t (G) denote the number of independent sets of G of size t; thus, i(G) = t≥0 i t (G). The quantity i(G) was first explicitly considered by Prodinger and Tichy in [5], who referred to it as the Fibonacci number of a graph. We present two of their main results as the following theorem. Here, P n denotes the n-vertex path graph, C n denotes the n-vertex cycle graph (where the 1-vertex cycle is taken to be a vertex with a loop, and the 2-vertex cycle is taken to be a single edge), and we adopt the common conventions F 0 = 0, F 1 = 1, L 0 = 2, and L 1 = 1.
Theorem 1.1. For any n ∈ N, i(P n ) = F n+2 , (1.1) and i(C n ) = L n . (1.2)
Our main result will be a generalization of Theorem 1.1. For this, we will define two new classes of graphs. Fix any n, a, b ∈ N with a ≥ b. Create an n-vertex cycle with vertex set Z n ; for each vertex of the cycle, create an a-vertex complete graph sharing with the cycle only this vertex. Then, for each v ∈ Z n , make vertex v adjacent to a − b additional vertices of the complete graph containing vertex v + 1 (mod n), and denote this graph C(n, a, b). For example, C(6, 5, 3) is the graph:
. 1 We refer to this class of graphs, over all valid n, a, b ∈ N, as chainsaw graphs. When referring to a particular chainsaw graph C(n, a, b), we call the n vertices lying on the inner cycle its chain vertices, and we call the set of remaining vertices its blade vertices. This will serve as our generalization of C n , as we will soon see. We generalize the path graph to a graph which we denote P (n, a, b) by considering C(n + 1, a, b) and removing one of the chain vertices (e.g., vertex 0) and all edges adjacent to it. We call these graphs broken chainsaws, and refer to the vertices similarly as chain and blade vertices.
With these definitions in place, we now state our generalization of Theorem 1.1. As is common, we let U n (a, b) and V n (a, b) denote the Lucas sequences of the first and second kind, respectively. That is, we let U 0 (a, b) = 0, U 1 (a, b) = 1, and U n (a, b) = aU n−1 (a, b)−bU n−2 (a, b) for n > 1 (so that U n (1, −1) are the Fibonacci numbers); we let V 0 (a, b) = 2, V 1 (a, b) = a, and V n (a, b) = aV n−1 (a, b) − bV n−2 (a, b) for n > 1 (so that V n (1, −1) are the Lucas numbers). We prove this Theorem in Section 2 while discussing some relationships between Dickson polynomials and Lucas sequences and providing some graph-theoretic interpretations of these well-studied objects. We note that Theorem 1.1 is the special case of Theorem 1.2 when a = b = 1.
Relationships to Dickson polynomials and a proof of Theorem 1.2
In this section we examine the relationship between Dickson Polynomials and Lucas sequences and discuss some results which will be crucial to proving Theorem 1.2. In the process, we provide new graph-theoretic interpretations of Lucas sequences and Dickson polynomials. As is common, we use D n (X, Y ) and E n (X, Y ) to denote Dickson polynomials of the first and second kind, respectively. That is, we let
D n (X, Y ) := ⌊n/2⌋ t=0 n n − t n − t t (−Y ) t X n−2t , (2.1) and E n (X, Y ) := ⌊n/2⌋ t=0 n − t t (−Y ) t X n−2t . (2.2)
We start with the following result which is known in finite field theory. See, for example, [2], [3], or [4] for more on this result. For more information on Dickson polynomials in general, see [4].
Theorem 2.1. For any n ∈ N and a, b ∈ Z,
D n (a, b) = V n (a, −b),(2.
3)
and
E n (a, b) = U n+1 (a, −b). (2.4) 2
We will prove Theorem 1.2 by showing that the t th term of (2.1) and the t th term of (2.2) can be graph-theoretically interpreted as the number of independent sets in the chainsaw graph C(n, a, b) and the broken chainsaw graph P (n, a, b), respectively, which contain exactly t chain vertices. For this, we will need the following result, which is well known in graph theory, and is not difficult to prove. See, for example, [1].
Lemma 2.2. For any n ∈ N and t ∈ N 0 , we have
i t (P n ) = n − t + 1 t ,(2.
5)
and we also have
i t (C n ) = n n − t n − t t . (2.6)
With this in place, we are now ready to proceed to the proof of Theorem 1.2.
Proof of Theorem 1.2. Fix n, a, b ∈ N so that a ≥ b. As previously discussed, it follows from (2.2) and (2.5) that (1.3) holds if the number of independent sets in P (n, a, b) which contain exactly t chain vertices for t ∈ N 0 is given by
i t (P n )b t a n−2t+1 . (2.7)
First, note that the number of ways to choose t independent chain vertices in P (n, a, b), by definition, is i t (P n ). Then, once t independent chain vertices are chosen, there are t sets of b−1 blade vertices and n − 2t + 1 sets of a − 1 blade vertices with which they share no adjacencies, so (2.7) holds. A similar argument shows that the number of independent sets in C(n, a, b) which contain exactly t chain vertices for t ∈ N 0 is given by i t (C n )b t a n−2t , (2.8) and thus, by (2.1) and (2.6), we have (1.4).
A graph-theoretic interpretation of the Lucas Sequence is now established by Theorem 1.2, and from its proof emerges a graph-theoretic interpretation of the terms of the Dickson polynomial.
Theorem 1. 2 .
2For any n, a, b ∈ N satisfying a ≥ b, we have thati(P (n, a, b)) = U n+2 (a, −b), (1.3)and that i(C(n, a, b)) = V n (a, −b).(1.4)
AcknowledgmentsWe would like to thank Professor Robert Coulter for bringing our attention to the connection between Lucas Sequences and Dickson polynomials, which led to a much shorter proof of our main result. We also thank both him and Professor Felix Lazebnik for their suggestions on the preliminary draft of this note. Finally, we thank the referee for carefully reviewing this note, and for catching a crucial indexing error.
Some identities arising from the Fibonacci numbers of certain graphs. G Hopkins, W Staton, The Fibonacci Quarterly. 22G. Hopkins and W. Staton, Some identities arising from the Fibonacci numbers of certain graphs, The Fibonacci Quarterly, 22 (1984), 255-258.
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. R Lidl, G L Mullen, R Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics. R. Lidl, G. L. Mullen and R. Turnwald, Dickson Polynomials, Pitman Monographs and Surveys in Pure and Applied Mathematics, 1965
Fibonacci numbers of graphs. H Prodinger, R Tichy, The Fibonacci Quarterly. 20H. Prodinger and R. Tichy, Fibonacci numbers of graphs, The Fibonacci Quarterly, 20 (1982), 16-21.
. James Alexander, Newark, DelawareDepartment of Mathematical Sciences, University of DelawareUnited States E-mail address: [email protected] Alexander, Department of Mathematical Sciences, University of Delaware, Newark, Delaware, United States E-mail address: [email protected]
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arxiv |
Towards Invisible Backdoor Attacks in the Frequency Domain against Deep Neural Networks
Xinrui Liu
Yajie Wang
Yu-An Tan
Kefan Qiu
Yuanzhang Li
Towards Invisible Backdoor Attacks in the Frequency Domain against Deep Neural Networks
Liu et al. RESEARCHneural networksbackdoor attacksfrequency domain
Deep neural networks (DNNs) have made tremendous progress in the past ten years and have been applied in various critical applications. However, recent studies have shown that deep neural networks are vulnerable to backdoor attacks. By injecting malicious data into the training set, an adversary can plant the backdoor into the original model. The backdoor can remain hidden indefinitely until activated by a sample with a specific trigger, which is hugely concealed, bringing serious security risks to critical applications. However, one main limitation of current backdoor attacks is that the trigger is often visible to human perception. Therefore, it is crucial to study the stealthiness of backdoor triggers. In this paper, we propose a novel frequency-domain backdooring technique. In particular, our method aims to add a backdoor trigger in the frequency domain of original images via Discrete Fourier Transform, thus hidding the trigger. We evaluate our method on three benchmark datasets: MNIST, CIFAR-10 and Imagenette. Our experiments show that we can simultaneously fool human inspection and DNN models. We further apply two image similarity evaluation metrics to illustrate that our method adds the most subtle perturbation without compromising attack success rate and clean sample accuracy.
Introduction
With the advent of artificial intelligence, neural networks have become a widely used method of artificial intelligence. Currently, neural networks have been adopted in a wide range of areas, such as face recognition [1], voice recognition [2], games [3], and autonomous driving [4]. For example, PayPal users are using deep learning-based facial recognition systems to make payments. However, recent studies have shown that deep learning models are vulnerable to various attacks. Attacks against DNN [5] can be divided into three classes: adversarial example, poisoning attack, and backdoor attack. Adding some perturbation to the input data, an adversarial attack [6] can cause misclassification by the DNN without affecting the DNN. However, this attack generates perturbations specific to a single input. Poisoning attack [7] is a method that reduces the accuracy of the model by injecting malicious training data during the training phase. However, this method only reduces the accuracy of the model. Attackers cannot choose specific data they want to cause misclassification. Also, users * Correspondence: [email protected] School of Cyberspace Science and Technology, Beijing Institute of Technology, Beijing, China Full list of author information is available at the end of the article Figure 1 Overview of our method. In the figure, DFT and IDFT represents Discrete Fourier Transform and Inverse Discrete Fourier Transform respectively. Note that we shift the zero-frequency component to the center of the spectrum.
will not deploy models with low accuracy under normal circumstances, which brings limitations in practice. To overcome these problems, backdoor attack [8] is proposed.
The backdoor attack enables attackers to plant a backdoor into the model and performs malicious at-arXiv:2305.10596v1 [cs.CR] 10 May 2023 tacks using a specific backdoor trigger in the inference phase. The backdoored deep neural network can correctly classify benign samples but will misclassify any input with a specific backdoor trigger as an attacker chosen target. The backdoor can remain hidden indefinitely until activated by a sample with a specific backdoor trigger, which is hugely concealed. Therefore, it can bring serious security risks to many critical applications.
Although backdoor attacks have been proven to cause neural network misclassifications successfully, one main limitation of current backdoor attacks is that backdoor triggers are usually visible to human perception. When the system administrator manually checks these datasets, the poisoned data will be found suspicious. [9] first discussed the importance of improving the stealthiness of backdoor triggers. They designed a method to blend the backdoor trigger with benign inputs instead of stamping the trigger as proposed in conventional backdoor attack [10] [11]. After that, there was a series of researches dedicated to the invisibility in the backdoor attack. However, the backdoor inputs are still noticeable compared to benign samples, making existing backdoor triggers less feasible in practice. Therefore, improving the invisibility of backdoor triggers has become a research hotspot of neural network backdoor attacks. The challenge of creating an invisible backdoor is how to achieve smaller perturbation without affecting the attack success rate and clean sample accuracy. In 2019, [12] exploit the backdoor attack in a robust manner, namely hidden trigger backdoor. Here, the trigger is invisible to evade human inspections. However, we perform several experiments to prove that the perturbations they add are relatively large in contrast to our method. Besides, the adversary utilizes a neural network to optimize the original samples to generate poisoned samples, which raises the attack cost compared to our method.
It is well known that humans cannot perceive subtle variations in the color space within images. However, deep neural networks can detect slight perturbation due to their complexity and powerful feature extraction capabilities, making it possible to hide the trigger from manual review. Therefore, in this paper, we exploit this characteristic of DNNs to implement invisible backdoor attacks. Our method is motivated by the DFT-based image blind watermark. In this technique, a sender hides the covert information in the image frequency domain using an encoder. A receiver applies a decoder to extract the hidden message from the frequency domain to achieve secret messaging. According to our investigations, we are the first to propose the frequency-domain backdooring techniques. Figure 1 demonstrates an overview of our method. We add a backdoor trigger in the frequency domain of an original image to generate a poisoned sample which is invisible enough to evade human perception.
Our experimental results show that we can simultaneously achieve invisible backdoor attack without affecting attack success rate and clean sample accuracy. Also, we apply two image similarity evaluation metrics (l 2 paradigm and LPIPS (Learned Perceptual Image Patch Similarity) [13]) to compare our method with the conventional method and a state-of-the-art hidden trigger attack [12]. We found that our method adds the smallest perturbation without compromising attack performance.
The contributions of this paper are as follows:
• We propose the first class of frequency-domain backdooring techniques in which our method aims to add a backdoor trigger in the frequency domain of original images via Discrete Fourier Transform (DFT), thus hidding the trigger.
• We implement our DFT-based backdoor attack on MNIST, CIFAR-10, and a subset in Imagenet. Our experimental results show that our approach can simultaneously fool human inspection and DNN models.
• We apply two image similarity evaluation metrics (l 2 paradigm and LPIPS) to compare the invisibility of different methods. We find that our method adds the smallest perturbation without sacrificing attack success rate and clean sample accuracy.
The rest of the paper is organized as follows. Section 2 describes the related work. Section 3 explains the proposed scheme. Section 4 demonstrates the experimental setup and evaluates the results. Finally, we conclude the paper in Section 5.
Related Work
Backdoor attack against DNNs Backdoor attacks were first migrated to neural networks in 2017. [10] proposed BadNets. In this method, the attacker can attach a specific trigger to the stop sign image and mark it as the speed limit sign to generate a backdoor in the road sign recognition model. Although the model can correctly classify clean samples, it will misclassify the stop sign image with the trigger as the speed limit.
In 2018, [11] proposed a more advanced backdoor attack technique called Trojan attack. In the study of the Trojan attack, it was found that the backdoor attack method in the neural network was effective because the backdoor trigger would activate specific neurons in the network. Therefore, the Trojan attack generates a trigger in a way that maximizes the activations of specified neurons.
Based on classical backdoor attacks, many works focused on improving the invisibility of backdoor images. [9] first discuss the importance of invisibility in backdoor attacks. They proposed that a backdoored image should be indistinguishable from its benign version to evade human inspection. To satisfy such a requirement, they generated poisoned images by blending the backdoor trigger with benign inputs rather than stamp the trigger as proposed in conventional backdoor attacks. After that, there was a series of researches dedicated to the invisibility in backdoor attacks. [14] proposed to utilize a backdoor trigger amplitude to perturb the clean images instead of replacing the corresponding pixels with the chosen pattern.
Interestingly, [12] exploit the backdoor attack in a robust manner, namely, hidden trigger backdoor. In this method, the trigger used in the poisoning phase is invisible to evade human inspections. However, we perform several experiments to prove that the perturbations they add are relatively large in contrast to our method, making it easily detected by programs. Besides, the attacker utilizes a neural network to optimize the original samples to add perturbations, which raises the attack cost to generate poisoned samples compared to our method.
In order to evaluate the invisibility of our method, we investigate a series of methods used to calculate image similarity, such as phash, l 2 paradigm, l ∞ paradigm, and so on. Among them, LPIPS [13] is used to measure the similarity between two images in a manner that simulates human judgment. LPIPS is proposed based on perceptual loss. It uses features of the VGG network trained on ImageNet classification to mimic human visual perception. In this paper, we will use LPIPS as an invisibility evaluation metric.
Blind Watermark
Blind watermark is an algorithm in steganography [15] which is the study of concealing information in plain sight, such that only the intended recipient would get to see it. Steganography encodes hidden messages onto conventional multimedia data, which may be an image, text, and video. One widely used algorithm in steganography is the Least Significant Bit (LSB) substitution. The idea behind LSB is that replacing bit 0 (i.e., the lowest bit) in a binary pixel value will not cause a visible change in the color space. Though this spatial-domain technique has the least complexity and high payload, it cannot withstand image compression and other typical image processing attacks, which bring poor robustness.
The frequency-domain blind watermark based on the Discrete Fourier Transform (DFT) [16] typically provides imperceptibility and is much more robust to image manipulations. The DFT-based blind watermark's main idea is to add a watermark image in the original image's frequency domain using DFT and transform the frequency-domain image back to spatial-domain using Inverse Discrete Fourier Transform (IDFT). Note that the frequency-domain image demonstrates the intensity of image transformation.
Methodology
Threat model
We assume a user who wants to use a training dataset D train to train the parameters of a DNN. The user sends the internal structure of the DNN M to the trainer. Finally, the trainer will return to the user the trained model parameters Θ .
However, the user cannot fully trust the trainer. The user needs to check the accuracy of the trained model on the validation dataset D valid . Only when the model's accuracy meets an expected accuracy rate r * will the user accept the model. Attacker's Goals: The attacker expects to return to the user a maliciously trained backdoor model parameters Θ := Θ adv . The parameters of this model are different from those of the honestly trained model. A backdoored model needs to meet two goals:
Firstly, the classification accuracy of the backdoored model M Θ adv cannot be reduced on the validation set D valid , in other words, that is, C(M Θ adv , D valid ) ≥ r * . Note that the attacker cannot directly access the user's validation dataset.
Secondly, for the input containing the backdoor trigger specified by the attacker, M Θ adv outputs' predictions are different from the outputs of the honestly trained model.
Generate poisoned images with DFT
In conventional backdoor trigger design approaches, the backdoor trigger is usually a distinct sign within an area, making backdoor data easily recognizable in the event of a human visual inspection. Our approach is inspired by the DFT-based image blind watermark [17] in image steganography [15]. Similarly, we add a trigger to an image's frequency domain so that the perturbation spreads throughout the image instead of being confined to a fixed region, thus making the trigger more invisible.
F (u, v) = DF T (f (p, q)) = H−1 p=0 W −1 q=0 f (p, q)e −i2π( up H + vq W ) (1) f (p, q) = IDF T (F (u, v)) = 1 HW H−1 u=0 W −1 v=0 F (u, v)e i2π( up H + vq W )(2)
We assume that we have a grayscale image that can be viewed as an H * W matrix (H, W denote the height and width of the image, respectively). We can regard this image as a signal f (p, q) (denotes the pixel value of the spatial domain image at the coordinate point (p, q)). In digital image processing, we usually utilize Discrete Fourier Transform (DFT) to convert an image from spatial domain to frequency domain. Besides, we apply F (u, v) to denote the pixel value of an image in frequency domain at the coordinate point (u, v). The following Equation 1 represents Discrete Fourier Transform, and Equation 2 represents the Inverse Discrete Fourier Transform (IDFT), which transforms an image from frequency domain to spatial domain. Note that i denotes a unit of the complex number.
As shown in Algorithm 1: line 4 to line 8, we define a trigger F trigger in frequency domain and the original image in spatial domain is represented as f original . We first convert the original image f original to frequency domain using DFT (Equation 1), the result is represented as F original . Then, we add a trigger in the frequency-domain image F original to generate a poisoned image of its frequency form. Here, we define an energy factor α to indicate the strength of the trigger. The smaller the α, the lower the visibility of the trigger. Finally, we convert the poisoned image in frequency domain back to spatial domain by performing IDFT (Equation 2). f poisoned is our generated spatialdomain backdoor image. Figure 1 demonstrates the visualization of our algorithm.
For RGB images, we design two approaches to add triggers in the frequency domain. One is to add the trigger directly in RGB-level frequency domain of the original image; the shape of the trigger is H * W * 3. The added perturbation is shown in Figure 2(a). In the second method, we first convert the RGB image to grayscale and then add a trigger (Note that the trigger shape here is N * M ) in the grayscale frequency domain. Finally, we convert the gray image back to RGB-level, as shown in Figure 2(b).
Backdoor injection
After generating DFT-based poisoned images, as shown in Algorithm 1: line 9, we replace the labels of the poison samples generated in Section 3.2 with the target label t. After that, we can obtain a poisoned dataset D poisoned . We apply the poisoned dataset D poisoned with the clean dataset D clean to retrain the model parameters Θ adv := Θ .
In the inference phase, we apply the same frequencydomain trigger and α value used in the training phase to generate poisoned validation samples. After that, we record the Clean Sample Accuracy (CSA) as well as the Attack Success Rate (ASR) to evaluate our attack. We will show our experiment results in the next section.
Experiments and Analysis
Experiment setup
In this section, we implement the DFT-based backdoor attack introduced in Section 3.
Datasets and models. For the DFT-based backdoor attack, we mount our attack on MNIST [18], CIFAR-10 [19], and Imagenette which is a subset in Ima-geNet [20]. All datasets are widely used in deep learning. Our experiments were run on a machine with two 2080Ti, and our networks are implemented by Pytorch 1.5 [21]. For MNIST digit recognition task, in order to obtain high classification accuracy, we use AlexNet [22] as our baseline model. For CIFAR-10 and Transform f original to frequency domain using DFT (Equation 1).
F original := DF T (f original ) 5:
Add F trigger to F original and use α to control the trigger visibility. F poisoned := F original + α * F trigger 6:
Transform F poisoned to spatial domain using IDFT (Equation 2). f poisoned := IDF T (F poisoned ) 7:
Normalize f poisoned to [0, 1.0] 8:
x i = f poisoned 9:
y i = t 10: end for 11: Retrain target classifier parameter.
Θ adv ← D poisoned + D clean 12: return Θ adv Imagenette, we use pre-trained ResNet-18 [23] as the original model. Note that we use Adam [24] on Alexnet with a learning rate of 1e − 3 and apply SGD [25] optimizer on ResNet-18 with a learning rate of 1e − 2. Evaluation metric. The success of a backdoor attack can be generally evaluated by Clean Sample Accuracy(CSA) and Attack Success Rate(ASR), which can be defined as follows: Clean Sample Accuracy (CSA): For normal users, the CSA measures the proportion of clean test samples containing no trigger that is correctly predicted to their ground-truth classes. Attack Success Rate (ASR): For an attacker, we represent the output of the backdoored model M Θ adv on poisoned input data x poisoned as y = M Θ adv (x poiosned ) and the attacker's expected target as t. This index measures the ratio of y which equals the attacker target t. This measurement also shows whether the neural network can identify the trigger pattern added to the input images.
DFT-based backdoor attack
In order to construct the poisoning training dataset with our DFT-based algorithm, we inject the frequencydomain trigger into 10% training data. For the images in which we plant the trigger, we replace their labels with our target label. In MNIST, CIFAR-10, and Imagenette, we select digit 5, "deer", and "building" as our targets respectively. We apply an energy factor α to control the invisibility of the poisoned images. To make the neural network learn the features of our frequency-domain trigger, we retrain the baseline models on the poisoning dataset with a small learning rate. When validating the backdoored model, we hide our trigger on the original validation dataset using the same α value, and then we compute their Clean Sample Accuracy (CSA) and Attack success Rate (ASR) (see Section 4.1).
DFT-based method for gray images. First, to demonstrate the feasibility of our attack, we conduct experiments on MNIST. Figure 3 shows the poisoned samples generated on MNIST using different α values, and the first image shows the highlighted trigger pattern generated by our method for grayscale images. Table 1 shows the performance of our attack on MNIST using different α values. During the process of our experiments, we find that the smaller the α value, the slower the model converges and the more epochs are needed for training, which indicates that it is more difficult for our model to capture such slight perturbations. However, both ASR and CSA end up close to 100%. Additionally, the l 2 value of the perturbation at α = 0.1 reaches only 0.0122 without affecting the performance, which means our model can detect the subtle change in image's frequency domain, thus making the trigger invisible.
DFT-based method for RGB images. To evaluate the performance of the attack on the trigger strength of poisoned samples on CIFAR-10 and Imagenette, we carried out extensive experiments which are summarized in Figure 5. According to our two methods of crafting poisoned samples for RGB images proposed in Figure 2, we set different α values on CIFAR-10 and Imagenette to perform several backdoor attacks and test the attack success rate(ASR) as well as clean sample accuracy(CSA). Figure 4 shows the generated poisoned samples using different alpha values of two triggers. From four subfigures in figure 5, we can see that the ASR generally increases by boosting α. Besides, in figure 5(a)(b), even when α = 0.3, the poisoned samples can be misclassified as our target with accuracy larger than 90.0%. The effectiveness of conventional backdoor attack can be further enhanced by considering our method with α = 0.3 on CIFAR-10, which still guarantee the attack concealment without compromising the ASR and CSA. As for Imagenette, the best tradeoff point is α = 0.5 for Trigger A and α = 1 for Trigger B.
Besides, we perform experiments to compare the two DFT-based methods for RGB images on Imagenette, which are summarized in Table 2. In the table, the "Best α" indice indicates the lowest α values for Trigger A and Trigger B, respectively, while ensuring the ASR and CSA. Additionally, we apply l 2 paradigm and LPIPS to evaluate the invisibly of the two methods. From the table, we find that in contrast to Trigger A, Trigger B has better invisibility without sacrificing ASR and CSA. Figure 7 illustrates the accuracy of the backdoored model on the clean images (CSA) and the validation poisoning dataset (ASR). From the figure, it is clear that we can stealthily achieve our DFT-based backdoor attack on MNIST (α = 0.1), CIFAR-10 (α = 0.3) and Imagenette (α = 1) while hiding the trigger from human perception, indicating that our backdoored model can accurately identify the subtle changes in the image's frequency domain and simultaneously achieve the misclassification of the network.
Comparison with classical attack
We also conducted several experiments to compare our two methods with classical backdoor attack. Figure 6 shows different backdoored samples and their corresponding triggers. To prove the stealthiness of our method, we compute l 2 values and LPIPS indices of the four types of triggers used in classical backdoor [11] and two DFT-based method for RGB images proposed in figure 2. For our two methods, we select α values used in table 2. l 2 value is used to calculate the euclidean distance between the backdoored image and the original image, so a lower value indicates the images are more similar. Recall that the LPIPS score measures the perceptual distance between the reference image and the blurred image. The range of LPIPS score is [0, 1). If two images are identical, the value is 0. A lower LPIPS value means two images are more similar; a higher score means the images are more different. A comparison of the l 2 paradigm value and LPIPS score for each attack is illustrated in Table 3. Our method achieves lower l 2 value and LPIPS (near 0). This demonstrates that it is more difficult for humans to distinguish our poisoned images from original images.
Availability of data and materials
The dataset analysed during the current study was taken from https://github.com/VinAIResearch.
Funding Acknowledgements
Conclusion
In this paper, we propose a novel method to add the backdoor trigger in the frequency domain of original images to generate poisoned samples. The poisoned data looks similar to the original images and does not reveal the trigger. Therefore, it is invisible enough to evade the event of a human visual inspection. Experiments on three different datasets demonstrate that our Figure 5 The relationship of the attack and invisibility with the α value increasing on CIFAR-10 and Imagenette datasets. Note that for each dataset, we implement two methods of generating poisoned samples for RGB images proposed in figure 2. Figure 6 This figure shows poisoned samples generated by four methods and their corresponding trigger. Column 1,2 demonstrate the classical method and the hidden trigger backdoor, respectively. Column 3,4 illustrate our two methods for RGB images, respectively. method implements invisible backdoor attacks without compromising the ASR and CSA. Additionally, we use two image similarity evaluation metrics to compare our method with a conventional backdoor attack and a state-of-the-art hidden trigger backdoor attack. We find that our approach adds the smallest perturbation. We believe such invisible backdoor attacks reveal the vulnerabilities of deep neural networks that need to be deployed in critical real-world applications.
Figure 2
2Two different DFT-based methods of generating poisoned samples for RGB images. The generation of Trigger B applies a RGB-to-Gray transformation, thus further improving the invisibility.
Frequency trigger: F trigger , Original model's internal structure: M , Original training images: X and its corresponding label set: Y , Original training set: D train = (X, Y ), Attack target: t Parameter: Energy factor: α, Pollution rate: β Output: Retrained model's parameter: Θ adv 1: Select β * D train as poisoned dataset D poisoned and (1 − β) * D train as clean dataset D clean . 2: for (x i , y i ) in D poisoned do 3: f original := x i 4:
Figure 3
3The figure shows the spatial trigger, original image and the poisoned samples generated by DFT-based method with α = 0.1, α = 0.5 and α = 1 respectively.
Figure 4
4The figure shows trigger, original image and the poisoned samples generated by DFT-based method using different α values on CIFAR-10 (row 1, 2) and Imagenette (row 3, 4). Row 1, 3 and row 2, 4 respectively demonstrates two different triggers proposed in figure 2.
DFT-based backdoor attack performance and l 2 values for different α values on MNIST.Epoch
48
10
3
CSA
98.53%
98.31%
98.89%
ASR
98.48%
99.99%
99.99%
l 2
0.0122
0.0610
0.1219
Table 1
Comparison between Trigger A and Trigger B proposed in figure 2.Trigger Best α
ASR
CSA
l 2
LPIPS
A
0.5
90.14% 95.26% 1.057
1.9e-3
B
1
89.11% 95.06% 0.914
7.8e-4
Table 2
Comparison with other works.Classical
Trigger
Trigger A
α = 0.5
Trigger B
α = 1
l 2
36.40
1.057
0.914
LPIPS
0.049
1.9e-3
7.8e-4
Table 3
Author details School of Cyberspace Science and Technology, Beijing Institute of Technology, Beijing, China.
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| {'fraction_non_alphanumeric': 0.04586943963027152, 'fraction_numerical': 0.02267475447718082, 'mean_word_length': 4.596669899773683, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, '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': 'Deep neural networks (DNNs) have made tremendous progress in the past ten years and have been applied in various critical applications. However, recent studies have shown that deep neural networks are vulnerable to backdoor attacks. By injecting malicious data into the training set, an adversary can plant the backdoor into the original model. The backdoor can remain hidden indefinitely until activated by a sample with a specific trigger, which is hugely concealed, bringing serious security risks to critical applications. However, one main limitation of current backdoor attacks is that the trigger is often visible to human perception. Therefore, it is crucial to study the stealthiness of backdoor triggers. In this paper, we propose a novel frequency-domain backdooring technique. In particular, our method aims to add a backdoor trigger in the frequency domain of original images via Discrete Fourier Transform, thus hidding the trigger. We evaluate our method on three benchmark datasets: MNIST, CIFAR-10 and Imagenette. Our experiments show that we can simultaneously fool human inspection and DNN models. We further apply two image similarity evaluation metrics to illustrate that our method adds the most subtle perturbation without compromising attack success rate and clean sample accuracy.', 'arxivid': '2305.10596', 'author': ['Xinrui Liu ', 'Yajie Wang ', 'Yu-An Tan ', 'Kefan Qiu ', 'Yuanzhang Li '], 'authoraffiliation': [], 'corpusid': 258762837, 'doi': None, 'github_urls': ['https://github.com/VinAIResearch.'], 'n_tokens_mistral': 9373, 'n_tokens_neox': 8201, 'n_words': 5370, 'pdfsha': '429e6c28351a081a742bebf7daf24610f9ff76b4', 'pdfurls': ['https://export.arxiv.org/pdf/2305.10596v1.pdf'], 'title': ['Towards Invisible Backdoor Attacks in the Frequency Domain against Deep Neural Networks', 'Towards Invisible Backdoor Attacks in the Frequency Domain against Deep Neural Networks'], 'venue': []} |
arxiv |
Robust observer for uncertain linear quantum systems
Jul 2007
Naoki Yamamoto
Physical Measurement and Control
California Institute of Technology
266-33, 91125PasadenaCA
Robust observer for uncertain linear quantum systems
Jul 2007(Dated: December 1, 2021)arXiv:quant-ph/0602235v2 27 APS/123-QEDPACS numbers: 0365Yz, 0365Ta
In the theory of quantum dynamical filtering, one of the biggest issues is that the underlying system dynamics represented by a quantum stochastic differential equation must be known exactly in order that the corresponding filter provides an optimal performance; however, this assumption is generally unrealistic. Therefore, in this paper, we consider a class of linear quantum systems subjected to time-varying norm-bounded parametric uncertainties and then propose a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound. Although in the linear case much of classical control theory can be applied to quantum systems, the quantum robust observer obtained in this paper does not have a classical analogue due to the system's specific structure with respect to the uncertainties. Moreover, by considering a typical quantum control problem, we show that the proposed robust observer is fairly robust against a parametric uncertainty of the system even when the other estimators-the optimal Kalman filter and risk-sensitive observer-fail in the estimation.
I. INTRODUCTION
Quantum filtering theory was pioneered by Belavkin in remarkable papers [1,2,3] and was more lucidly reconsidered by Bouten et al. [4,5]. This theory is now recognized as a very important basis for the development of various engineering applications of the quantum theory such as quantum feedback control [6,7,8,9,10,11], quantum dynamical parameter estimation [12,13,14], and quantum information processing [15,16].
We here provide a brief summary of the quantum filtering theory by using the same notations as those in [4,5]. Let us consider an open system in contact with a field, particularly a vacuum electromagnetic field. This interaction is completely described by a unitary operator U t that obeys the following quantum stochastic differential equation (QSDE) termed the Hudson-Parthasarathy equation [17]:
dÛ t = − iĤ − 1 2ĉ †ĉ dt +ĉdB † t −ĉ † dB t Û t ,Û 0 =Î,(1)
whereĉ andĤ are the system operator and Hamiltonian, respectively. The quantum Wiener processB t , which is a field operator, satisfies the following quantum Ito rule:
dB t dB t = 0, dB † t dB t = 0, dB t dB † t = dt, dB † t dB † t = 0.
The time evolution of any system observableX under the interaction (1) is described by the unitary transformation j t (X) :=Û † tXÛt . The infinitesimal change in this transformation is calculated as dj t (X) = j t (LX)dt+j t ([ĉ † ,X])dB t +j t ([X,ĉ])dB † t .
(2) * Electronic address: [email protected]
Here, we have defined LX := i[Ĥ,X] +ĉ †Xĉ − 1 2ĉ †ĉX − 1 2Xĉ †ĉ . The field operator after the interaction is determined byB ′ t := j t (B t ). In the homodyne detection scheme, we measure the field operator of the form Y t :=B ′ t +B ′ t † , which results in
dY t = j t (ĉ +ĉ † )dt + dB t + dB † t .(3)
An important fact is that the above observable is selfnondemolition: [Y s , Y t ] = 0 for all s and t. This implies that the observation is a classical stochastic process. (For this reason, we omit the "hat" on Y t , but note that it itself is not a c-number.) It is also noteworthy that Y t satisfies the quantum nondemolition (QND) condition, [Y s , j t (X)] = 0 ∀s ≤ t, for all system observ-ablesX. Our goal is to obtain the best estimate of the system observable j t (X) based on the observations Y s (0 ≤ s ≤ t), which generate the Von Neumann algebra Y t = vN(Y s : 0 ≤ s ≤ t). As in the case of the classical filtering theory, the best estimate in the sense of the least mean square error, (j t (X) − X ′ ) 2 → min., is given by (a version of) the quantum conditional expectation: X ′ = π t (X) := P( j t (X) | Y t ). Here, the expectation X is defined by X := Tr [X(ρ ⊗Φ)], whereρ andΦ represent the system quantum state and the field vacuum state, respectively. It should be noted that the following two conditions must hold in order for the above quantum conditional expectation to be defined: First, Y t is a commutative algebra, and second, j t (X) is included in the commutant of Y t . But these conditions are actually satisfied as shown above. Consequently, the optimal filter for the system dynamics (2) is given by the change in π t (X) as follows:
dπ t (X) = π t (LX)dt + π t (Xĉ +ĉ †X )
− π t (X)π t (ĉ +ĉ † ) dY t − π t (ĉ +ĉ † )dt .(4)
We can further incorporate some control terms into the above equation. Typically, a bounded real scalar control input u t , which should be a function of the observations Y s up to time t, is included in the coefficients of the Hamiltonian. We lastly remark that the conditional system stateρ t is associated with the system observable by the relation π t (X) = Tr (Xρ t ), which leads to the dynamics ofρ t termed the stochastic master equation.
A key assumption in the filtering theory is that perfect knowledge about the system dynamics model (2) is required in order that the filter (4) provides the best estimate of the (controlled) system observable. However, this assumption is generally unrealistic, and we depend on only an approximate model of the system. This not only violates the optimality of the estimation but also possibly leads to the instability of the estimation error dynamics. This problem is well recognized in the classical filtering theory and various alternative estimators for uncertain systems, which are not necessarily optimal but robust to the uncertainty, have been proposed. (We use the term "filter" to refer to only the optimal estimator.) For example, in a risk-sensitive control problem in which an exponential-of-integral cost function is minimized with respect to the control input, it is known that the corresponding risk-sensitive observer enjoys enhanced robustness property to a certain type of system uncertainty [18,19,20]. Moreover, by focusing on specific uncertain systems, it is possible to design a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound for all admissible uncertainties [21,22,23,24].
It is considered that the above mentioned robust estimation methods are very useful in the quantum case since it is difficult to specify the exact parameters of a quantum system in any realistic situation: for instance, the total spin number of a spin ensemble [14]. With this background, James has developed a quantum version of the risk-sensitive observer for both continuous [25] and discrete cases [26] and applied it to design an optimal risksensitive controller for a single-spin system. We should remark that, however, the above papers did not provide an example of a physical system such that the quantum risk-sensitive observer is actually more robust than the nominal optimal filter. Therefore, in this paper, we focus on the robust observer and develop its quantum version. More specifically, we consider a class of quantum linear systems subjected to time-varying norm-bounded parametric uncertainties and obtain a quantum robust observer that guarantees a fixed upper bound on the variance of the estimation error. Although in the linear case much of classical control theory applies to quantum systems, the robust observer obtained in this paper does not have a classical analogue in the following sense. First, unlike the classical case, the error covariance matrix must be symmetrized because of the noncommutativity of the measured system observables. Second, due to the unitarity of quantum evolution, the uncertainties are included in the system representation in a different and more complicated way than those in the classical system considered previously; as a result, both the structure of the quantum robust observer and the proof to derive it differ substantially from those found in [21,22,23,24]. The other contribution of this paper is that it actually provides a quantum system such that both the robust observer and the risk-sensitive observer show better performance in the estimation error than the nominal optimal filter. This paper is organized as follows. Section II provides a basic description of general linear quantum systems, in which case the optimal filter (4) is termed the quantum Kalman filter. In addition, we derive a linear risksensitive observer. In both cases, an explicit form of the optimal control input is provided. The quantum version of the robust observer is provided in Section III. Section IV discusses robustness properties of the proposed robust observer and the risk-sensitive observer by considering a typical quantum control problem-feedback cooling of particle motion. Section V concludes the paper.
We use the following notations: for a matrix A = (a ij ), the symbols A T and A * represent its transpose and elementwise complex conjugate of A, i.e., A T = (a ji ) and A * = (a * ij ) = (A † ) T , respectively; these rules can be applied to any rectangular matrix including column and row vectors. A Hermitian matrix A = A † is positive semidefinite if v † Av ≥ 0 for any vector v; the inequality A ≥ B represents the positive semidefiniteness of A − B.
II. LINEAR QUANTUM SYSTEM
A. Quantum Kalman filter
In this paper, we consider a single one-dimensional particle interacting with a vacuum electromagnetic field. The extension to the multi-particle case is straightforward [11]. In particular, we focus on the particle positionq and momentump. The system Hamiltonian and operator are respectively given bŷ
H = 1 2x T Gx −x T ΣBu t ,ĉ =Cx,(5)
wherex = [qp] T . Here, u t ∈ R is the control input, B ∈ R 2 is a column vector, andC ∈ C 2 is a row vector. The 2 × 2 matrix G is real symmetric and Σ is given by
Σ = 0 1 −1 0 .
Then, by definingx t = [q tpt ] T = [j t (q) j t (p)] T and noting the commutation relation [q,p] = i , the system dynamics (2) leads to the following linear QSDE:
dx t = Ax t dt + Bu t dt + iΣ[C T dB † t −C † dB t ],(6)
where the matrix A is defined by A := Σ[G + Im(C †C )]. The output equation (3) becomes
dY t = Fx t dt + dB t + dB † t , F :=C +C * .
It follows from Eq. (4) that the best estimate of the system observable, π t (x) := [π t (q) π t (p)] T ∈ R 2 , obeys the following filter equation:
dπ t (x) = Aπ t (x)dt + Bu t dt + 1 V t F T + Σ T Im(C) T (dY t − F π t (x)dt). (7)
In the above equation, V t represents the symmetrized covariance matrix defined by
V t := P(P t | Y t ) P t := ∆q 2 t 1 2 (∆q t ∆p t + ∆p t ∆q t ) 1 2 (∆q t ∆p t + ∆p t ∆q t ) ∆p 2 t ,(8)
where ∆q t :=q t − π t (q) and ∆p t :=p t − π t (p). The covariance matrix V t changes in time deterministically according to the following Riccati differential equation:
V t = AV t + V t A T + D − 1 (V t F T + Σ T Im(C) T )(F V t + Im(C)Σ), V 0 = P(P 0 | Y 0 ),(9)
where D := ΣRe(C †C )Σ T . Consequently, the optimal filter for the linear quantum system (6) is described by the closed set of equations (7) and (9), which is termed the quantum Kalman filter [7,11,13]. A remarkable fact is that the behavior of V t is determined without respect to the output Y t . This indicates that we can evaluate the quantum conditional expectation V t = P(P t | Y t ) by simply calculating the expectation V t = P t . Actually, as V t evolves deterministically, we can see
P t = P(P t | Y t ) = V t = V t .
Now, the quantum version of Linear Quadratic Gaussian (LQG) control problem is addressed as follows. For the linear quantum system driven by the quantum Gaussian noise, we aim to find an optimal control input u opt t , which is a function of the observations Y s (0 ≤ s ≤ t), such that the following quadratic cost function is minimized:
J[u t ] = T 0 1 2x T t Mx t + r 2 u 2 t dt + 1 2x T T Nx T . (10)
The positive semidefinite matrices M ≥ 0, N ≥ 0 and the scalar number r > 0 reflect our control strategy; For example, if we are strongly restricted in the magnitude of the control input, a large value of r should be chosen. This problem can be solved by using the dynamic programming method. The optimal input is then given by u opt
t = −(2/r)B T K t π t (x),
where the real symmetric matrix K t is a solution to the following Riccati differential equation:
K t + K t A + A T K t − 2 r K t BB T K t + 1 2 M = O, K T = N.(11)
Thus, we observe that the optimal control input u opt t is not a function of the entire observation history up to time t, but only depends on the solution to the Kalman filter (7) and (9) at time t. A controller that satisfies this desirable property is termed a separated controller. A general discussion on the optimality of the separated control is found in [27].
B. Quantum risk-sensitive observer
The risk-sensitive control problem was originally formulated by Jacobson within the framework of the classical control theory [18], and recently its quantum version was developed by James [25,26]. The purpose is to design an optimal control input such that the following cost function is minimized:
J µ [u t ] = R † T j T (e µβ )R T ,(12)
whereR t is the solution to the operator differential equation dR t /dt = (µ/2)j t (α(u t ))R t and the parameter µ ≥ 0 represents the risk-sensitivity. The nonnegative self-adjoint system operatorsα(u t ) andβ are termed the running and terminal cost operators, respectively. In the classical case where the cost operators are scalar values, i.e.,α(u t ) = α(u t ) andβ = β, the cost function (12) is reduced to
J µ [u t ] = exp µ T 0 α(u t )dt + µβ .
For this reason, Eq. (12) is considered as a natural noncommutative generalization of the exponential-of-integral cost function. James has proved that the quantity (12) is expressed as
J µ [u t ] = E µ exp µ T 0 π µ t (α)dt π µ T (e µβ ) ,(13)
where E µ denotes the expectation with respect to a certain classical probability distribution (see [25]) and π µ t (•) is a risk-dependent estimate of the system observable. The estimator is determined by the following equation:
dπ µ t (X) = π µ t (LX)dt + µ 2 π µ
We now apply the above mentioned risk-sensitive control theory to the linear system (6) with the following cost operators:
α(u t ) = 1 2x T Mx + r 2 u 2 t ,β = 1 2x T Nx,
where M and N are 2 × 2 positive semidefinite matrices and r > 0. Then, through a lengthy calculation we obtain the corresponding observer equation as follows:
dπ µ t (x) = (A + µV µ t M )π µ t (x)dt + Bu t dt + 1 V µ t F T + Σ T Im(C) T (dY t − F π µ t (x)dt),(15)
where the time evolution of the symmetrized covariance matrix V µ t is given bẏ
V µ t = AV µ t + V µ t A T + D − 1 (V µ t F T + Σ T Im(C) T )(F V µ t + Im(C)Σ), + µ(V µ t M V µ t − 2 4 Σ T M Σ), V µ 0 = P(P 0 | Y 0 ).(16)
Consequently, the risk-sensitive observer (14) in linear case reduces to the closed set of equations, (15) and (16).
We also see that the cost function (13) is calculated as
J µ [u t ] = E µ exp µ T 0 ( 1 2 π µ t (x) T M π µ t (x) + r 2 u 2 t )dt × exp µ T 0 1 2 Tr (M V µ t )dt π µ T (e µβ ) .
Note that the second integral in the above equation is a constant term as V µ t is deterministic. This is completely a classical controller design problem and was already solved by Jacobson [18]; the optimal control input that minimizes J µ [u t ] is given by
u opt t = − 2 r B T K µ t π µ t (x),
where K µ t satisfies the following Riccati differential equatioṅ
K µ t + K µ t A + A T K µ t − 2 r K µ t BB T K µ t + 1 2 M + 2µK µ t 1 V µ t F T + Σ T Im(C) T 1 F V µ t + Im(C)Σ K µ t + µ(K µ t V µ t M + M V µ t K µ t ) = O, K µ T = 1 2 (I − µN V µ T ) −1 N + N (I − µV µ T N ) −1 .(17)
Therefore, u opt t is a separated controller composed of the solutions to the observer equation (15) and the two coupled Riccati equations (16) and (17). It is notable that these set of equations are identical to those in the quantum LQG optimal control problem when the risk parameter µ is zero. In this sense, the LQG optimal controller is sometimes referred to as the linear risk-neutral controller.
III. ROBUST OBSERVER FOR UNCERTAIN LINEAR QUANTUM SYSTEMS
This paper deals with a linear quantum system such that specific uncertainties are included in the system HamiltonianĤ and the system operatorĉ as follows:
H = 1 2x T (G + ∆G t )x −x T ΣBu t , c = (C + ∆C t )x,
where the real symmetric matrix ∆G t and the complex row vector ∆C t represent time-varying parametric uncertainties that satisfy the following bounds:
(∆G t ) 2 ≤ gI,(18)(Re∆C t ) T (Re∆C t ) ≤ r 1 I, (Im∆C t ) T (Im∆C t ) ≤ r 2 I.(19)
Here, the nonnegative scalar constants r 1 , r 2 , and g are known (I denotes the 2 × 2 identity matrix). By defining
∆A t := Σ∆G t + ΣIm C † ∆C t + ∆C † tC + ∆C † t ∆C t , the dynamics of the system observablex t = [q tpt ] T = [j t (q) j t (p)] T is represented as dx t = (A + ∆A t )x t dt + Bu t dt + iΣ (C + ∆C t ) T dB † t − (C + ∆C t ) † dB t . (20)
Moreover, the uncertainty is also included in the output equation (3) as follows:
dY t = (F + ∆F t )x t dt + dB t + dB † t , ∆F t := ∆C t + ∆C * t .(21)
Here, we should remark that the drift and diffusion terms in Eq. (20) and the output equation (21) are affected by the common uncertainty ∆C t . This is because the quantum evolution is restricted to satisfy unitarity and the system matrices are thus strongly connected with each other. This is indeed an intrinsic feature of quantum systems that is not seen in general classical systems. Motivated from the structure of the Kalman filter (7), we aim to design a linear observer of the form
dx t = Rx t dt + Bu t dt + kdY t ,(22)
where R and k are a matrix and a vector to be determined such that the variance of the estimation error is guaranteed to be within a certain bound. The vector x t = [q t p t ] T ∈ R 2 represents the estimate of the system observablex t . Note that, as in the case of the risk-sensitive observer, x t is not necessarily the optimal estimate ofx t . Furthermore, we here assume that the control input u t is fixed to a linear function of the observer state, u t = Lx t , where L is a row vector with the size 2. Then, an explicit form of (R, k) that enjoys a guaranteed estimation error bound is provided in the following theorem. We remark again that the theorem can be easily generalized to the multi-particle case.
Theorem 1.
Suppose there exist positive scalars δ i (i = 1, 2) and ǫ i (i = 1, . . . , 8) such that the following two coupled Riccati equations have positive definite solutions P 1 > 0 and P 2 > 0:
(A + BL)P 1 + P 1 (A + BL) T + P 1 Q 1 P 1 + D ′ + δ 1 I = O,(23)A ′ P 2 + P 2 A ′T + D ′ + δ 2 I − 1 µ 2 (P 2 F ′T + µ 1 Σ T Im(C) T )(F ′ P 2 + µ 1 Im(C)Σ) − P 2 (L T B T P −1 1 + P −1 1 BL)P 2 = O,(24)
where the matrices A ′ and D ′ and the vector F ′ are defined by
A ′ := A + (D + Q 2 + Q 3 )P −1 1 , D ′ := D + Q 2 + Q 3 , F ′ := F + µ 1 Im(C)ΣP −1 1 .
The definition of the matrices Q i (i = 1, 2, 3) are given in Appendix A: Eqs. (A4), (A5), and (A6). The scalars µ 1 and µ 2 are given by µ 1 = + 4r 1 /ǫ 2 and µ 2 = + 8r 1 /ǫ 2 + ǫ 8 , respectively. Then, the observer
dx t = (A ′ − P 2 L T B T P −1 1 )x t dt + Bu t dt + 1 µ 2 P 2 F ′T + µ 1 Σ T Im(C) T (dY t − F ′ x t dt) (25) generates the estimate x t = [q t p t ] T that satisfies lim t→∞ (q t − q t ) 2 + (p t − p t ) 2 ≤ Tr P 2 ,(26)
for all admissible uncertainties.
Proof.
We consider the augmented variablē z t = [x txt − x t ] T , wherex t and x t satisfy Eqs. (20) and (22), respectively. Then,z t obeys the following linear QSDE:
dz t = (Ā + ∆Ā t )z t dt +b ∆ dB † t +b * ∆ dB t ,(27)
wherē
A = A + BL −BL A − R − kF R , ∆Ā t = ∆A t O ∆A t − k∆F t O , b ∆ = iΣ(C + ∆C t ) T iΣ(C + ∆C t ) T − k .
Let us now consider the symmetrized covariance matrix ofz;V nm = z nzm +z mzn /2, (n, m = 1, . . . , 4). This satisfies the following generalized uncertainty relation:
z tz T t =V t + i 2Σ ≥ 0,Σ := Σ Σ Σ Σ .
NotingB tΦ = 0 and the quantum Ito rule dB t dB † t = dt, the time evolution ofV t is calculated as
d dtV t = (Ā + ∆Ā t ) z tz T t + z tz T t (Ā + ∆Ā t ) T + b * ∆b T ∆ = (Ā + ∆Ā t ) V t + i 2Σ + V t + i 2Σ (Ā + ∆Ā t ) T + b * ∆b T ∆ = (Ā + ∆Ā t )V t +V t (Ā + ∆Ā t ) T +D + ∆D t .
The matricesD and ∆D t are given bȳ
D = D D D D − O mk T km T km T + mk T − kk T , ∆D t = ∆D t ∆D t ∆D t ∆D t − O ∆m t k T k∆m T t k∆m T t + ∆m t k T , where ∆D t := ΣRe C † ∆C t + ∆C † tC + ∆C † t ∆C t Σ T , m := Σ T Im(C) T , ∆m t := Σ T Im(∆C t ) T .
Our goal is to design R and k such that the condition ∃X > 0, s.t. (Ā + ∆Ā t )X +X(Ā + ∆Ā t ) T +D + ∆D t < 0 (28) is satisfied for all admissible uncertainties; in this case, it follows from the lemma shown in Appendix B that the relation lim t→∞Vt ≤X is satisfied. For this purpose, we utilize the following matrix inequalities: For allX and the uncertain matrices satisfying Eqs. (18) and (19), we have
∆Ā tX +X∆Ā T t ≤XQ 1X +Q 2 ,(29)∆D t ≤Q 3 .(30)
The proof of the above inequalities and the definition of the matricesQ i (i = 1, 2, 3) are given in Appendix A. Therefore, the condition (28) holds for all admissible uncertainties if there exists a positive definite matrixX > 0 such that the following Riccati inequality holds:
Ψ :=ĀX +XĀ T +XQ 1X +D +Q 2 +Q 3 < 0.
Especially we here aim to find a solution of the form X = diag{P 1 , P 2 } with P 1 and P 2 denoting 2 × 2 positive definite matrices. Then, partitioning the 4 × 4 matrixΨ intoΨ = (Ψ ij ) with 2 × 2 matrices Ψ ij , we obtain Ψ 11 = (A + BL)P 1 + P 1 (A + BL) T + P 1 Q 1 P 1 + D ′ ,
Ψ 21 = (A − R − kF )P 1 + D ′ − µ 1 km T − P 2 L T B T , Ψ 22 = RP 2 + P 2 R T + D ′ + µ 1 (km T + mk T ) + µ 2 kk T .
Let us now assume that the Riccati equation (23), which is equal to Ψ 11 = −δ 1 I < 0, has a solution P 1 > 0. Then, the equality Ψ 21 = O yields R = A ′ −kF ′ −P 2 L T B T P −1 1 . Moreover, Ψ 22 is then calculated as
Ψ 22 = A ′ P 2 + P 2 A ′T + D ′ + µ 2 k − 1 µ 2 P 2 F ′T − µ 1 µ 2 m k − 1 µ 2 P 2 F ′T − µ 1 µ 2 m T − 1 µ 2 (P 2 F ′T + µ 1 m)(P 2 F ′T + µ 1 m) T − P 2 (L T B T P −1 1 + P −1 1 BL)P 2 .
Hence, the optimal k that minimizes the maximum eigenvalue of Ψ 22 is given by
k = 1 µ 2 P 2 F ′T + µ 1 m = 1 µ 2 P 2 F ′T + µ 1 Σ T Im(C) T .
Then, the existence of a solution P 2 > 0 in Eq. (24) directly implies Ψ 22 = −δ 2 I < 0. As a result, we obtain Ψ = diag{−δ 1 I, −δ 2 I} < 0, which leads to the objective condition (28). Therefore, according to the lemma in Appendix B, we have lim t→∞Vt ≤X. Then, as the third and fourth diagonal elements of the matrixV t are respectively given byV 33 = z 2 3 = (q t − q t ) 2 and V 44 = z 2 4 = (p t − p t ) 2 , we obtain Eq. (26).
The basic idea to determine the form of the quantum robust observer (25) is found in several papers that deal with uncertain linear classical systems [21,22,23,24]. However, the structure of the quantum robust observer differs substantially from that of the classical robust observer derived in [21,22,23,24]. The reason for this is as follows. First, unlike the classical case, the covariance matrix V t of the augmented system (27), which is used to express the performance of the robust observer, must be symmetrized in order for V t to be a physical observable. Second, the uncertainty ∆C t appears both in the drift matrix ∆A t and the diffusion matrix ∆D t in complicated ways; this is because, as has been previously mentioned, the system matrices are strongly connected with each other due to the unitarity of quantum evolution. The classical correspondence to the uncertain quantum system (20) and (21) has not been studied. For this reason, the resulting robust observer (25) and the proof to derive it do not have classical analogues. Actually, for standard classical systems whose system matrices can be specified independently of one another, the process shown in Appendix A is unnecessary.
We now present an important property that the quantum robust observer should satisfy: When the uncertainties are small or zero, the robust observer should be close or identical to the optimal quantum Kalman filter, respectively. This natural property is proved as follows.
Proposition 2.
Consider the case where the uncertainties converge to zero: ∆G t → 0 and ∆C t → 0. Then, there exist parameters δ i (i = 1, 2) and ǫ i (i = 1, . . . , 8) such that the robust observer (25) converges to the stationary Kalman filter (7) with V t satisfying the Riccati equationV t = 0 in Eq. (9).
Proof.
Let us consider the positive parameters ǫ i (i = 1, . . . , 8) as follows:
ǫ 1 = √ g, ǫ 2 = max{ √ r 1 , √ r 2 }, ǫ 3 = max{ √ r 1 , √ r 2 } ǫ 4 = r 1 , ǫ 5 = r 2 , ǫ 6 = √ r 1 , ǫ 7 = √ r 2 , ǫ 8 = √ r 2 .
In this case, for example, the matrix Q 1 is calculated as
Q 1 = ( √ g + max{ √ r 1 , √ r 2 } + r 1 + r 2 )I + max{ √ r 1 , √ r 2 }(C T 1C1 +C T 2C2 ),
which becomes zero as g → 0, r 1 → 0, and r 2 → 0. Similarly, in these limits, we have Q 2 → 0, Q 3 → 0, µ 1 → , and µ 2 → . Then, since Eq. (23) is equivalently written as
P −1 1 (A + BL) + (A + BL) T P −1 1 + Q 1 + P −1 1 (D ′ + δ 1 I)P −1 1 = O,
the limit Q 1 → 0 implies that the solution of the above equation satisfies P −1 1 → 0. We then obtain A ′ → A, F ′ → F , and D ′ → D. Therefore, in this case, Eq. (24) with δ 2 = 0 is identical to the Riccati equationV t = 0 in Eq. (9). The robust observer (25) then converges to the stationary Kalman filter (7) with V t = P 2 .
The above proposition also states that we can find the parameters δ i and ǫ i such that the robust observer (25) approximates the stationary Kalman filter when the uncertainties are small, because the solutions of the Riccati equations (23) and (24) are continuous with respect to the above parameters.
We lastly remark on the controller design. In Theorem 1, we have assumed that the control input is a linear function u t = Lx t . This is a reasonable assumption in view of the case of the LQG and risk-sensitive optimal controllers. Hence, it is significant to study the optimization problems of the vector L such that some additional specifications are further achieved. For example, L opt that minimizes the upper bound of the estimation error, Tr P 2 , is highly desirable. However, it is difficult to solve this problem, since the observer dynamics depends on L in a rather complicated manner. Therefore, the solution to this problem is beyond the scope of this paper.
IV. EXAMPLE-FEEDBACK COOLING OF PARTICLE MOTION
The main purpose of this section is to show that there actually exists an uncertain quantum system such that both the robust observer and the risk-sensitive observer perform more effectively than the Kalman filter, which is no longer optimum for uncertain systems. Moreover, we will carry out a detailed comparison of the above three observers by considering each estimation error. This is certainly significant from a practical viewpoint.
First, let us describe the system. The control objective is to stabilize the particle positionq at the origin by continuous monitoring and control. In other words, we aim to achieve π t (q) = q = 0 with a small error variance. The system observable is thus given bŷ c =q, i.e.,C = [1 0].
For the Hamiltonian part,Ĥ =Ĥ free +Ĥ control , we assume the following: The control Hamiltonian is proportional to the position operator:
H control = −u tq , i.e., B = 0 1 ,(31)
where u t = Lx t is the input, and the free Hamiltonian is of the formĤ free = 2p 2 + V (q), where V (q) denotes the potential energy of the particle. In general, the potential energy can assume a complicated structure. For example, Doherty et al. [28] have considered a nonlinear feedback control problem of a particle in a double-well potential V (q) =q 4 −q 2 . Since the present paper deals with only linear quantum systems, we approximate V (q) to the second order around the origin and consider a spatially local control of the particle. In particular, we examine the following two approximated free Hamiltonians:
H free 1 = 2p 2 − 0.05q 2 ,Ĥ free 2 = 2p 2 + 0.05q 2 .
The former is sometimes referred to as an anti-harmonic oscillator, while the latter is a standard harmonic oscillator approximation. The system matrices corresponding toĤ free are respectively given by
G 1 = −0.05 0 0 2 , G 2 = 0.05 0 0 2 .
In the case of the harmonic oscillator Hamiltonian, the system is autonomously stable at the origin. In contrast, in the case of the anti-harmonic oscillator, the system becomes unstable when we do not invoke any control. However, it is observed that the control Hamiltonian (31) with an appropriate control input can stabilize the system. An example is the LQG optimal controller with the following tuning parameters of the cost function (10): Figure I illustrates an estimate of the particle position in both the unstable autonomous trajectory and the controlled stable trajectory; in the latter case, the control objective π t (q) = 0 is actually satisfied. Second, we describe the uncertainty included in the system. In particular, we consider two situations in which uncertain Hamiltonians ∆Ĥ 1 = − √ d tq 2 and ∆Ĥ 2 = √ d tq 2 are added toĤ 1 andĤ 2 , respectively. The unknown time-varying parameter d t is bounded by the known constant g ≥ 0, i.e., d t ∈ [0, g]. Regarding the uncertainty in the system operatorĉ, on the other hand, we assume ∆C t = 0, ∀t. In this case, we can set Q 1 = ǫ 1 I, Q 2 = (g/ǫ 1 )I, and Q 3 = 0 by choosing the parameters shown in the proof of Proposition 2.
M = 3 0 0 1 , r = 1 5 , N = 2 0 0 0 .(32)
The comparison of the three observers is performed based on the following evaluation. For the Kalman filter and the risk-sensitive observer, we evaluate the stationary mean square error between the "true" system and the estimator for the "nominal" system corresponding to d t = 0 (see Appendix C). In both cases, the tuning parameters in the cost function are set to Eq. (32). Next, for the robust observer, we evaluate the guaranteed upper bound of the estimation error Tr P 2 in Eq. (26). The control input in the robust observer is set to the stationary LQG controller for the nominal system:
u t = Lx t = −(2/r)B T K ∞ x t ,
where K ∞ is the stationary solution of Eq. (11).
Let us now describe the simulation results. First, we consider the case in which the total system Hamiltonian is given byĤ =Ĥ free 1 +Ĥ control + ∆Ĥ 1 . Table I lists the three estimation errors mentioned above for several values of g. Here, the uncertainty d t is set to the "worst case" d t = g for each value of g. In the first row of the table, the notation "N/A" indicates that the solution of the Lyapunov equation (C1) does not satisfyW + i Σ /2 ≥ 0. This implies that the error dynamics between the uncertain actual system and the nominal Kalman filter is unstable. In other words, the Kalman filter fails in the estimation. It should be noted that two excessively large values of the estimation error, which appear in the first and second rows, indicate that the error dynamics is nearly unstable. Therefore, it can be concluded that the Kalman filter and the risk-sensitive observer for the nominal system do not work well when the uncertainty d t (= g) assumes a large. On the other hand, as shown in the third row in Table I, the robust observer is not very sensitive to the magnitude of the uncertainty and provides a good estimation even when g is large. The above discussion suggests that the robust observer is possibly TABLE I: Comparison of the Kalman, risk-sensitive, and robust observers, denoted by "KAL", "RSK", and "ROB", respectively. The free Hamiltonian of the system is approximated by the anti-harmonic oscillator. In order to calculate the guaranteed upper bound of the estimation error of the robust observer, Tr P2, parameters δ1 and δ2 are fixed to 0.1, and ǫ1 is selected such that Tr P2 takes the minimum value. The risk-sensitive parameter is µ = 0.3, and the Planck constant is set to unity: = 1. Note that both the robust observer and the risk-sensitive observer are not identical to the Kalman filter even when g = 0, because the parameters δ2 and µ are now set to non-zero values. the best option for dealing with a large uncertainty. In other cases, the risk-sensitive observer should be used. Next, we consider the second example, in which the total system Hamiltonian is given by the harmonic os-cillatorĤ =Ĥ free 2 +Ĥ control + ∆Ĥ 2 . In this case, it is immediately observed in Table II that the estimation errors of the robust observer are always greater than those of the others, while the risk-sensitive observer shows a good performance, particularly when g assumes a large value. Hence, in this case the risk-sensitive observer is the most appropriate.
An interesting feature of the robust observer is that in the case of both the harmonic and anti-harmonic Hamiltonians, it provides almost the same trend in the estimation errors with respect to g, whereas the Kalman filter and the risk-sensitive observer produce drastically different trends in the errors. This indicates that the structure of the robust observer is designed such that the estimation error is insensitive to the stability property of the system. However, this design policy sometimes leads to the over conservative stability of the error dynamics, and the estimation performance eventually reduces.
V. CONCLUSION
In this paper, we have considered a linear quantum system subjected to time-varying norm-bounded parametric uncertainties and developed a quantum version of the robust observer. Although in the linear case much of classical control theory can be applied to quantum systems, due to the unitarity of quantum evolution, the quantum uncertain system must have a specific structure with respect to the uncertainties, and its classical correspondence has not been studied; the resulting quantum robust observer has thus no classical analogue. The observer differs from both the optimal Kalman filter and the risk-sensitive observer; however, it guarantees the upper bound of the variance of the estimation error. We then investigated the robustness property of the three estimators mentioned above by considering a typical quantum control problem-feedback cooling of particle motion. This examination clarified that the robust observer is superior to the others when the autonomous system is unstable and is subjected to an unknown perturbation with a large magnitude. Therefore, we can conclude that the robust filtering method originally developed for classical systems is actually very effective for quantum systems as well. This fact implies that several robust control techniques in classical control theory (e.g., [29]) will be applicable to uncertain quantum systems.
whereΣ = −[Σ Σ] T ∈ R 4×2 ,Ē = [I O] ∈ R 2×4 and Θ 1 = ΣC T 1 −ΣC T 2 ΣC T 1 −ΣC T 2 − 2k ,Θ 2 = C 2 0 T −C 1 0 T , ∆J 1 = ∆C 2 ∆C 1 , ∆J 2 = [∆C T 1 ∆C T 2 ].
We here denoted 0 T = [0 0]. Accordingly, the matrix ∆ĀX +X∆Ā T is now represented by
∆ĀX +X∆Ā T = (Σ∆GĒ)X +X(Σ∆GĒ) T + (Θ 1 ∆J 1Ē )X +X(Θ 1 ∆J 1Ē ) T + (Σ∆J 2Θ2 )X +X(Σ∆J 2Θ2 ) T + (Σ∆C T 1 ∆C 2Ē )X +X(Σ∆C T 1 ∆C 2Ē ) T − (Σ∆C T 2 ∆C 1Ē )X −X(Σ∆C T 2 ∆C 1Ē ) T . (A2)
We are then able to apply Eq. (A1) to evaluate bounds of each line in the above equation. For example, the second line has the following bound:
(Σ∆GĒ)X +X(Σ∆GĒ) T ≤ ǫ 1XĒ TĒX + 1 ǫ 1Σ ∆G 2ΣT ≤ ǫ 1XĒ TĒX + g ǫ 1ΣΣ T ,
where here the assumption on the uncertainty (18) was used. The free parameter ǫ 1 > 0 should be tuned appropriately. Next, for evaluating the third line of Eq. (A2), we remark the following:
∆J 1 ∆J T 1 ≤ diag{2r 2 , 2r 1 }.(A3)
This inequality is easily seen; the relations ∆C i ∆C T i ≤ r i (i = 1, 2) lead to det diag{2r 2 , 2r 1 } − ∆J 1 ∆J T 1 = ∆C 1 2 ∆C 2 2 − ∆C 1 , ∆C 2 ≥ 0.
Here, • 2 and •, • denote the standard Euclidean norm and inner product, respectively. By using Eqs. (A1) and (A3), we then obtain the following inequality:
(Θ 1 ∆J 1Ē )X +X(Θ 1 ∆J 1Ē ) T ≤ ǫ 2XĒ TĒX + 1 ǫ 2Θ 1 ∆J 1 ∆J T 1Θ T 1 ≤ ǫ 2XĒ TĒX + 2 ǫ 2Θ 1 diag{r 2 , r 1 }Θ T 1 .
For the other lines of Eq. (A2), we can use the same manner to have their bounds that do not depend on the uncertainties. As a result, we obtain the objective inequality ∆ĀX +X∆Ā T ≤XQ 1X +Q 2 , wherē holds. Then, Eq. (B1) has a unique stationary solution that satisfies lim t→∞ P t ≤ X.
Proof. We readily see that the matrix A is strictly stable; any eigenvalue of A has a negative real part. Now, let us define δP t := X − P t . Then, by using the assumption we havė δP t = −(AX + XA T + BB T ) + AδP t + δP t A T ≥ AδP t + δP t A T , which yields δP t ≥ e At δP 0 e A T t . We then obtain lim t→∞ δP t ≥ 0 since A is strictly stable. This shows the assertion.
APPENDIX C: NOMINAL-TRUE SYSTEMS DIFFERENCE
The objective here is to characterize the stationary mean square error between the "true" system (20) and the risk-sensitive observer (15) specifically designed for the "nominal" system (6). For this purpose, we calculate the symmetrized covariance matrix of the error vector e t :=x t − π µ t (x), wherex t and π µ t (x) are generated from Eqs. (20) and (15), respectively. Particularly, we now focus on the stationary observer. Thus, let us assume that the two Riccati equations (16) and (17) have unique steady solutions V µ ∞ and K µ ∞ , respectively. Then, defining
b o := 1 V µ ∞ F T + Σ T (ImC) T , L o := − 2 r B T K µ ∞ ,
the stationary risk-sensitive observer is described by
dπ µ t (x) = (A + µV µ ∞ M + BL o )π µ t (x)dt + b o (dY t − F π µ t (x)dt).
We then see that the augmented vectorζ t = [x têt ] T satisfies dζ t =Ā oζt dt +b o dB t +b * o dB † t , wherē
A o = A + ∆A + BL o −BL o ∆A − µV µ ∞ M A + µV µ ∞ M − b o F , b o = iΣC T iΣC T − b o .
LetV t be the symmetrized covariance matrix ofζ t . As mentioned in the proof of Theorem 1, this matrix satisfies ζ tζ T t =V t + i Σ /2. By using this relation, we obtain dV t /dt =Ā oVt +V tĀ T o +D o , whereD o is given bȳ
D o = D D D D − O [b o Im(C)Σ] T b o Im(C)Σ b o Im(C)Σ + [b o Im(C)Σ] T − b o b T o .
As a result, the variance of the estimation error is given by lim t→∞ ê T têt =W 33 +W 44 , whereW is the stationary solution of the following Lyapunov equation:
A oW +WĀ T o +D o = O.(C1)
The estimation error between the true system and the Kalman filter designed for the nominal system is immediately evaluated by setting µ = 0 in the above discussion.
FIG. 1 :
1An example of the unstable autonomous trajectory (dot line) and the controlled stable trajectory (solid line) shown by πt(q).
TABLE II :
IIComparison of the three types of estimators in the
case of the harmonic oscillator Hamiltonian. All parameters
of the estimators are set to the same values in Table I.
g
0.00
0.20
0.40
0.60
0.80
1.00
KAL
1.40
1.37
1.40
1.44
1.47
1.50
RSK
1.44
1.38
1.38
1.39
1.40
1.41
ROB
1.68
3.23
4.79
6.84
9.80
14.48
t (Xα +αX) − 2π µ t (X)π µ t (α) dt+ π µ t (Xĉ +ĉ †X ) − π µ t (X)π µ t (ĉ +ĉ † ) × dY t − π µ t (ĉ +ĉ † )dt .(14)This differs from the filtering equation(4) in that the risk-dependent term is added to it. Therefore, π µ t (X) is no longer the optimal estimate of the system observable. However, the risk-dependent term is indeed necessary in order for the cost function(12) to be expressed only in terms of quantities defined on the system space that is driven by the output Y t . This implies that our knowledge about the system is tempered by purpose.
AcknowledgmentsThe author wishes to thank R. van Handel, L. Bouten, and H. Mabuchi for their helpful comments. This work was supported in part by the Grants-in-Aid for JSPS fellows No.06693.At first, we derive a simple yet useful matrix inequality. For any real matrices X and Y , we obviously havewhere ǫ > 0 is a free parameter. The above inequality immediately leads toNext, let us definẽ(In this appendix, we omit the suffix t for simplicity.) Then, the conditions(19)are represented by ∆C T i ∆C i ≤ r i I (i = 1, 2). Note that they lead to the scalar inequalities: ∆C i ∆C T i ≤ r i (i = 1, 2). Now we are at the point to prove. Let us first derive the inequality(29). By a straightforward calculation, we obtainThe matrices Q 1 and Q 2 are defined as follows.The parameters ǫ i > 0 (i = 1, . . . , 5) should be chosen appropriately.Let us next derive Eq. (30). Similar to the previous case, we use Eq. (A1) to obtain a bound that does not depend on the uncertainty. First, we immediately obtainwhere ǫ 6 > 0 and ǫ 7 > 0 are free parameters. This readily leads toAlso, setting X = [0 T k T ] and Y = [−∆m T − ∆m T ] in Eq. (A1), we obtain the following inequality:andWe consider a matrix-valued differential equation of the formṖ t = AP t + P t A T + BB T .(B1)Lemma. Suppose there exists a positive definite matrix X > 0 such that the inequality AX + XA T + BB T < 0
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| {'fraction_non_alphanumeric': 0.07369319202442967, 'fraction_numerical': 0.03181695706843902, 'mean_word_length': 3.42801749850865, 'pattern_counts': {'":': 0, '<': 6, '<?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': 50, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "In the theory of quantum dynamical filtering, one of the biggest issues is that the underlying system dynamics represented by a quantum stochastic differential equation must be known exactly in order that the corresponding filter provides an optimal performance; however, this assumption is generally unrealistic. Therefore, in this paper, we consider a class of linear quantum systems subjected to time-varying norm-bounded parametric uncertainties and then propose a robust observer such that the variance of the estimation error is guaranteed to be within a certain bound. Although in the linear case much of classical control theory can be applied to quantum systems, the quantum robust observer obtained in this paper does not have a classical analogue due to the system's specific structure with respect to the uncertainties. Moreover, by considering a typical quantum control problem, we show that the proposed robust observer is fairly robust against a parametric uncertainty of the system even when the other estimators-the optimal Kalman filter and risk-sensitive observer-fail in the estimation.", 'arxivid': 'quant-ph/0602235', 'author': ['Naoki Yamamoto \nPhysical Measurement and Control\nCalifornia Institute of Technology\n266-33, 91125PasadenaCA\n'], 'authoraffiliation': ['Physical Measurement and Control\nCalifornia Institute of Technology\n266-33, 91125PasadenaCA'], 'corpusid': 17174678, 'doi': '10.1103/physreva.74.032107', 'github_urls': [], 'n_tokens_mistral': 15027, 'n_tokens_neox': 13338, 'n_words': 8533, 'pdfsha': '2aa1ee1bb1fb180daa2ec8b133e678b716b9a774', 'pdfurls': ['https://export.arxiv.org/pdf/quant-ph/0602235v2.pdf'], 'title': ['Robust observer for uncertain linear quantum systems', 'Robust observer for uncertain linear quantum systems'], 'venue': []} |
arxiv |
28 Mar 2023
Zuan Liu
Zihao Qi [email protected]
Yufei
Guodong Zhou [email protected]
Zuan Liu
Yufei Qin
Guodong Zhou
Zihao Qi
School of Mathematical Sciences
School of Mathematical Sciences
Shanghai Key Laboratory of PMMP
East China Normal University
200241ShanghaiChina
Fudan University
200433ShanghaiChina
28 Mar 2023GRÖBNER-SHIRSHOV BASES AND LINEAR BASES FOR FREE MULTI-OPERATED ALGEBRAS OVER ALGEBRAS WITH APPLICATIONS TO DIFFERENTIAL ROTA-BAXTER ALGEBRAS AND INTEGRO-DIFFERENTIAL ALGEBRAS
Quite much recent studies has been attracted to the operated algebra since it unifies various notions such as the differential algebra and the Rota-Baxter algebra. An Ω-operated algebra is a an (associative) algebra equipped with a set Ω of linear operators which might satisfy certain operator identities such as the Leibniz rule. A free Ω-operated algebra B can be generated on an algebra A similar to a free algebra generated on a set. If A has a Gröbner-Shirshov basis G and if the linear operators Ω satisfy a set Φ of operator identities, it is natural to ask when the union G ∪ Φ is a Gröbner-Shirshov basis of B. A previous work answers this question affirmatively under a mild condition, and thereby obtains a canonical linear basis of B.In this paper, we answer this question in the general case of multiple linear operators. As applications we get operated Gröbner-Shirshov bases for free differential Rota-Baxter algebras and free integro-differential algebras over algebras as well as their linear bases. One of the key technical difficulties is to introduce new monomial orders for the case of two operators, which might be of independent interest.Contents1 2 ZUAN LIU, ZIHAO QI, YUFEI QIN AND GUODONG ZHOU 4.2. Case of nonunital algebras with λ = 0 24 4.3. Case of unital algebras 25 4.4. Differential Rota-Baxter algebras vs integro-differential algebras 25 References 26
Introduction
This paper extends the results of [17] to algebras endowed with several operators, with applications to differential Rota-Baxter algebras and integro-differential algebras. 0.1. Operated GS basis theory: from a single operator to multiple operators.
Since its introduction by Shirshov [20] and Buchberger [4] in the sixties of last century, Gröbner-Shirshov (=GS) basis theory has become one of the main tools of computational algebra; see for instance [10,1,3]. In order to deal with algebras endowed with operators, Guo and his coauthors introduced a GS basis theory in a series of papers [11,23,15,6] (see also [2]) with the goal to attack Rota's program [19] to classify "interesting" operators on algebras. Guo et al. considered operators satisfying some polynomial identities, hence called operated polynomial identities (aka. OPIs) [11,23,15,6]. Via GS basis theory and the somewhat equivalent theory: rewriting systems, they could define when OPIs are GS. They are mainly interested into two classes of OPIs: differential type OPIs and Rota-Baxter type OPIs, which are carefully studied in [15,23,6]. For the state of art, we refer the reader to the survey paper [8] and for recent development, see [22,12,17,21].
In these papers [11,23,15,6], the operated GS theory and hence Rota's classification program have been carried out only for algebras endowed with a single operator. It would be very interesting to carry out further Rota's program for the general case of multiple linear operators.
The paper [2] contains a first step of this program by developing the GS basis theory in this generalised setup. We will review and update the GS basis theory in the multi-operated setup in Section 2.
Another direction is to generalise from operated algebras over a base field to operated algebras over a base ring. While previous papers [17,18] considered this aspect for single operator case, this paper is aimed to deal with this aspect for multiple linear operator case. In particular, some new monomial orders for the two operator case will be constructed which enable us to study operated GS bases for free operated algebras generated by algebras, while it seems that the monomial orders appeared in previous papers can be applied directly when the base ring is not a field any more.
Free operated algebras over algebras.
Recently, there is a need to develop free operated algebras satisfying some OPIs over a fixed algebras and construct GS bases and linear bases for these free algebras as long as a GS basis is known for the given algebra. Ebrahimi-Fard and Guo [5] used rooted trees and forests to give explicit constructions of free noncommutative Rota-Baxter algebras on modules and sets; Lei and Guo [16] constructed the linear basis of free Nijenhuis algebras over associative algebras; Guo and Li [12] gave a linear basis of the free differential algebra over associative algebras by introducing the notion of differential GS bases.
In a previous paper [17], the authors considered a question which can be roughly stated as follows:
Question 0.1. Given a (unital or nonunital) algebra A with a GS basis G and a set Φ of OPIs, assume that these OPIs Φ are GS in the sense of [2,15,23,6]. Let B be the free operated algebra satisfying Φ over A. When will Φ ∪ G be a GS basis for B?
They answer this question in the affirmative under a mild condition in [17,Theorem 5.9]. When this condition is satisfied, Φ ∪ G is a GS basis for B and as a consequence, we also get a linear basis of B. This result has been applied to all Rota-Baxter type OPIs, a class of differential type OPIs, averaging OPIs and Reynolds OPI in [17]. It was also applied to differential type OPIs by introducing some new monomial orders [18].
In this paper, we consider a similar question for multi-operated algebras.
Let Ω be a nonempty set which will be the index set of operators. Algebras endowed with operators indexed by Ω will be called Ω-algebras. OPIs can be extended to the multi-operated setup and one can introduce the notion of Ω-GS for OPIs.
Question 0.2. Let Φ be a set of OPIs of a set of operators indexed by Ω. Let A be a (unital) algebra together with a GS basis G. Assume that these OPIs Φ are GS in the sense of Section 2. Let B be the free Ω-algebra over A such that the operators satisfy Φ. When will Φ ∪ G be an Ω-GS basis for B?
We extend the main result of [17] to multi-operated cases; see Theorem 2.12 for unital algebras and Theorem 2.13 for nonunital algebras. 0.3. Differential Rota-Baxter algebras and integro-differential algebras.
The main motivation of this paper comes, in fact, from differential Rota-Baxter algebras and integro-differential algebras. Differential Rota-Baxter algebras were introduced by Guo and Keigher [13] which reflect the relation between the differential operator and the integral operator as in the First Fundamental Theorem of Calculus. Free differential Rota-Baxter algebras were constructed by using various tools including angularly decorated rooted forests and GS basis theory [13,2].
Integro-differential algebras (of zero weight) were defined for the algebraic study of boundary problems for linear systems of linear ordinary differential equations. Guo, Regensburger and Rosenkranz [14] introduced Integro-differential algebras with weight. Free objects and their linear bases were constructed by using GS basis theory [14,9,7] The main goal of this paper is to study free differential Rota-Baxter algebras and free integrodifferential algebras over algebras from the viewpoint of operated GS basis theory. In particular, when the base algebra is reduced to k, our results also give GS bases and linear bases for free differential Rota-Baxter algebras and free integro-differential algebras.
However, the original monomial orders used in [2,14,9,7] do not satisfy the hypothesis in Theorems 2.12 and 2.13 for free multi-operated algebras over algebras, and we have to introduce a new monomial order ≤ PD (resp. ≤ uPD ) to overcome the problem; see Section 1.3.
In contrast to the use different monomial orders while dealing with free differential Rota-Baxter algebras and free integro-differential algebras in [2] and [7] respectively, we will demonstrate that our monomial ordering ≤ PD can be applied to both types of algebras simultaneously, as we shall see in Sections 3 and 4. Moreover, since the case of the unital algebras was not discussed in [2], this aspect is addressed in Subsection 3.3 by using our monomial order ≤ uPD .
Outline of the paper.
This paper is organized as follows.
The first section contains remainder on free objects in multi-operated setting and on the construction of free Ω-semigroups and related structures, and introduces some new monomial orders for the case of two operators, which will be the key technical tool of this paper.
In the second section, we recall the theory of GS bases for the multi-operated setting. After introducing OPIs, GS property for OPIs and Ω-GS bases for multi-operated algebras are defined; after giving some facts about free multi-operated Φ-algebras on algebras, answers to Question 0.2 are presented.
In the third section, multi-operated GS bases and linear bases for free differential Rota-Baxter algebras on algebras are studied and the fourth section contains our investigation for free integrodifferential algebras on algebras.
Notation: Throughout this paper, k denotes a base field. All the vector spaces and algebras are over k.
New monomial orders on free multi-operated semigroups and monoids
In this section, we recall free objects in multi-operated setting and the construction of free Ω-semigroups and related structures, and define two new monomial orders ≤ PD and ≤ uPD on free multi-operated semigroups and monoids. The main results of this paper will highly depend on these new monomial orders.
For a set Z, denote by kZ (resp. S(Z), M(Z)) the free k-vector space (resp. free semigroup, free monoid) generated by Z. Denote the category of sets (resp. semigroups, monoids) by Set (resp. Sem, Mon). Denote the categories of k-algebras and unital k-algebras by Alg and uAlg respectively.
Throughout this section, let Ω be a nonempty set which will be the index set of operators.
1.1. Free objects in the multi-operated setup.
Definition 1.
1. An operated set with an operator index set Ω or simply an Ω-set is a set S endowed with a family of maps P ω : S → S indexed by ω ∈ Ω. The morphisms between Ω-sets can be defined in the obvious way. Denote the category of Ω-sets by Ω-Set. Similarly, we can define Ω-semigroups and Ω-monoids. Their categories are denoted by Ω-Sem and Ω-Mon respectively.
Ω-vector spaces, nonunital or unital Ω-algebras can be defined in a similar way, except asking, moreover, that all the operators are k-linear maps. Denote the category of Ω-vector spaces, (resp. nonunital Ω-algebras, unital Ω-algebras) by Ω-Vect (resp. Ω-Alg, Ω-uAlg) with obvious morphisms.
As in [17], there exists the following diagram of functors:
Ω-Vect / / { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ Ω-Alg / / o o { { ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ Ω-uAlg z z ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ o o Ω-Set ; ; ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ / / Ω-Sem ; ; ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ / / o o Ω-Mon : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ o o Vect O O ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ / / ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ { { ✈ ✈ ✈ ✈ ✈ ✈ Alg O O ✤ ✤ ✤ ✤ ✤ ✤ ✤ ✤ / / ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ { { ✈ ✈ ✈ ✈ ✈ uAlg O O o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ z z ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ Set O O ; ; ✈ ✈ ✈ ✈ ✈ ✈ / / Sem / / O O ; ; ✈ ✈ ✈ ✈ ✈ o o Mon : : ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ O O o o
In this diagram, all functors from right to left, from below to above and from southwest to northeast are the obvious forgetful functors. The other functors are free object functors which are left adjoint to the forgetful functors.
Our notations for free object functors are analogous to those in [17]. For instance, F Ω-Alg Alg denotes the free object functor from the category of algebras to that of nonunital Ω-algebras.
We could give similar constructions of these free object functors as in Sections 1-3 of [17]. However, as we don't need the details, we will not repeat them. The curious readers could consult [17] and extend the constructions in [17] without essential difficulties.
Free multi-operated semigroups and monoids.
Now we explain the construction of the free Ω-semigroup generated by a set Z.
For ω ∈ Ω, denote by ⌊Z⌋ ω the set of all formal elements ⌊z⌋ ω , z ∈ Z and put ⌊Z⌋ Ω = ⊔ ω∈Ω ⌊Z⌋ ω . The inclusion into the first component Z ֒→ Z ⊔ ⌊Z⌋ Ω induces an injective semigroup homomorphism i 0,1 : S Ω,0 (Z) := S(Z) ֒→ S Ω,1 (Z) := S(Z ⊔ ⌊Z⌋ Ω ). For n ≥ 2, assume that we have constructed S Ω,n−2 (Z) and S Ω,n−1 (Z) = S(Z ⊔ ⌊S Ω,n−2 (Z)⌋ Ω ) endowed with an injective homomorphism of semigroups i n−2,n−1 : S Ω,n−2 (Z) ֒→ S Ω,n−1 (Z). We define the semigroup S Ω,n (Z) := S(Z ⊔ ⌊S Ω,n−1 (Z)⌋ Ω ) and the natural injection
Id Z ⊔ ⌊i n−2,n−1 ⌋ Ω : Z ⊔ ⌊S Ω,n−2 (Z)⌋ Ω ֒→ Z ⊔ ⌊S Ω,n−1 (Z)⌋ Ω induces an injective semigroup homomorphism i n−1,n : S Ω,n−1 (Z) = S(Z ⊔ ⌊S Ω,n−2 (Z)⌋ Ω ) ֒→ S Ω,n (Z) = S(Z ⊔ ⌊S Ω,n−1 (Z)⌋ Ω ). Define S Ω (Z) = lim − − →
S Ω,n (Z) and the maps sending u ∈ S Ω,n (Z) to ⌊u⌋ ω ∈ S Ω,n+1 (Z) induces a family of operators P ω , ω ∈ Ω on S Ω (Z).
The construction of the free Ω-monoid M Ω (M) over a set Z is similar, by just replacing S(Z) by M(Z) everywhere in the construction. Remark 1.2. We will use another construction of M Ω (Z). In fact, add some symbols ⌊1⌋ Ω = {⌊1⌋ ω , ω ∈ Ω} to Z and form S Ω (Z ⊔ ⌊1⌋ Ω ), then M Ω (Z) can be obtained from S Ω (Z ⊔ ⌊1⌋ Ω ) by just adding the empty word 1.
It is easy to see that kS Ω (Z)(resp. kM Ω (Z)) is the free nonunital (resp. unital) Ω-algebra generated by Z.
Monomial orders.
In this subsection, we introduce some new monomial orders on free Ω-semigroups and free Ω-monoids. We only consider the case of two operators, say Ω = {P, D} as the main examples in mind are differential Rota-Baxter algebras and integro-differential algebras following the convention from [7].
We first recall the definitions of well orders and monomial orders.
Definition 1.3. Let Z be a nonempty set.
(a) A preorder ≤ is a binary relation on Z that is reflexive and transitive, that is, for all x, y, z ∈ Z, we have
(i) x ≤ x; and (ii) if x ≤ y, y ≤ z, then x ≤ z.
In the presence of a preoder ≤, we denote x = Z y if x ≤ y and x ≥ y; if x ≤ y but x y, we write x < y or y > x. (b) A pre-linear order ≤ on Z is a preorder ≤ such that either x ≤ y or x ≥ y for all x, y ∈ Z. (c) A linear order or a total order ≤ on Z is a pre-linear order ≤ such that ≤ is symmetric, that is, x ≤ y and y ≤ x imply x = y. (d) A preorder ≤ on Z is said to satisfy the descending chain condition, if for each descending chain
x 1 ≥ x 2 ≥ x 3 ≥ · · · , there exists N ≥ 1 such that x N = Z x N+1 = Z · · · .
A linear order satisfying the descending chain condition is called a well order.
Before giving the definition of monomial orders, we need to introduce the following notions generalising the case of one operator. (a) A monomial order on S(Z) is a well-order ≤ on S(Z) such that u < v ⇒ uw < vw and wu < wv for any u, v, w ∈ S(Z);
(a') a monomial order on M(Z) is a well-order ≤ on M(Z) such that u < v ⇒ wuz < wvz for any u, v, w, z ∈ M(Z);
(b) a monomial order on S Ω (Z) is a well-order ≤ on S Ω (Z) such that
u < v ⇒ q| u < q| v for all u, v ∈ S Ω (Z) and q ∈ S ⋆ Ω (Z); (b') a monomial order on M Ω (Z) is a well-order ≤ on M Ω (Z) such that u < v ⇒ q| u < q| v for all u, v ∈ M Ω (Z) and q ∈ M ⋆ Ω (Z).
Let us recall some known preorders.
Definition 1.6. For two elements u, v ∈ S Ω (Z), (a) define u ≤ D v ⇔ deg D (u) ≤ deg D (v), where the D-degree deg D (u) of u is the number of occurrence of ⌊ ⌋ D in u; (b) define u ≤ P v ⇔ deg P (u) ≤ deg P (v), where the P-degree deg P (u) of u is the number of occurrence of ⌊ ⌋ P in u; (c) define u ≤ dZ v ⇔ deg Z (u) ≤ deg Z (v), where the Z-degree deg Z (u)
is the number of elements of Z occurring in u counting the repetitions;
Definition 1.7. Let Z be a set endowed with a well order ≤ Z . Introduce the degree-lexicographical order ≤ dlex on S(Z) by imposing, for any u v ∈ S(Z), u < dlex v if (a) either deg Z (u) < deg Z (v), or (b) deg Z (u) = deg Z (v), and u = mu i n, v = mv i n ′ for some m, n, n ′ ∈ M(Z) and u i , v i ∈ Z with u i < Z v i .
It is obvious that the degree-lexicographic order ≤ dlex on S(Z) is a well order . We now define a preorder ≤ Dlex on S Ω (Z), by the following recursion process:
(a) For u, v ∈ S Ω,0 (Z) = S(Z), define u ≤ Dlex 0 v ⇔ u ≤ dlex v.
(b) Assume that we have constructed a well order ≤ Dlex n on S Ω,n (Z) for n ≥ 0 extending all ≤ Dlex i for any 0 ≤ i ≤ n − 1. The well order ≤ Dlex n on S Ω,n (Z) induces a well order on ⌊S Ω,n (Z)⌋ P (resp. ⌊S Ω,n (Z)⌋ D ), by imposing ⌊u⌋ P ≤ ⌊v⌋ P (resp. ⌊u⌋ D ≤ ⌊v⌋ D ) whenever u ≤ Dlex n v ∈ S Ω,n (Z). By setting u < v < w for all u ∈ Z, v ∈ ⌊S Ω,n (Z)⌋ D , and w ∈ ⌊S Ω,n (Z)⌋ P , we obtain a well order on Z ⊔ ⌊S Ω,n (Z)⌋ P ⊔ ⌊S Ω,n (Z)⌋ D . Let ≤ Dlex n+1 be the degree lexicographic order on S Ω,n+1 (Z) = S(Z ⊔ ⌊S Ω,n (Z)⌋ P ⊔ ⌊S Ω,n (Z)⌋ D ) induced by that on Z ⊔ ⌊S Ω,n (Z)⌋ P ⊔ ⌊S Ω,n (Z)⌋ D . Obviously ≤ Dlex n+1 extends ≤ Dlex n . By a limit process, we get a preorder on S Ω (Z) which will be denoted by ≤ Dlex . As is readily seen, ≤ Dlex is a linear order. Remark 1.8. It is easy to see that the above construction of ≤ Dlex can be extended to the case of more than two operators.
In fact, for a given well order ≤ Ω in the index set Ω, the defining process of ≤ Dlex on S Ω (Z) is the same as above except one detail in step (b), where we need to put u < v < w for all u ∈ Z, v ∈ ⌊S Ω,n (Z)⌋ ω 1 and w ∈ ⌊S Ω,n (Z)⌋ ω 2 with ω 1 ≤ Ω ω 2 ∈ Ω. Definition 1.9. For any u ∈ S Ω (Z), let u 1 , . . . , u n ∈ Z be all the elements occurring in u from left to right. If a right half bracket ⌋ D locates in the gap between u i and u i+1 , where 1 ≤ i < n, the GD-degree of this right half bracket is defined to be n − i; if there is a right half bracket ⌋ D appearing on the right of u n , we define the GD-degree of this half bracket to be 0. We define the GD-degree of u, denoted by deg GD (u), to be the sum of the GD-degrees of all the half right brackets in u.
For example, the GD-degrees of the half right brackets in u = ⌊x⌋ D ⌊y⌋ D with x, y ∈ Z are respectively 1 and 0 from left to right, so deg GD (u) = 1 by definition.
For u, v ∈ S Ω (Z), define the GD-degree order ≤ GD by
u ≤ GD v ⇔ deg GD (u) ≤ deg GD (v).
Definition 1.10. For any u ∈ S Ω (Z), let u 1 , . . . , u n ∈ Z be all the elements occurring in u from left to right. If there are i elements in Z contained in a bracket ⌊ ⌋ P , the GP-degree of this bracket is defined to be n − i. We denote by deg GP (u) the sum of the GP-degree of all the brackets ⌊ ⌋ P in u.
For example, the the GP-degrees of the brackets ⌊ ⌋ P of u = ⌊xy⌋ P ⌊z⌋ P with x, y, z ∈ Z are respectively 1 and 2 from left to right, so deg GP (u) = 3 by definition.
For u, v ∈ S Ω (Z), define the GD-degree order ≤ GD by
u ≤ GP v ⇔ deg GP (u) ≤ deg GP (v).
It is easy to obtain the following lemma whose proof is thus omitted.
Lemma 1.11. The orders ≤ D , ≤ P , ≤ dZ , ≤ GD and ≤ GP are pre-linear orders satisfying the descending chain condition.
Combining all the orders above, we can now construct an order ≤ PD of S Ω (Z):
u ≤ PD v ⇔ u ≤ D v, or u = D v and u ≤ P v, or u = D v, u = P v and u ≤ dZ v, or u = D v, u = P v, u = dZ v and u ≤ GD v, or u = D v, u = P v, u = dZ v, u = GD v and u ≤ GP v, or u = D v, u = P v, u = dZ v, u = GD v, u = GP v and u ≤ Dlex v.
To prove that the ≤ PD is a well-order, we need some preparation.
Definition 1.12.
(a) Given some preorders ≤ α 1 , . . . , ≤ α k on a set Z with k ≥ 2, introduce another preorder ≤ α 1 ,...,α k by imposing recursively
u ≤ α 1 ,...,α k v ⇔ u < α 1 v, or u = α 1 v and u ≤ α 2 ,...,α k v. (b) Let k ≥ 2 and let ≤ α i be a pre-linear order on Z i , 1 ≤ i ≤ k. Define the lexicographical product order ≤ clex on the cartesian product Z 1 × Z 2 × · · · × Z k by defining (x 1 , · · · , x k ) ≤ clex (y 1 , · · · , y k ) ⇔ x 1 < α 1 y 1 , or x 1 = Z 1 y 1 and (x 2 , · · · , x k ) ≤ clex (y 2 , · · · , y k ) ,
where (x 2 , · · · , x k ) ≤ clex (y 2 , · · · , y k ) is defined by induction, with the convention that ≤ clex is the trivial relation when k = 1. (a) For k ≥ 2, let ≤ α 1 , . . . , ≤ α k−1 be pre-linear orders on Z, and ≤ α k a linear order on Z. Then ≤ α 1 ,...,α k is a linear order on Z. (b) Let ≤ α i be a well order on Z i , 1 ≤ i ≤ k. Then the lexicographical product order ≤ clex is a well order on the cartesian product Z 1 × Z 2 × · · · × Z k .
Proposition 1.14. The order ≤ PD is a well order on S Ω (Z).
Proof. Since ≤ Dlex is a linear order, so is ≤ PD by Lemma 1.11 and Lemma 1.13(a). It suffices to verify that ≤ PD satisfies the descending chain condition.
Let
v 1 ≥ PD v 2 ≥ PD v 3 ≥ PD · · · ∈ S Ω (Z)
be a descending chain. By Lemma 1.11, there exist N ≥ 1 such that
deg D (v N ) = deg D (v N+1 ) = deg D (v N+2 ) = · · · =: k, deg P (v N ) = deg P (v N+1 ) = deg P (v N+2 ) = · · · =: p, deg Z (v N ) = deg Z (v N+1 ) = deg Z (v N+2 ) = · · · deg GD (v N ) = deg GD (v N+1 ) = deg GD (v N+2 ) = · · · , and deg GP (v N ) = deg GP (v N+1 ) = deg GP (v N+2 ) = · · · . Thus all v i with i ≥ N belong to S Ω,k+p (Z)
. The restriction of the order ≤ Dlex to S Ω,k+p (Z) equals to the well order ≤ Dlex k+p , which by definition satisfies the descending chain condition, so the chain v 1 ≥ PD v 2 ≥ PD v 3 ≥ PD · · · stabilizes after finite steps. Proof. Let u ≤ PD v. It is obvious that preorders ≤ D , ≤ P and ≤ dZ are bracket compatible, left compatible and right compatible. This solves the three cases u
Definition 1.15 ([23, Definition 5.6]). A preorder ≤ α on S Ω (Z) is called bracket compatible (resp. left compatible, right compatible) if u ≤ α v ⇒ ⌊u⌋ D ≤ α ⌊v⌋ D and ⌊u⌋ P ≤ α ⌊v⌋ P , (resp. wu ≤ α wv, uw ≤ α vw, for all w ∈ S Ω (Z))< D v; u = D v, u < dgp v; u = D v, u = dgp v and u < dgx v. If u = D v, u = P v, u = dZ v and u < GD v, obviously ⌊u⌋ D < GD ⌊v⌋ D , ⌊u⌋ P < GD ⌊v⌋ P uw < GD vw and wu < GD wv for w ∈ S Ω (Z). So ⌊u⌋ D < PD ⌊v⌋ D , ⌊u⌋ P < PD ⌊v⌋ P , uw < PD vw and wu < PD wv.
The case that u = D v, u = P v, u = dZ v, u = GD v and u < GP v is similar to the above one. It remains to consider the case that
u = D v, u = P v, u = dZ v, u = GD v, u = GP v and u < Dlex v. Let n ≥ deg D (u), deg P (u). Since u, v ∈ S Ω,n (Z), thus u ≤ Dlex n v. By the fact that the restriction of ≤ Dlex n+1 to ⌊S Ω,n (Z)⌋ D is induced by ≤ Dlex n , we have ⌊u⌋ D ≤ Dlex n+1 ⌊v⌋ D , ⌊u⌋ D ≤ Dlex ⌊v⌋ D , and ⌊u⌋ D ≤ PD ⌊v⌋ D . Similarly ⌊u⌋ P ≤ PD ⌊v⌋ P . Let w ∈ S Ω,m (Z).
One can obtain uw ≤ Dlex r vw and wu ≤ Dlex r wv for r = max {m, n}, so uw ≤ PD vw and wu ≤ PD wv.
We are done. Now let's move to the unital case. Now we extend ≤ PD from S Ω (Z) to M Ω (Z) by using Remark 1.2. Definition 1.18. Let Z be a set with a well order. Let † P (resp. † D ) be a symbol which is understood to be ⌊1⌋ P (resp. ⌊1⌋ D ) and write Z ′ = Z ⊔ { † P , † D }. Consider the free operated semigroup S Ω (Z ′ ) over the set Z ′ . The well order on Z extends to a well order ≤ on Z ′ by setting † P > z > † D , for any z ∈ Z. Besides, we impose deg P ( † P ) = 1 and deg GP ( † P ) = 0. Then the monomial order ≤ PD on S Ω (Z ′ ) induces a well order ≤ uPD on M Ω (Z) = S Ω (Z ′ ) ⊔ {1} (in which ⌊1⌋ P and ⌊1⌋ D is identified with † P and † D respectively), by setting u > uPD 1 for any u ∈ S Ω (Z ′ ). Proof. Obviously, the well order ≤ uPD is bracket compatible on M Ω (Z)\{1}. Let x ∈ M Ω (Z)\{1}. By definition, x > uPD 1. We have ⌊x⌋ P > Dlex ⌊1⌋ P which implies ⌊x⌋ P > uPD † P . It is ready to see that ⌊x⌋ D > uPD x > uPD † D . Thus ≤ uPD is bracket compatible.
Clearly, ≤ uPD is left and right compatible.
We record several important conclusions which will be useful later.
Proposition 1.20. For any u, v ∈ M Ω (Z)\{1}, we have (a) ⌊u⌋ P ⌊1⌋ P > uPD ⌊u⌊1⌋ P ⌋ P ≥ uPD ⌊⌊u⌋ P ⌋ P , (b) ⌊1⌋ P ⌊v⌋ P > uPD ⌊⌊v⌋ P ⌋ P ≥ uPD ⌊⌊1⌋ P v⌋ P , (c) ⌊1⌋ P ⌊1⌋ P > uPD ⌊⌊1⌋ P ⌋ P , (d) ⌊1⌋ P ⌊v⌋ D > uPD ⌊⌊1⌋ P v⌋ D , (e) ⌊u⌋ D ⌊1⌋ P > uPD ⌊u⌊1⌋ P ⌋ D . Proof. Let u, v ∈ M Ω (Z)\{1} = S Ω (Z ′ ). (a) It is easy to see ⌊⌊u⌋ P ⌋ P have lowest deg Z ′ among ⌊u⌋ P ⌊1⌋ P , ⌊u⌊1⌋ P ⌋ P , ⌊⌊u⌋ P ⌋ P , and we also have deg GP (⌊u⌋ P ⌊1⌋ P ) > deg GP (⌊u⌊1⌋ P ⌋ P ). (b) It is similar to (a). (c) It follows from deg Z ′ (⌊1⌋ P ⌊1⌋ P ) > deg Z ′ (⌊⌊1⌋ P ⌋ P ). (d) It can be deduced from ⌊1⌋ P ⌊v⌋ D > Dlex ⌊⌊1⌋ P v⌋ D by Definition 1.7. (e) It holds because deg GD (⌊u⌋ D ⌊1⌋ P ) > deg GD (⌊u⌊1⌋ P ⌋ D ).
Operator polynomial identities and multi-operated GS bases
In this section, we extend the theory of operated GS bases due to [2,15,23,6] from the case of single operator to multiple operators case. The presentation is essentially contained in [7].
Operator polynomial identities.
In this subsection, we give some basic notions and facts related to operator polynomial identities. Throughout this section, X denotes a set.
Definition 2.1. We call an element φ(x 1 , . . . , x n ) ∈ kS Ω (X) (resp. kM Ω (X)) with n ≥ 1, x 1 , . . . , x n ∈ X an operated polynomial identity (aka OPI).
From now on, we always assume that OPIs are multilinear, that is, they are linear in each x i . (r 1 , . . . , r n ) = 0, for all r 1 , . . . , r n ∈ A. In this case, (A, {P ω } ω∈Ω ) is called a (unital) φ-algebra.
Definition 2.2. Let φ(x 1 , . . . , x n ) be an OPI. A (unital) Ω-algebra A = (A, {P ω } ω∈Ω ) is said to satisfy the OPI φ(x 1 , . . . , x n ) if φ
Generally, for a family Φ of OPIs, we call a (unital) Ω-algebra (A, {P ω } ω∈Ω ) a (unital) Φ-algebra if it is a (unital) φ-algebra for any φ ∈ Φ. Denote the category of Φ-algebras (resp. unital Φalgebras) by Φ-Alg (resp. Φ-uAlg).
Definition 2.3.
An Ω-ideal of an Ω-algebra is an ideal of the associative algebra closed under the action of the operators. The Ω-ideal generated by a subset S ⊆ A is denoted by S Ω-Alg (resp. S Ω-uAlg ).
Obviously the quotient of an Ω-algebra (resp. unital Ω-algebra) by an Ω-ideal is naturally an Ω-algebra (resp. Ω-unital algebra).
From now on, Φ denotes a family of OPIs in kS Ω (X) or kM Ω (X). For a set Z and a subset Y of
M Ω (Z), introduce the subset S Φ (Y) ⊆ kM Ω (Z) to be S Φ (Y) := {φ(u 1 , . . . , u k ) | u 1 , . . . , u k ∈ Y, φ(x 1 , . . . , x k ) ∈ Φ}.
2.2. Multi-operated GS bases for Φ-algebras.
In this subsection, operated GS basis theory is extended to algebras with multiple operators following closely [2]. Obviously, each subset S ⊆ M Ω (Z) can be made monicized if we divide each nonzero element by its leading coefficient.
We need another notation. Let Z be a set. For u ∈ M Ω (Z) with u 1, as u can be uniquely written as a product u 1 · · · u n with u i ∈ Z ∪⌊M Ω (Z)⌋ Ω for 1 ≤ i ≤ n, call n the breadth of u, denoted by |u|; for u = 1, we define |u| = 0.
Definition 2.5. Let ≤ be a monomial order on S Ω (Z) (resp. M Ω (Z)) and f, g ∈ kS Ω (Z) (resp. kM Ω (Z)) be monic.
(a) If there are p, u, v ∈ S Ω (Z) (resp. M Ω (Z)) such that p =f u = vḡ with max |f |, |ḡ| < |p| < |f | + |ḡ|, we call ( f, g) u,v p := f u − vg the intersection composition of f and g with respect to p. (b) If there are p ∈ S Ω (Z) (resp. M Ω (Z)) and q ∈ S ⋆ Ω (Z) (resp. M ⋆ Ω (Z)) such that p =f = q|ḡ, we call ( f, g) q p := f − q| g the inclusion composition of f and g with respect to p. Definition 2.6. Let Z be a set and ≤ a monomial order on S Ω (Z) (resp. M Ω (Z)). Let G ⊆ kS Ω (Z) (resp. kM Ω (Z)).
(a) An element f ∈ kS Ω (Z) (resp. kM Ω (Z)) is called trivial modulo (G, p)
for p ∈ S Ω (Z) (resp. M Ω (Z)) if f = i c i q i | s i with q i |s i < p, where c i ∈ k, q i ∈ S ⋆ Ω (Z) (resp. M ⋆ Ω (Z)) and s i ∈ G.
If this is the case, we write ( f, g) p ≡ 0 mod (G, p).
In general, for any u, v ∈ S Ω (Z) (resp. M Ω (Z)), u ≡ v mod (G, p) means that u − v = c i q i | s i , with q i |s i < p, where c i ∈ k, q i ∈ S ⋆ Ω (Z) (resp. M ⋆ Ω (Z)) and s i ∈ G.
(b) The subset G is called a GS basis in kS Ω (Z) (resp. kM Ω (Z)) with respect to ≤ if, for all pairs f, g ∈ G monicized with respect to ≤, every intersection composition of the form ( f, g) u,v p is trivial modulo (G, p), and every inclusion composition of the form ( f, g) q p is trivial modulo (G, p).
To distinguish from usual GS bases for associative algebras, from now on, we shall rename GS bases in multi-operated contexts by Ω-GS bases. We call Φ Ω-GS on kM Ω (Z) with respect to ≤ if S Φ (M Ω (Z)) is an Ω-GS basis in kM Ω (Z) with respect to ≤.
2.3.
Multi-operated GS basis for free Φ-algebras over algebras.
In this subsection, we consider multi-operated GS basis for free Φ-algebras over algebras and generalise the main result of [17] to multi-operated cases.
We will use the following results without proof as they are counterparts in multi-operated setup of [17,Propositions 4.8]. As in [17], we consider the following question:
Question 2.11. Let A be a (unital) algebra together with a Gröbner-Shirshov basis G. Assume that a set Φ of operated polynomial identities is Ω-GS in the sense of Definition 2.9. Considering the free (unital) Φ-algebra B over A, when will the union "Φ ∪ G" be a Ω-GS basis for B?
It is surprising that the answer of the corresponding question given in [17] can be generalised to multi-operated case without much modifications.
Theorem 2.12. Let X be a set and Φ ⊆ kM Ω (X) a system of OPIs. Let A = kM(Z)/I A be a unital algebra with generating set Z. Assume that Φ is Ω-GS on Z with respect to a monomial order ≤ in M Ω (Z) and that G is a GS basis of I A in kM(Z) with respect to the restriction of ≤ to M(Z).
Suppose that the leading monomial of any OPI φ(x 1 , . . . , x n ) ∈ Φ has no subword in M(X)\X, and that φ(u 1 , . . . , u n ) vanishes or its leading monomial is φ(u 1 , . . . , u n ) for all u 1 , . . . , u n ∈ M Ω (Z).
Then S Φ (M Ω (Z)) ∪ G is an Ω-GS basis of S Φ (M Ω (Z)) ∪ I A Ω-uAlg in kM Ω (Z) with respect to ≤.
Proof. The proof of [17, Theorem 5.9] carries verbatim over multi-operated case, because it reveals that the key point is that the leading monomial of any OPI φ(x 1 , . . . , x n ) ∈ Φ has no subword in M(X)\X.
For details, see the proof of [17,Theorem 5.9].
There exists a nonunital version of the above result, which is also a multi-operated version of [18, Theorem 2.15].
Theorem 2.13. Let X be a set and Φ ⊆ kS Ω (X) a system of OPIs. Let A = kS(Z)/I A be an algebra with generating set Z. Assume that Φ is Ω-GS on Z with respect to a monomial order ≤ in S Ω (Z) and that G is a GS basis of I A in kS(Z) with respect to the restriction of ≤ to S(Z).
Suppose that the leading monomial of any OPI φ(x 1 , . . . , x n ) ∈ Φ has no subword in S(X)\X, and that for all u 1 , . . . , u n ∈ S Ω (Z), φ(u 1 , . . . , u n ) vanishes or its leading monomial is φ(u 1 , . . . , u n ). Then S Φ (S Ω (Z)) ∪ G is an Ω-GS basis of S Φ (S Ω (Z)) ∪ I A Ω-Alg in kS Ω (Z) with respect to ≤.
Free differential Rota-Baxter algebras over algebras
In this section, we apply Theorems 2.12 and 2.13 to differential Rota-Baxter algebras. From now on, let Ω = {D, P}, fix a set X = {x, y} with two elements such that variables in OPIs will take values in X. When talking about algebras or reductions of OPIs, fix a set Z and we understand that variables in OPIs will be replaced by elements of S Ω (Z) or M Ω (Z).
We first recall the definition of differential Rota-Baxter algebras. We use D( ) and P( ) instead of the linear operators ⌊ ⌋ D and ⌊ ⌋ P . (a) A (unital) differential k-algebra of weight λ (also called a (unital) λ-differential k-algebra) is a (unital) associative k-algebra R together with a linear operator D : R → R such that
D(uv) = D(u)v + uD(v) + λD(u)D(v) for all u, v ∈ R;
when R has a unity 1, it is asked that D(1) = 0. (b) A Rota-Baxter k-algebra of weight λ is an associative k-algebra R together with a linear operator P : R → R such that
P(u)P(v) = P(uP(v)) + P(P(u)v) + λP(uv) for all u, v ∈ R.
(c) A (unital) differential Rota-Baxter k-algebra of weight λ (also called a (unital) λ-differential Rota-Baxter k-algebra) is a (unital) differential k-algebra (R, D) of weight λ and a Rota-Baxter operator P of weight λ such that
D • P = id .
When we consider free differential Rota-Baxter algebras on algebras, it is disappointing to see that the traditional order (see [2]) would not meet the condition of Theorems 2.12 and 2.13. This is the intention of the new monomial orders ≤ PD and ≤ uPD introduced in Section 1.3.
Case of nonunital algebras with λ 0.
Assume in this subsection that λ 0. Denote (1) φ 1 (x, y) = P(x)P(y) − P(xP(y)) − P(P(x)y) − λP(xy), D(P(x)) − x. We first consider nonunital free differential Rota-Baxter algebras on algebras. Proof. Let u 1 , · · · , u n and v 1 , · · · , v m be all the elements of Z occurring in u and v from left to right.
(2) φ 2 (x, y) = D(x)D(y) + λ −1 D(x)y + λ −1 xD(y) − λ −1 D(xy), (3) φ 3 (x) =
For
φ 1 (u, v) = P(u)P(v) − P(uP(v)) − P(P(u)v) − λP(uv)
, we have deg P (P(uv)) is smaller than those of the other three terms, while the deg D , deg Z and deg GD of the other elements are the same. And one can see
deg GP (P(u)P(v)) − deg GP (P(uP(v))) = m > 0, deg GP (P(u)P(v)) − deg GP (P(P(u)v)) = n > 0, so the leading monomial of φ 1 (u, v) is P(u)P(v).
The statements about φ 2 (u, v) and φ 3 (u) are obvious by comparing deg D .
Now let
Φ DRB ′ := {φ 1 (x, y), φ 2 (x, y), φ 3 (x)} .
However, Φ DRB ′ is not Ω-GS in kS Ω (Z) with respect to ≤ PD .
Example 3.3. For u, v ∈ S Ω (Z), let f = φ 2 (P(u), v) = D(P(u))D(v) + λ −1 D(P(u))v + λ −1 P(u)D(v) − λ −1 D(P(u)v), g = φ 3 (u) = D(P(u)) − u, q = ⋆D(v), p = D(P(u))D(v) =f = q|¯g . Then ( f, g) q p = f − q| g ≡ λ −1 (P(u)D(v) − D(P(u)v) + uv + λuD(v)). Let φ 4 (x, y) = P(x)D(y) − D(P(x)y) + xy + λxD(y)
. It is clear that the leading monomial of φ 4 (u, v) is P(u)D(v) with respect to ≤ PD which cannot be reduced further.
Example 3.4. For u, v ∈ S Ω (Z), let f = φ 2 (u, P(v)) = D(u)D(P(v)) + λ −1 D(u)P(v) + λ −1 uD(P(v)) − λ −1 D(uP(v)), g = φ 3 (v) = D(P(v)) − v, q = D(u)⋆, p = D(u)D(P(v)) =f = q|¯g . Then ( f, g) q p = f − q| g ≡ λ −1 (D(u)P(v) − D(uP(v)) + uv + λD(u)v).
Let φ 5 (x, y) = D(x)P(y) − D(xP(y)) + xy + λD(x)y. It is clear that the leading monomial of φ 5 (u, v) is D(u)P(v) with respect to ≤ PD which cannot be reduced further. Now denote Φ DRB to be the set of the following OPIs:
(1) φ 1 (x, y) = P(x)P(y) − P(xP(y)) − P(P(x)y) − λP(xy),
(2) φ 2 (x, y) = D(x)D(y) + λ −1 D(x)y + λ −1 xD(y) − λ −1 D(xy), (3) φ 3 (x) = D(P(x)) − x, (4) φ 4 (x, y) = P(x)D(y) − D(P(x)y) + xy + λxD(y), (5) φ 5 (x, y) = D(x)P(y) − D(xP(y)) + xy + λD(x)y. It is obvious that S Φ DRB ′ (Z) Ω-Alg = S Φ DRB (Z) Ω-Alg for each set Z.
Next we will show that Φ DRB is Ω-GS with respect to ≤ PD . Before that, we need the following lemma to simplify our proof. Proof. We write i ∧ j the composition of OPIs of φ i and φ j , which means φ i lies on the left and φ j lies on the right for intersection composition or φ j is included in φ i for inclusion composition. The ambiguities of all possible compositions in Φ DRB are listed as below: for arbitrary u, v, w ∈ S Ω (Z) and q ∈ S Ω ⋆ (Z),
1 ∧ 1 P(u)P(v)P(w), P q| P(u)P(v) P (w), P (u) P q| P(v)P(w) 1 ∧ 2 P q| D(u)D(v) P (w), P (u) P q| D(u)D(v) , 1 ∧ 3 P q| D(P(u)) P (v), P (u) P q| D(P(v)) , 1 ∧ 4 P(u)P(v)D(w), P q| P(u)D(v) P (w), P (u) P q| P(v)D(w) , 1 ∧ 5 P q| D(u)P(v) P (w), P (u) P q| D(v)P(w) , 2 ∧ 1 D q| P(u)(P(v) D (w), D (u) D q| P(v)P(w) , 2 ∧ 2 D(u)D(v)D(w) , D q| D(u)D(v) D (w), D (u) D q| D(v)D(w) ,
∧ 3 D(P(u))D(v), D(u)D(P(v)), D q| D(P(u)) D (v), D (u) D q| D(P(v)) , ∧ 4 D q| P(u)D(v) D (w), D (u) D q| P(v)D(w) , ∧ 5 D(u)D(v)P(w), D q| D(u)P(v) D (w), D (u) D q| D(v)P(w)
,
3 ∧ 1 D P q| P(u)P(v) , 3 ∧ 2 D P q| D(u)D(v) , 3 ∧ 3 D P q| D(P(u)) , 3 ∧ 4 D P q| P(u)D(v) , 3 ∧ 5 D P q| D(u)P(v) , 4 ∧ 1 P q| P(u)P(v) D (w), P (u) D q| P(v)P(w) , 4 ∧ 2 P(u)D(v)D(w), P q| D(u)D(v) D (w), P (u) D q| D(v)D(w) , 4 ∧ 3 P(u)D(P(v)), P q| D(P(u)) D (v), P (u) D q| D(P(v)) , 4 ∧ 4 P q| P(u)D(v) D (w), P (u) D q| P(v)D(w) , 4 ∧ 5 P(u)D(v)P(w), P q| D(u)P(v) D (w), P (u) D q| D(v)P(w) , 5 ∧ 1 D(u)P(v)P(w), D q| P(u)P(v) P (w), D (u) P q| P(v)P(w) , 5 ∧ 2 D q| D(u)D(v) P (w), D (u) P q| D(v)D(w) , 5 ∧ 3 D(P(u))P(v), D q| D(P(u)) P (v), D (u) P q| D(P(v)) , 5 ∧ 4 D(u)P(v)D(w), D q| P(u)D(v) P (w), D (u) P q| P(v)D(w) , 5 ∧ 5 D q| D(u)P(v) P (w), D (u) P q| D(v)P(w) .
Notice that all compositions above but the underlined ones can be dealt with by Lemma 3.5. There remains to consider the underlined compositions. We only give the complete proof for the case 5 ∧ 1, the other cases being similar. For the case 5 ∧ 1, write f = φ 5 (u, v), g = φ 1 (v, w) and p = D(u)P(v)P(w) . So we have
( f, g) P(w),D(u) p = −D(uP(v))P(w) + uvP(w) + λD(u)vP(w) + D(u)P(vP(w)) + D(u)P(P(v)w) + λD(u)P(vw) ≡ −D(uP(v)P(w)) + uP(v)w + λD(uP(v))w + uvP(w) + λD(u)vP(w) + D(uP(P(v)w)) − uP(v)w − λD(u)P(v)w + D(uP(vP(w))) − uvP(w) − λD(u)vP(w) + λD(uP(vw)) − λuvw − λ 2 D(u)vw ≡ −D(uP(v)P(w)) + D(uP(vP(w))) + D(uP(P(v)w)) + λD(uP(vw)) − λD(u)P(v)w + λD(uP(v))w − λuvw − λ 2 D(u)vw = −D(uφ 1 (v, w)) − λφ 5 (u, v)w ≡ 0 mod S Φ DRB (Z), p .
We are done.
Theorem 3.8. Let Z be a set, A = kS(Z)/I A a nonunital k-algebra. Then we have:
F Φ DRB -Alg Alg (A) = kS Ω (Z)/ S Φ DRB (Z) ∪ I A Ω-Alg .
Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex . Then S Φ DRB (Z) ∪ G is an operated GS basis of S
Φ DRB (Z) ∪ I A Ω-Alg in kS Ω (Z) with respect to ≤ PD .
Proof. Since the leading monomial in Φ DRB has no subword in S(X)\X, the result follows immediately from Theorem 3.7 and Theorem 2.13.
As a consequence, we obtain a linear basis. Theorem 3.9. Let Z be a set, A = kS(Z)/I A a nonunital k-algebra with a GS basis G with respect to ≤ dlex . Then the set Irr(S Φ DRB (Z) ∪ G) which is by definition the complement of
q|s, q| P(u)P(v) , q| D(u)D(v) , q| D(P(u)) , q| P(u)D(v) , q| D(u)P(v) , s ∈ G, q ∈ S ⋆ Ω (Z), u, v ∈ S Ω (Z) in S Ω (Z) is a linear basis of the free nonunital λ-differential Rota-Baxter algebra F Φ DRB -Alg Alg (A) over A.
Proof. It can be induced directly by Theorem 2.8.
Remark 3.10. Since the monomial order used in [2] does not satisfy the conditions of Theorem 2.13, we have to make use of a new monomial order while treating free differential Rota-Baxter algebras over an algebra. In fact, since the leading monomials are different, even for free differential Rota-Baxter algebras over a field, our monomial order will provide new operated GS basis and linear basis.
3.2.
Case of nonunital algebras with λ = 0. Now we consider nonunital free differential Rota-Baxter algebras on algebras with λ = 0. This case can be studied similarly to the case λ 0, so we omit the details in this subsection.
Denote φ 1 (x, y) with λ = 0 by φ 0 1 (x, y) . Let φ 0 2 (x, y) = D(x)y + xD(y) − D(xy). We also write φ 0 3 (x) = φ 3 (x) for convenience. Proposition 3.11. For any u, v ∈ S Ω (Z), the leading monomials of φ 0 1 (u, v), φ 0 2 (u, v) and φ 0 3 (u) with respect to ≤ PD are P(u)P(v), D(u)v and D(P(u)) respectively.
Let Φ 0 DRB ′ := φ 0 1 (x, y), φ 0 2 (x, y), φ 0 3 (x)
. By the following example, one can see that Φ 0 DRB ′ is not Ω-GS in kS Ω (Z) with respect to ≤ PD .
Example 3.12. For u, v ∈ S Ω (Z), let f = φ 0 2 (P(u), v) = D(P(u))v + P(u)D(v) − D(P(u)v), g = φ 0 3 (u) = D(P(u)) − u, q = ⋆v, p = D(P(u))v =f = q|¯g . Then ( f, g) q p = f − q| g ≡ P(u)D(v) − D(P(u)v) + uv. Let φ 0 4 (x, y) = P(x)D(y) − D(P(x)y) + xy.
It is clear that the leading monomial of φ 0 4 (u, v) with u, v ∈ S Ω (Z) is P(u)D(v) with respect to ≤ PD which cannot be reduced further. Now denote Φ 0 DRB to be the set of the following OPIs:
(1) φ 0 1 (x, y) = P(x)P(y) − P(xP(y)) − P(P(x)y),
(2) φ 0 2 (x, y) = D(x)y + xD(y) − D(xy), (3) φ 0 3 (x) = D(P(x)) − x, (4) φ 0 4 (x, y) = P(x)D(y) − D(P(x)y) + xy. It is obvious that S Φ 0 DRB ′ (Z) Ω-Alg = S Φ 0 DRB (Z) Ω-Alg
for arbitrary set Z.
Similar to the case λ 0, it can be proved that Φ 0 DRB is Ω-GS with respect to ≤ PD . Remark 3.13. Note that φ 0 4 (x, y) is just φ 4 (x, y) with λ = 0, and for u, v ∈ S Ω (Z)
φ 0 2 (u, P(v)) = D(u)P(v) + uD(P(v)) − D(uP(v)) ≡ D(u)P(v) + uv − D(uP(v)), which is exactly φ 5 (u, v) with λ = 0. So φ 5 (x, y) (λ = 0) does not appear in Φ 0 DRB . Theorem 3.14. Φ 0 DRB is Ω-GS in kS Ω (Z) with respect to ≤ PD . Proof.
As in the proof of Theorem 3.7, we write i ∧ j the composition of OPIs of φ i and φ j . There are two kinds of ambiguities of all possible compositions in Φ 0 DRB . Since φ 0 1 (x, y), φ 0 3 (x), and φ 0 4 (x, y) have the same leading monomials as in the case λ 0, the corresponding ambiguities i ∧ j with i, j ∈ {1, 3, 4} are the same in the proof of Theorem 3.7. Since φ 0 2 (x, y) has a different leading monomial, the ambiguities of the case i ∧ j with i = 2 or j = 2 are the following: for arbitrary u, v, w ∈ S Ω (Z), q ∈ S Ω ⋆ (Z) and s ∈ S Ω (Z) or ∅,
1 ∧ 2 P q| D(u)v P (w), P (u) P q| D(v)w ; 2 ∧ 1 D q| P(u)P(v) w, D (u) q| P(u)P(v) ; 2 ∧ 2 D(u)sD(v)w, D q| D(u)v w, D (u) q| D(v)w ; 2 ∧ 3 D (P(u)) v, D q| D(P(u)) v, D (u) q| D(P(v)) ; 2 ∧ 4 D(u)sP(v)D(w), D q| P(u)D(v) w, D (u) q| P(v)D(w) ; 3 ∧ 2 D P q| D(u)v ; 4 ∧ 2 P(u)D(v)w, P q| D(u)v D (w), P (u) D q| D(v)w .
Almost all the cases can be treated similarly as in the proof of Theorem 3.7, except a slight difference in the case 2 ∧ 2. In fact, let f = φ 0 2 (u, sD(v)), g = φ 0 2 (v, w) and
p = D(u)sD(v)w. So we have ( f, g) D(u)s,D(w) p = uD(sD(v))w − D(usD(v))w − D(u)svD(w) + D(u)sD(vw) ≡ −usD(v)D(w) + uD(sD(v)w) + usD(v)D(w) − D(usD(v)w) + uD(svD(w)) − D(usvD(w)) − uD(sD(vw)) + D(usD(vw)) = uD(sφ 0 2 (v, w)) − D(usφ 0 2 (v, w)) ≡ 0 mod S Φ 0 DRB (Z), p . We are done.
Theorem 3.15. Let Z be a set and A = kS(Z)/I A a nonunital k-algebra. Then we have:
F Φ 0 DRB -Alg Alg (A) = kS Ω (Z)/ S Φ 0 DRB (Z) ∪ I A Ω-Alg .
Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex .
Then S Φ 0 DRB (Z) ∪ G is an operated GS basis of S Φ 0 DRB (Z) ∪ I A Ω-Alg in kS Ω (Z) with respect to ≤ PD .
As a consequence, we obtain a linear basis.
Theorem 3.16. Let Z be a set and A = kS(Z)/I A a nonunital k-algebra with a GS basis G with respect to ≤ dlex . Then the set Irr(S Φ 0 DRB (Z) ∪ G) which is by definition the complement of q|s, q| P(u)P(v) , q| D(u)
v , q| D(P(u)) , q| P(u)D(v) , s ∈ G, q ∈ S ⋆ Ω (Z), u, v ∈ S Ω (Z) in S Ω (Z) is a linear basis of the free nonunital 0-differential Rota-Baxter algebra F Φ 0 DRB -Alg Alg (A) over A.
Case of unital algebras.
Now we consider unital differential Rota-Baxter algebras. Since the proofs are similar to those in the previous subsections, we omit most of them. The study still divided into cases of λ 0 and λ = 0.
When λ 0, since unital differential Rota-Baxter algebras have the condition D(1) = 0, put Φ uDRB to be the union of Φ DRB with {D(1)}, but by abuse of notation, in Φ DRB , x, y take their values in M Ω (Z) instead of S Ω (Z).
Remark 3.17. We have:
φ 2 (u, v) ≡ 0 when u = 1 or v = 1; φ 4 (u, v) ≡ −D(P(u)) + u = −φ 3 (u) when v = 1; φ 5 (u, v) ≡ −D(P(v)) + v = −φ 3 (v) when u = 1.
So adding of the unity 1 will not produce new OPIs. Moreover, it is clear that except the above cases, the leading monomials of OPIs in Φ DRB are the same with respect to ≤ PD and ≤ uPD by Proposition 1.20.
With similar proofs as in Subsection 3.1, we can prove the following results.
F Φ uDRB -uAlg uAlg (A) = kM Ω (Z)/ S Φ uDRB (Z) ∪ I A Ω-uAlg .
Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex . Then S Φ uDRB (Z) ∪ G is an operated GS basis of S Φ uDRB (Z) ∪ I A Ω-uAlg in kM Ω (Z) with respect to ≤ uPD . Theorem 3.20. Let Z be a set and A = kM(Z)/I A a unital k-algebra with a GS basis G with respect to ≤ dlex . Then the set Irr(S Φ uDRB (Z) ∪ G) which is by definition the complement of
q|s, q| P(u)P(v) , q| D(u)D(v) , q| D(P(u)) , q| P(u)D(v) , q| D(u)P(v) , q| D(1) , s ∈ G, q ∈ M ⋆ Ω (Z), u, v ∈ M Ω (Z) in M Ω (Z) is a linear basis of the free unital λ-differential Rota-Baxter algebra F Φ uDRB -uAlg uAlg (A) over A.
When λ = 0, denote Φ 0 uDRB := Φ 0 DRB (again by abuse of notation, Φ 0 DRB is understood that u, v take their values in M Ω (X) instead of S Ω (X)). By using similar proofs in Subsection 3.2, one can show the following results.
Theorem 3.22. Φ 0 uDRB is Ω-GS in kM Ω (Z) with ≤ uPD . Theorem 3.23. Let Z be a set and A = kM(Z)/I A a unital k-algebra. Then we have:
F Φ 0 uDRB -uAlg uAlg (A) = kM Ω (Z)/ S Φ 0 uDRB (Z) ∪ I A Ω-uAlg .
Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex .
Then S Φ 0 uDRB (Z) ∪ G is an operated GS basis of S Φ 0 uDRB (Z) ∪ I A Ω-uAlg in kM Ω (Z) with respect to ≤ uPD .
Theorem 3.24. Let Z be a set and A = kM(Z)/I A a unital k-algebra with a GS basis G with respect to ≤ dlex . Then the set Irr(
S Φ 0 uDRB (Z) ∪ G) which is by definition the complement of q|s, q| P(u)P(v) , q| D(u)v , q| D(P(u)) , q| P(u)D(v) , s ∈ G, q ∈ M ⋆ Ω (Z), u, v ∈ M Ω (Z) in M Ω (Z) is a linear basis of the free unital 0-differential Rota-Baxter algebra F Φ 0 uDRB -uAlg uAlg (A) over A.
So far, we have completed the study of differential Rota-Baxter algebras.
Free integro-differential algebras over algebras
In this section, we carry the study of GS bases of free integro-differential algebras over algebras. It reveals that integro-differential algebras can be investigated by using a method similar to differential Rota-Baxter algebras, but the details are more difficult.
We first recall the definition of integro-differential algebras.
Definition 4.1. Let λ ∈ k. An integro-differential k-algebra of weight λ (also called a λ-integrodifferential k-algebra) is a differential k-algebra (R, d) of weight λ with a linear operator P : R → R which satisfies (c) in Definition 3.1:
D • P = id , and such that P(D(u)P(v)) = uP(v) − P(uv) − λP(D(u)v) for all u, v ∈ R, P(P(u)D(v)) = P(u)v − P(uv) − λP(uD(v)) for all u, v ∈ R.
Recall that
(1) φ 1 (x, y) = P(x)P(y) − P(xP(y)) − P(P(x)y) − λP(xy), (2) φ 2 (x, y) = D(x)D(y) + λ −1 D(x)y + λ −1 xD(y) − λ −1 D(xy), (3) φ 3 (x) = D(P(x)) − x,(4)
φ 4 (x, y) = P(x)D(y) − D(P(x)y) + xy + λxD(y), (5) φ 5 (x, y) = D(x)P(y) − D(xP(y)) + xy + λD(x)y, and denote (6) φ 6 (x, y) = P(D(x)P(y)) − xP(y) + P(xy) + λP(D(x)y), (7) φ 7 (x, y) = P(P(x)D(y)) − P(x)y + P(xy) + λP(xD(y)). Notice that for u, v ∈ S Ω (Z), since P(D(u)P(v)) (resp. P(P(u)D(v))) has the largest P-degree in φ 6 (u, v) (resp. φ 7 (u, v)), the leading monomial of φ 6 (u, v) (resp. φ 7 (u, v)) with respect to ≤ PD is P(D(u)P(v)) (resp. P(P(u)D(v))).
4.1.
Case of nonunital algebras with λ 0.
Assume in this subsection that λ 0. We first consider nonunital free integro-differential k-algebras over algebras.
According to the definition of integro-differential algebras, define
Φ ID ′ := { φ 2 (x, y), φ 3 (x), φ 6 (x, y), φ 7 (x, y) } .
By Example 3.3, Example 3.4, Example 4.2 and Example 4.3, Φ ID ′ is not Ω-GS in kS Ω (X) with respect to ≤ PD .
Example 4.2. For u, v ∈ S Ω (Z), let f = φ 7 (u, v) = P(P(u)D(v)) − P(u)v + P(uv) + λP(uD(v)), g = φ 4 (u, v) = P(u)D(v) − D(P(u)v) + uv + λuD(v), q = P(⋆), p = P(P(u)D(v)) =f = q|¯g . Then ( f, g) q p = f − q| g ≡ −P(D(P(u)v)) + P(u)v. Let φ 8 (x, y) = P(D(P(x)y)) − P(x)y.
It is clear that the leading monomial of φ 8 (u, v) is P(D(P(u)v)) with respect to ≤ PD which cannot be reduced further.
Example 4.3. For u, v ∈ S Ω (Z), let f = φ 6 (u, v) = P(D(u)P(v)) − uP(v) + P(uv) + λP(D(u)v), g = φ 5 (u, v) = D(u)P(v) − D(uP(v)) + uv + λD(u)v, q = P(⋆), p = P(D(u)P(v)) =f = q|¯g .
Then
( f, g) q p = f − q| g ≡ −P(D(uP(v))) + uP(v). Let φ 9 (x, y) = P(D(xP(y))) − xP(y). It is clear that the leading monomial of φ 9 (u, v) is P(D(uP(v))) with respect to ≤ PD which cannot be reduced further.
Remark 4.4. Note that the OPI φ 1 (x, y) can be induced by φ 3 (x, y) and φ 6 (x, y). So an integrodifferential algebra can be seen as a differential Rota-Baxter algebra. Explicitly, for u, v ∈ S Ω (Z),
let f = φ 6 (P(u), v) = P(D(P(u))P(v)) − P(u)P(v) + P(P(u)v) + λP(D(P(u))v), g = φ 3 (u) = D(P(u)) − u, q = P(⋆P(v)), p = P(D(P(u))P(v)) =f = q|¯g . Then ( f, g) q p = f − q| g ≡ P(u)P(v) − P(uP(v)) − P(P(u)v) − λP(uv) = φ 1 (u, v).
Now denote Φ ID to be the set of the following OPIs:
(1) φ 1 (x, y) = P(x)P(y) − P(xP(y)) − P(P(x)y) − λP(xv),
(2) φ 2 (x, y) = D(x)D(y) + λ −1 D(x)y + λ −1 xD(y) − λ −1 D(xy), (3) φ 3 (x) = D(P(x)) − x,(4)
φ 4 (x, y) = P(x)D(y) − D(P(x)y) + xy + λxD(y), (5) φ 5 (x, y) = D(x)P(y) − D(xP(y)) + xy + λD(x)y, (8) φ 8 (x, y) = P(D(P(x)y)) − P(x)y, (9) φ 9 (x, y) = P(D(xP(y))) − xP(y). Notice that Φ ID = Φ DRB ∪ {φ 8 (x, y), φ 9 (x, y)}.
Proposition 4.5. S Φ ID ′ (Z) Ω-Alg = S Φ ID (Z) Ω-Alg for each set Z.
Proof. We firstly prove S Φ ID (Z) Ω-Alg ⊆ S Φ ID ′ (Z) Ω-Alg , which follows from
φ 1 (u, v) ∈ φ 3 (u, v), φ 6 (u, v) Ω-Alg by Remark 4.4, φ 4 (u, v) ∈ φ 2 (u, v), φ 3 (u) Ω-Alg by Example 3.3, φ 5 (u, v) ∈ φ 2 (u, v), φ 3 (u) Ω-Alg by Example 3.4, φ 8 (u, v) ∈ φ 4 (u, v), φ 7 (u, v) Ω-Alg by Example 4.2, φ 9 (u, v) ∈ φ 5 (u, v), φ 6 (u, v) Ω-Alg by Example 4.3, where u, v ∈ S Ω (Z). Next we show S Φ ID ′ (Z) Ω-Alg ⊆ S Φ ID (Z) Ω-Alg . Note that P(φ 4 (u, v)) = P(P(u)D(v)) − P(D(P(u)v)) + P(uv) + λP(uD(v)) = P(P(u)D(v)) − P(u)v + P(uv) + λuD(v) − P(D(P(u)v)) + P(u)v = φ 7 (u, v) − φ 8 (u, v), and P(φ 5 (u, v)) = P(D(u)P(v)) − P(D(uP(v))) + P(uv) + λP(D(u)v) = P(D(u)P(v)) − uP(v) + P(uv) + λD(u)v − P(D(uP(v))) + uP(v) = φ 6 (u, v) − φ 9 (u, v). So we have φ 6 (u, v) ∈ φ 5 (u, v), φ 9 (u, v) Ω-Alg , φ 7 (u, v) ∈ φ 4 (u, v), φ 8 (u, v) Ω-Alg . It proves S Φ ID ′ (Z) Ω-Alg ⊆ S Φ ID (Z) Ω-Alg .
We are done. Now we can prove Φ ID is Ω-GS.
Theorem 4.6. Φ ID is Ω-GS in kS Ω (Z) with respect to ≤ PD .
Proof. Since the ambiguities i ∧ j with i, j = 1, 2, 3, 4, 5 in Φ ID are the same as in Theorem 3.7, we only need to consider the ambiguities involving φ 8 and φ 9 . The cases that cannot be dealt with directly by Lemma 3.5 are listed below: for arbitrary u, v, w ∈ S Ω (Z), q ∈ S Ω ⋆ (Z) and s ∈ S Ω (Z) or ∅,
1 ∧ 8 P (D (P (u) v)) P (w), P (u) P (D (P (v) w)), 3 ∧ 8 D (P (D (P (u) v))), 4 ∧ 8 P D (P(u)v) D (w), 5 ∧ 8 D (u) P (D (P (v) w)),
1 ∧ 9 P D (uP(v)) P (w), P (u) P D (vP(w)) , 3 ∧ 9 D P D (uP(v)) , 4 ∧ 9 P D (uP(v)) D (w), 5 ∧ 9 D (u) P (D (vP (w))),
8 ∧ 1 P D (P(u)P(v)s) , 8 ∧ 4 P D (P(u)D(v)s) , 9 ∧ 1 P D (sP(u)P(v)) , 9 ∧ 5 P D (sD(u)P(v)) , 8 ∧ 8 P D P D (P(u)v) w , 8 ∧ 9 P D P D (uP(v)) w , 9 ∧ 8 P D uP D (P(v)w) ,
9 ∧ 9 P D uP D vP(w) . All these compositions can be treated similarly as in the proof of Theorem 3.7. We only give the complete proof for the case 8 ∧ 1. Take f = φ 8 (u, P(v)s), g = φ 1 (u, v), p = P(D(P(u)P(v)s)) and q = P(D(⋆s)). Then we have ( f, g) q p = −P(u)P(v)s + P(D(P(uP(v))s)) + P(D(P(P(u)v)s)) + λP(D(P(uv)s))
≡ −P(uP(v))s − P(P(u)v)s − λP(uv)s + P(uP(v))s + P(P(u)v)s + λP(uv)s ≡ 0 mod S Φ ID (Z), p .
We are done.
Theorem 4.7. Let Z be a set and A = kS(Z)/I A a nonunital k-algebra. Then we have:
F Φ ID -Alg Alg (A) = kS Ω (Z)/ S Φ ID (Z) ∪ I A Ω-Alg .
Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex .
Then S Φ ID (Z) ∪ G is an operated GS basis of S Φ ID (Z) ∪ I A Ω-Alg in kS Ω (Z) with respect to ≤ PD .
Proof. Since the leading monomial in Φ ID has no subword in S(X)\X, the result follows immediately from Theorem 4.6 and Theorem 2.13.
As a consequence, we obtain a linear basis . for arbitrary s ∈ G, q ∈ S ⋆ Ω (Z), u, v ∈ S Ω (Z). Proof. It can be induced directly from Theorem 2.8. Remark 4.9. Since the monomial order ≤ PD is different from that used in [7], our operated GS basis and linear basis are different from theirs. The reason is that the monomial order in [7] does not satisfy the condition of Theorem 2.13, thus cannot enable us to discuss free integro-differential algebras over algebras. A Φ IID -algebra is just a nonunital λ-integro-differential algebra with the operators P and D being the inverse operator of each other, so we call such an operated algebra an invertible integrodifferential algebra. One can show that Φ IID ∪ {φ 4 (x, y), φ 5 (x, y)} is Ω-GS in kS Ω (Z) with respect to ≤ PD .
Case of unital algebras.
Now we consider unital integro-differential algebras. Since the proofs are similar to those in the previous subsections, we omit most of them. The study still is divided into cases of λ 0 and λ = 0.
When λ 0, since unital integro-differential algebras have the condition D(1) = 0, we put Φ uID := Φ ID ∪ {D(1)}. Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex . Then S Φ uID (Z) ∪ G is an operated GS basis of S Φ uID (Z) ∪ I A Ω-uAlg in kM Ω (Z) with respect to ≤ uPD . Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex . Then S Φ 0 uID (Z) ∪ G is an operated GS basis of S Φ 0 uID (Z) ∪ I A Ω-uAlg in kM Ω (Z) with respect to ≤ uPD . Theorem 4.21. Let Z be a set and A = kM(Z)/I A a unital k-algebra with a GS basis G with respect to ≤ dlex . Then the set Irr(S Φ 0 uID (Z) ∪ G) which is by definition the complement of q|s, q| P(u)P(v) , q| D(u)v , q| D(P(u)) , q| P(u)D(v) , q| P(D(P(u)v)) , q| P(D(uP(v))) , s ∈ G, q ∈ M ⋆ Ω (Z), u, v ∈ M Ω (Z) in M Ω (Z) is a linear basis of the free unital 0-integro-differential algebra F Φ 0 uID -uAlg uAlg (A) over A.
Differential Rota-Baxter algebras vs integro-differential algebras.
Since integro-differential algebras have one more defining relation than differential Rota-Baxter algebras, by Proposition 2.10, the free integro-differential algebra over an algebra A is a quotient of the free differential Rota-Baxter algebra over A in general. However, by using the descriptions of Φ DRB and Φ ID and Theorems 3.7 and 4.6, we could also show that the former one is a differential Rota-Baxter algebra subalgebra of the later one.
Theorem 4.22. The free nonunital λ-integro-differential algebra F Φ ID -Alg Alg (A) over an algebra A is differential Rota-Baxter subalgebra of the free nonunital λ-differential Rota-Baxter algebra F Φ DRB -Alg Alg (A) over A.
Proof. We have the observation mentioned before Φ ID = Φ DRB ∪ {φ 8 (x, y), φ 9 (x, y)}.
That is to say, the operated Gröbner-Shirshov basis of the free nonunital λ-differential Rota-Baxter algebra F Φ DRB -Alg Alg (A) over an algebra A is a subset of that of the free nonunital λ-integrodifferential algebra F Φ ID -Alg Remark 4.23. Gao and Guo [7] also studied GS bases of the free integro-differential algebras and free differential Rota-Baxter algebra both generated by sets, and they deduced that the free integro-differential algebra generated by a set is a subalgebra of the free differential Rota-Baxter algebra generated by the same set. Theorem 4.22 proves an analogous fact for these free algebras generated by algebras. However, our method is completely different from theirs.
Remark 4.24. By using the descriptions of Φ 0 DRB and Φ 0 ID (resp. Φ uDRB and Φ uID , Φ 0 uDRB and Φ 0 uID ) and Theorems 3.14 and 4.13 (resp. Theorems 3.18 and 4.16, Theorems 3.22 and 4.19), we always have the same result in both unital and nonunital cases with any λ (zero or nonzero).
.
Multi-operated GS bases for Φ-algebras 11 2.3. Multi-operated GS basis for free Φ-algebras over algebras 12 3. Free differential Rota-Baxter algebras over algebras13 3.1. Case of nonunital algebras with λ 0 14 3.2. Case of nonunital algebras with λ = 0 17 3.3. Case of unital algebras 19 4. Free integro-differential algebras over algebras 20 4.1. Case of nonunital algebras with λ 0 21 Date: March 29, 2023. 2010 Mathematics Subject Classification. 16Z10 03C05 08B20 12H05 16S10 17B38 .
Definition 1 . 4 .
14Let Z be a set and ⋆ a symbol not in Z. (a) Define M ⋆ Ω (Z) to be the subset of M Ω (Z ∪ ⋆) consisting of elements with ⋆ occurring only once. (b) For q ∈ M ⋆ Ω (Z) and u ∈ M Ω (Z), we define q| u ∈ M Ω (Z) to be the element obtained by replacing the symbol ⋆ in q by u. In this case, we say u is a subword of q| u . (c) For q ∈ M ⋆ Ω (Z) and s = i c i u i ∈ kM Ω (Z) with c i ∈ k and u i ∈ M Ω (Z), we define q| s := i c i q| u i . (d) Define S ⋆ Ω (Z) to be the subset of S Ω (Z ∪ ⋆) consisting of elements with ⋆ occurring only once. It is easy to see S ⋆ Ω (Z) is a subset of M ⋆ Ω (Z), so we also have notations in (a)-(c) for S ⋆ Ω (Z) by restriction. Definition 1.5. Let Z be a set.
Lemma 1.16 ([23, Lemma 5.7]). A well order ≤ is a monomial order on S Ω (Z) if and only if ≤ is bracket compatible, left compatible and right compatible. Now we can prove the main result of this section which is the main technical point of this paper. Theorem 1.17. The well order ≤ PD is a monomial order on S Ω (Z).
Theorem 1 . 19 .
119The well order ≤ uPD is a monomial order on M Ω (Z).
Definition 2. 4 .
4Let Z be a set, ≤ a linear order on M Ω (Z) and f ∈ kM Ω (Z). (a) Let f k. The leading monomial of f , denoted byf , is the largest monomial appearing in f . The leading coefficient of f , denoted by c f , is the coefficient off in f . We call f monic with respect to ≤ if c f = 1. (a') Let f ∈ k (including the case f = 0). We define the leading monomial of f to be 1 and the leading coefficient of f to be c f = f . (b) A subset S ⊆ kM Ω (Z) is called monicized with respect to ≤, if each nonzero element of S has leading coefficient 1.
Theorem 2. 7 .
7(Composition-Diamond Lemma) Let Z be a set, ≤ a monomial order on M Ω (Z) and G ⊆ kM Ω (Z). Then the following conditions are equivalent:(a) G is an Ω-GS basis in kM Ω (Z).(b) Denote Irr(G) := M Ω (Z)\ q| s | s ∈ G, q ∈ M ⋆ Ω (Z) . As a k-space, kM Ω (Z) = kIrr(G) ⊕ G Ω-uAlg and Irr(G) is a k-basis of kM Ω (Z)/ G Ω-uAlg .Theorem 2.8. (Composition-Diamond Lemma) Let Z be a set, ≤ a monomial order on S Ω (Z) and G ⊆ kS Ω (Z). Then the following conditions are equivalent: (a) G is an Ω-GS basis in kS Ω (Z). (b) Denote Irr(G) := S Ω (Z)\ q| s | s ∈ G, q ∈ S ⋆ Ω (Z) . As a k-space, kS Ω (Z) = kIrr(G) ⊕ G Ω-Alg and Irr(G) is a k-basis of kS Ω (Z)/ G Ω-Alg . Definition 2.9 ([6, Definiton 2.21(a)]).(a) Let Φ ⊆ kS Ω (X) be a family of OPIs. Let Z be a set and ≤ a monomial order on S Ω (Z). We call Φ Ω-GS on kS Ω (Z) with respect to ≤ if S Φ (S Ω (Z)) is an Ω-GS basis in kS Ω (Z) with respect to ≤. (b) Let Φ ⊆ kM Ω (X) be a family of OPIs. Let Z be a set and ≤ a monomial order on M Ω (Z).
Proposition 2 .
210.(a) Let Φ ⊂ kS Ω (X) and A = kS(Z)/I A an algebra. ThenF Φ-Alg Alg (A) := kS Ω (Z)/ S Φ (S Ω (Z)) ∪ I A Ω-Alg is the free Φ-algebra generated by A. (b) Let Φ ⊂kM Ω (X) and A = kM(Z)/I A a unital algebra. Then
F
Φ-uAlg uAlg (A) := kM Ω (Z)/ S Φ (M Ω (Z)) ∪ I A Ω-uAlg is the free unital Φ-algebra over A.
Definition 3.1 ([7, Definition 2.1]). Let λ ∈ k be fixed.
Proposition 3 . 2 .
32For any u, v ∈ S Ω (Z), the leading monomials of φ 1 (u, v), φ 2 (u, v) and φ 3 (u) under ≤ PD are respectively P(u)P(v), D(u)D(v) and D(P(u)).
Lemma 3 . 5 .
35Let φ(x 1 , . . . , x n ) and ψ(y 1 , . . . , y m ) be two OPIs. Let Z be a set. Suppose that, for any u 1 , . . . , u n , v 1 , . . . , v m ∈ S Ω (Z), the leading monomial of φ(u 1 , . . . , u n ) isφ(u 1 , . . . , u n ) and leading monomial ofψ(v 1 , . . . , v m ) isψ(v 1 , . . . , v m ). Now write f = φ(u 1 , . . . , u n ) and g = ψ(v 1 , . . . , v m ) for fixed u 1 , . . . , u n , v 1 , . . . , v m ∈ S Ω (Z). If there exists i (1 ≤ i ≤ n) and r ∈ S Ω ⋆ (Z) such that u i = r|¯g,then the inclusion composition ( f, g) q p = f − q| g with p =f and q =φ(u 1 , . . . , u i−1 , r, u i+1 , . . . , u n ), is trivial modulo (S {φ,ψ} (Z), w). We call this type of inclusion composition as complete inclusion composition. Proof. The assertion follows from ( f, g) q p = f − q| g = (φ −φ)(u 1 , . . . , u i−1 , r|ḡ, u i+1 , . . . , u n ) −φ(u 1 , . . . , u i−1 , r| g−ḡ , u i+1 , . . . , u n ) = (φ −φ)(u 1 , . . . , u i−1 , r| g , u i+1 , . . . , u n ) − φ(u 1 , . . . , u i−1 , r| g−ḡ , u i+1 , . . . , u n ).
Remark 3. 6 .
6Lemma 3.5 extends [6, Theorem 4.1(b)] to the case of multiple operators. Now we can prove Φ DRB is Ω-GS.Theorem 3.7. Φ DRB is Ω-GS in kS Ω (Z) with respect to ≤ PD .
Theorem 3 .
318. Φ uDRB is Ω-GS in kM Ω (Z) with respect to ≤ uPD .Theorem 3.19. Let Z be a set and A = kM(Z)/I A a unital k-algebra. Then we have:
Remark 3 . 21 .
321In Φ 0 uDRB , we have φ 0 2 (1, 1) = D(1) + D(1) − D(1) = D(1), so it is not necessary to add D(1) into Φ 0 uDRB . Note that φ 0 4 (u, 1) ≡ −D(P(v)) + v = −φ 0 3 (v), so adding the unity 1 will not induce any new OPI.
Theorem 4. 8 .
8Let Z be a set and A = kS(Z)/I A a nonunital k-algebra with a GS basis G with respect to ≤ dlex . Then a linear basis of the free nonunital λ-integro-differential algebra F Φ DRB -AlgAlg (A) over A is given by the set Irr(S Φ ID (Z) ∪ G), which is by definition the complement in S Ω (Z) of the subset consisting of q| w where w runs through s, P(u)P(v), D(u)D(v), D(P(u)), P(u)D(v), D(u)P(v), P(D(P(u)v)), P(D(uP(v)))
Remark 4 . 10 .
410Define a new OPI φ 10 (x) = P(D(x)), and letΦ IID = { φ 1 (x, y), φ 2 (x, y), φ 3 (x), φ 10 (x) }.
Theorem 4 .
416. Φ uID is Ω-GS in kM Ω (Z) with respect to ≤ uPD . Theorem 4.17. Let Z be a set and A = kM(Z)/I A a unital k-algebra. Then we have:F Φ uID -uAlg uAlg (A) = kM Ω (Z)/ S Φ uID (Z) ∪ I A Ω-uAlg .
Theorem 4 . 18 .
418Let Z be a set, A = kM(Z)/I A a unital k-algebra with a GS basis G with respect to ≤ dlex . Then a linear basis of the free unital λ-integro-differential algebra F Φ uID -uAlg uAlg (A) over A is given by the set Irr(S Φ uID (Z) ∪ G), which is by definition the complement in M Ω (Z) of the subset consisting of q| w where w runs throughs, P(u)P(v), D(u)D(v), D(P(u)), P(u)D(v), D(u)P(v), P(D(P(u)v)), P(D(uP(v))), D(1)for arbitrary s ∈ G, q ∈ M ⋆ Ω (Z), u, v ∈ M Ω (Z). When λ = 0, denote Φ 0 uID := Φ 0 ID . Theorem 4.19. Φ 0 uID is Ω-GS in kM Ω (Z) with respect to ≤ uPD .Theorem 4.20. Let Z be a set and A = kM(Z)/I A a unital k-algebra. Then we have: F Φ 0 uID -uAlg uAlg (A) = kM Ω (Z)/ S Φ 0 uID (Z) ∪ I A Ω-uAlg .
Alg (A) over A. So by Diamond Lemma, F Φ ID -Alg Alg (A) is a subspace of F Φ DRB -Alg Alg (A). It is obvious that F Φ ID -Alg Alg (A) is also differential Rota-Baxter subalgebra of F Φ DRB -Alg Alg (A).
Acknowledgements:The authors were supported by NSFC(No. 11771085, 12071137)and by STCSM (22DZ2229014).Again, Φ 0 ID ′ is not Ω-GS in kS Ω (Z) with respect to ≤ PD .Remark 4.11. By Example 4.2, we can get φ 0 8 (u, v) from φ 0 4 (u, v) and φ 0 7 (u, v). One can not obtain φ 0However, we can still generate φ 0 9 (u, v) as follows:Now denote Φ 0 ID to be the set of the following OPIs:φ 0 9 (x, y) = P(D(xP(y))) − xP(y). As in the previous subsection, one can prove the following results. Theorem 4.13. Φ 0 ID is Ω-GS in kS Ω (Z) with respect to ≤ PD . Theorem 4.14. Let Z be a set and A = kS(Z)/I A a nonunital k-algebra. Then we have:Moreover, assume I A has a GS basis G with respect to the degree-lexicographical order ≤ dlex .Theorem 4.15. Let Z be a set and A = kS(Z)/I A a nonunital k-algebra with a GS basis G with respect to ≤ dlex . Then the set Irr(S Φ 0 ID (Z) ∪ G) which is by definition the complement of q|s, q| P(u)P(v) , q| D(u)v , q| D(P(u)) , q| P(u)D(v) , q| P(D(P(u)v)) , q| P(D(uP(v))) , s ∈ G, q ∈ S ⋆ Ω (Z), u, v ∈ S Ω (Z)in S Ω (Z) is a linear basis of the free nonunital 0-integro-differential algebra F Φ 0 ID -Alg Alg (A) over A.
consider nonunital free integro-differential algebras on algebras with λ = 0. This case can be studied similarly as the case λ 0, so we omit the details in this subsection. As in Subsection 3.2, for a OPI φ. Zuan Liu, Zihao Qi, Yufei Qin, Guodong Zhou, ZUAN LIU, ZIHAO QI, YUFEI QIN AND GUODONG ZHOU we consider nonunital free integro-differential algebras on algebras with λ = 0. This case can be studied similarly as the case λ 0, so we omit the details in this subsection. As in Subsection 3.2, for a OPI φ, we denote φ 0 for φ with λ = 0 and also write φ 0 = φ when λ
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| {'fraction_non_alphanumeric': 0.12553346369592214, 'fraction_numerical': 0.025634803956552148, 'mean_word_length': 3.0293305728088336, 'pattern_counts': {'":': 0, '<': 38, '<?xml version=': 0, '>': 20, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 111, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Quite much recent studies has been attracted to the operated algebra since it unifies various notions such as the differential algebra and the Rota-Baxter algebra. An Ω-operated algebra is a an (associative) algebra equipped with a set Ω of linear operators which might satisfy certain operator identities such as the Leibniz rule. A free Ω-operated algebra B can be generated on an algebra A similar to a free algebra generated on a set. If A has a Gröbner-Shirshov basis G and if the linear operators Ω satisfy a set Φ of operator identities, it is natural to ask when the union G ∪ Φ is a Gröbner-Shirshov basis of B. A previous work answers this question affirmatively under a mild condition, and thereby obtains a canonical linear basis of B.In this paper, we answer this question in the general case of multiple linear operators. As applications we get operated Gröbner-Shirshov bases for free differential Rota-Baxter algebras and free integro-differential algebras over algebras as well as their linear bases. One of the key technical difficulties is to introduce new monomial orders for the case of two operators, which might be of independent interest.Contents1 2 ZUAN LIU, ZIHAO QI, YUFEI QIN AND GUODONG ZHOU 4.2. Case of nonunital algebras with λ = 0 24 4.3. Case of unital algebras 25 4.4. Differential Rota-Baxter algebras vs integro-differential algebras 25 References 26', 'arxivid': '2302.14221', 'author': ['Zuan Liu ', 'Zihao Qi [email protected] ', 'Yufei ', 'Guodong Zhou [email protected] ', 'Zuan Liu ', 'Yufei Qin ', 'Guodong Zhou ', 'Zihao Qi ', '\nSchool of Mathematical Sciences\nSchool of Mathematical Sciences\nShanghai Key Laboratory of PMMP\nEast China Normal University\n200241ShanghaiChina\n', '\nFudan University\n200433ShanghaiChina\n'], 'authoraffiliation': ['School of Mathematical Sciences\nSchool of Mathematical Sciences\nShanghai Key Laboratory of PMMP\nEast China Normal University\n200241ShanghaiChina', 'Fudan University\n200433ShanghaiChina'], 'corpusid': 257232445, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 32259, 'n_tokens_neox': 28058, 'n_words': 14368, 'pdfsha': '45918484d2f98456b20b1706ad6842025ef47efa', 'pdfurls': ['https://export.arxiv.org/pdf/2302.14221v3.pdf'], 'title': [], 'venue': []} |
arxiv |
Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems
21 Feb 2023
Xuhui Meng
Institute of Interdisciplinary Research for Mathematics and Applied Science
School of Mathematics and Statistics
Huazhong University of Science and Technology
430074WuhanChina
Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems
21 Feb 2023Preprint submitted to Journal Name February 22, 2023* Corresponding authoruncertainty quantificationphysics-informed neural networksdeep operator networksgenerative adversarial networksnormalizing flowsdifferential equations
Physics-informed deep learning have recently emerged as an effective tool for leveraging both observational data and available physical laws. Physics-informed neural networks (PINNs) and deep operator networks (DeepONets) are two such models. The former encodes the physical laws via the automatic differentiation, while the latter learns the hidden physics from data. Generally, the noisy and limited observational data as well as the overparameterization in neural networks (NNs) result in uncertainty in predictions from deep learning models. In [1], a Bayesian framework based on the Generative Adversarial Networks (GAN) has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets. Specifically, the proposed approach in [1] has two stages: (1) prior learning, and (2) posterior estimation. At the first stage, the GANs are employed to learn a functional prior either from a prescribed function distribution, e.g., Gaussian process, or from historical data and available physics. At the second stage, the Hamiltonian Monte Carlo (HMC) method is utilized to estimate the posterior in the latent space of GANs. However, the vanilla HMC does not support the mini-batch training, which limits its applications in problems with big data. In the present work, we propose to use the normalizing flow (NF) models in the context of variational inference, which naturally enables the minibatch training, as the alternative to HMC for posterior estimation in the latent space of GANs. A series of numerical experiments, including a nonlinear differential equation problem and a 100-dimensional Darcy problem, are conducted to demonstrate that NF with full-/mini-batch training are able to achieve similar accuracy as the "gold rule" HMC. Moreover, the minibatch training of NF makes it a promising tool for quantifying uncertainty in solving high-dimensional partial differential equation (PDE) problems with big data.
Introduction
Physics-informed deep learning, capable of leveraging both data and physics, has emerged as an effective tool for diverse applications in the area of scientific computing, e.g., data fusion [2][3][4], solving forward and inverse differential equations [5][6][7][8][9][10]. Generally, the data noise associated with the measurement error in realworld applications and the overparametrization of NNs result in uncertainties in model predictions [11][12][13]. Quantifying the uncertainty propagation in deep learning approaches is crucial for critical applications involving physical and biological systems [12,13].
In the context of uncertainty quantification (UQ) in deep learning, the Bayesian neural network (BNN) has been one of the most popular and successful models for decades [14]. Recently, this approach has also been applied to solving forward and inverse ordinary/partial differential equation (ODE/PDE) problems [15], which is referred to as Bayesian physics-informed neural network (B-PINN) in [15]. It is shown that B-PINN is capable of quantifying both the aleatoric uncertainty associated with the data noise as well as the epistemic uncertainty associated with the overparameterization of NNs. In addition to the BNN, extensions of standard neural networks, such as deep ensembles [16,17] and dropout [18], has also been proposed to quantify uncertainties in deep learning. However, it is challenging for these models to achieve similar accuracy in terms of quality of the predicted uncertainties when comparing to BNNs [19,20]. Interested readers are directed to [19,20] for comprehensive comparisons among different UQ methods.
Despite the success of BNN, there are still limitations to be addressed. For instance, (1) the prior in BNN is specified for the weights/biases (i.e., hyperparameters), but the effect of the prior distribution for the hyperparamters on the prior in the functional space (i.e., output of BNNs) remains unclear, and (2) the number of dimensions for the hyperparameters in the BNN is generally very high, resulting in difficulties in posterior estimation. To address these issues, [1] proposed to directly learn priors in functional space from data using generative adversarial networks (GANs) or physics-informed GANs (PI-GANs) if we have physical laws or PDEs. More specifically, the method proposed in [1] has two stages, i.e., (1) train a GAN/PI-GAN given data and/or physics to learn the prior in functional space, and (2) estimate the posterior distribution in the latent space of GAN/PI-GAN using Hamiltonian Monte Carlo (HMC) [14]. Note that the latent space of GAN/PI-GAN is generally characterized as low dimensional, which alleviates the difficulties in posterior estimation in BNNs/B-PINNs. It is reported that the approach proposed in [1] is capable of learning flexible functional priors, e.g., both Gaussian and non-Gaussian process, and it can also predict reasonable uncertainty bounds in regression as well as PDE problems. In the present work, we refer to the learned functional prior using GAN/PI-GAN as neural functional prior (NFP) since the prior is represented by NNs once the GAN/PI-GAN is well-trained.
Although HMC is capable of providing accurate posterior estimation as reported in [1], the vanilla HMC does not support mini-batch training which restricts its applications in problems with big data. In addition to HMC, the variational inference (VI) is also a widely used approach for posterior estimation. In particular, the mean-field variational inference (MFVI), in which the true posterior is approximated by a Gaussian distribution, is one of the most commonly employed models [21]. The MFVI is a unified method supporting both the full-and mini-batch training since it is based on stochastic gradient descent optimization algorithms. It can naturally handle problems with big data. However, as reported in [19,20], it is challenging for MFVI to achieve similar accuracy as the "gold rule" HMC in terms of the quality of the predicted uncertainties due to the fact that true posterior can be highly non-Gaussian. It has also been reported that employments of richer posterior approximations do result in better performance in VI [22]. Recently, the NN-based generative models, i.e., normalizing flows (NF), have been proposed to define a wide range of probability distributions, e.g., Gaussian and non-Gaussian. Rezende et al. [22] utilized the NF to approximate the true posterior in VI, which clearly improves the accuracy compared to MFVI. Furthermore, the training of NF is also based on stochastic optimization as in the MFVI, indicating that it also enables the minibatch training. In this study, we propose to employ the NF rather than HMC to compute the posterior in the latent space of NFP, which is expected to be a unified approach to enable both full-and mini-batch training, and thus making it a promising tool for solving high-dimensional parametric partial differential equation (PDE) problems with big data.
We organize the rest of this paper as follows. In Sec. 2, we review the neural functional prior for learning the priors from data and/or physics in functional space, and we also introduce how to use NF to estimate the posterior in NFP. In Sec. 3, we present results for three numerical experiments including regression and differential equation problems. We then summarize the findings of this study in Sec. 4. Finally, a brief overview for DeepONets and the details for the employed NNs as well as training strategy are provided in Appendix A and Appendix B, respectively.
Methodology
Consider a nonlinear ODE/PDE problem for a certain physical system as follows:
F λ,ζ [u(x; ζ)] = f (x; ζ), x ∈ Ω, ζ ∈ Z, B λ,ζ [u(x; ζ)] = b(x; ζ), x ∈ ∂Ω(1)
where x is the D x -dimensional temporal-spatial coordinate, ζ is a random event in a probability space Z, F λ,ζ is a general differential operator, B λ,ζ is the boundary operator, Ω is the computational domain, ∂Ω is the temporal-spatial boundary, f is the forcing term, b represents the boundary/initial condition, and λ are the problem parameters.
Our particular interest in the present study is focused on the following two scenarios, i.e., Case (I) inverse or mixed ODE/PDE problems, where F are deterministic and specified. That is, ζ is fixed. u and f are partially known, B is either known or unknown, and λ are either partially known or unknown. The objective is to predict u, f , as well as λ in the entire domain Ω; and Case (II) operator learning problems, in which F are stochastic and unknown (The stochasticity of F λ,ζ arises from the stochasticity in λ(x; ζ) with ζ ∈ Z), B and λ can be either known or unknown. We would like to employ DeepONets to learn the operator mapping f to u if we have paired data on (f, u), or we can learn the operator mapping (f, b, λ) to u if we have paired data on (f /b/λ, u). For simplicity, we will use λ to represent all the inputs for DeepONets. In addition, we assume that we have two types of data in all problems, i.e., historical data D for prior learning, and testing data D for posterior estimation, similar as in [1,20,23].
Prior learning in function space
In this section, we introduce how to learn the functional prior using GANs given historical data D for solving problems of interest in Equation (1).
For Case (I), we employ the generator of GANs as the surrogate for u, which takes as inputs the coordinates x and the samples ξ from a prescribed distribution P (ξ), e.g., uniform or Gaussian distribution. We refer to ξ as input noise in the present study. Also, the standard multivariate Gaussian distribution is utilized for P (ξ) in this work. As shown in Fig. 1, we denote the generator as G θ , where θ represents the parameters in NNs, and the output of the generator is represented byũ(x, ξ). For Figure 1: Schematic of the neural functional prior. G θ is the generator in GANs parametized by θ, which takes x (spatial-temporal coordinates) and ξ (input noise in the latent space) as inputs; φ(x, ξ) and ψ(x, ξ) can be the solution u and the source f in Equation (1), or vice-versa; M phy represents the available physics, which is either the differential operators in a certain ODE/PDE implemented by the automatic differentiation, or a pretrained DeepONet that encodes the hidden physics. Finally,λ(x, ξ) are either unknown or partially known problem parameters in inverse problems. Adapted from [20,23].
problems with multiple fields, we can either use multiple generators with one output each or a generator with multiple outputs as the corresponding surrogates. With specified F, we can then obtain the surrogate for f , i.e.,f (x, ξ) = F[ũ(x, ξ)], using the automatic differentiation as in PINNs [5]. We note that we can get the surrogate for b in the similar way as obtainingf if B is specified. For the inverse problems we consider here, we also need surrogates for the problem parameters, i.e., λ. In the current study, we employ another generator with the outputλ(x; ξ) to approximate λ.
We now discuss the historical data for training GANs. On the one hand, we have a certain number of samples from a hidden distribution P r , e.g.,
D = D u i ∪ D f i ∪ D b i ∪ D λ i N i=1
, where N is the number of samples for u/f /b/λ. With the generatorsũ(x; ξ),f (x; ξ), b(x; ξ), andλ(x; ξ), we can then train G θ (x, ξ) to generate "fake" samples that match samples from the hidden distribution P r . In particular, we employ a certain numbers of discrete points to resolve each sample in D. Details will be presented in each test case of Sec. 3. In addition, we denote the discriminator neural network in GANs by D η , which takes a real or fake sample as input, and outputs a real number. The loss function for optimizing the generator parameters θ and the discriminator parameters η are as follows:
L G = −E ξ∼P (ξ) [D η (G θ (x, ξ))], L D = E ξ∼P (ξ) [D η (G θ (x, ξ))] − E T ∼Pr [D η (T )] + ωET ∼P i ( ∇T D η (T ) 2 − 1) 2 .(2)
where P i is the distribution induced by uniform sampling on interpolation lines be-tween independent samples of T and G θ (x, ξ), and ω is the gradient penalty coefficient. Here we set ω = 0.1. During the training, we update η and θ iteratively with the ratio of 5 : 1 [1]. For Case (II), i.e., operator learning problems, we assume that we have paired data on λ and u, we can then train a DeepONet to approximate the operator mapping λ to u, which is denoted by u = S[λ](x). A brief overview of DeepONets is presented in Appendix A. Once the DeepONet is well-trained, it is capable of encoding the hidden physics represented by data. Afterwards, we employ the generatorλ(x, ξ) in GANs as the surrogate for λ, and perform the training using the historical data on λ or the same data for training the DeepONet, in a similar way as in Case (I). Upon the completion of training GANs, we can obtain the functional priors for both λ and u, which are denoted byλ(x, ξ) andũ(x, ξ) = S[λ(x, ξ)](x), respectively.
Posterior estimation using normalizing flows
Once the GANs are well-trained, we can then obtainũ(x, ξ), andf (x, ξ)/b(x, ξ)/λ η (x; ξ) represented by NNs to serve as the functional priors for new tasks in the future. We refer to the functional priors here as neural functional priors because we employ NNs to represent them. With the neural functional priors, the objective is to infer the posterior distribution in the latent space of GANs, i.e., ξ, given data on a new task, i.e., D. Specifically, the posterior for ξ can be expressed as follows based on the Bayes' rule:
P (ξ|D) = P (D|ξ)P (ξ) P (D) ,(3)
where P (ξ) is the prior distribution for ξ, which is Gaussian, i.e, P (ξ) = (2π) −d ξ /2 exp(− ξ 2 /2) (d ξ is the dimensionality of ξ), here. In addition, P (D|ξ) is the likelihood for the testing data on the new task, i.e., D = D u ∪ D f ∪ D b ∪ D λ . We can then write the likelihood as follows if we assume the measurement errors are Gaussian:
P (D|ξ) = P (D u |ξ)P (D f |ξ)P (D b |ξ)P (D λ |ξ), P (D α |ξ) = Nα i=1 1 2πσ (i) α 2 exp − (α(x (i) α ; ξ) − α (i) ) 2 2σ (i) α 2 , α = u, f, b, λ,(4)
where N α and x α are the number of discrete points and the corresponding coordinate for the field of u/f /b/λ, respectively; and σ α is the standard deviation for the noise of the corresponding filed. Finally, the marginal likelihood P (D) in Equation (3) is in general analytically intractable, we will thus compute P (ξ|D) numerically using variational inference with normalizing flows.
In the variational inference, the posterior density function for the unknown parameter ξ = (ξ 1 , ξ 2 ...ξ d ξ ), i.e., P (ξ|D), is approximated by another density function, which is from a normalizing flow model parameterized by ρ, as illustrated in Fig. 2. We denote the samples and the corresponding density function from NF as z n = G ρ (z) and Q ρ (z n ) = Q ρ (G ρ (z)), respectively. We can then tune ρ to minimize
D KL (Q ρ (z n )||P (ξ|D)) E zn∼Qρ(zn) [ln Q ρ (z n ) − ln P (z n ) − ln P (D|z n )] = 1 N z Nz j=1 [ln Q ρ (z n,j ) − ln P (z n,j ) − ln P (D|z n,j )],(5)
where D KL denotes the Kullback-Leibler (KL) divergence, and N z is the number of posterior samples for ξ used to compute the loss at each training step. Similar as in Equation (4), D represents all available measurements on u/f /b/λ. Further, the loss function can be written as follows if the minibatch training is used:
D KL (Q ρ (z n )||P (ξ|D)) 1 N z Nz j=1 [ln Q ρ (z n,j ) − ln P (z n,j ) − N α M α Mα i=1 ln P (D α i |z n,j )], α = u, f, b, λ,(6)
where N α and M α are the numbers of all measurements and the minibatch size for the field u/f /b/λ, respectively. The detailed algorithm for VI with normalizing flows is presented in Algorithm 1. Note that Various variants of NF [24][25][26][27] have been developed recently. Here we employ the inverse autoregressive flow model proposed in [26] due to its efficiency in generating posterior samples. The NF consists of n(n ≥ 1) blocks, and each block is a NN. z denotes the input of NF, which is a multivariate standard normal distribution. In addition, z has the same dimensionality as ξ. z n is the output of n th block; ρ n represents the hyperparameters in the NN of n th block, and G ρ (z) is the final output of NF, where ρ = (ρ 1 , ..., ρ n ) is the collection of hyperparameters in all blocks.
former represents the prediction of u(x) while the latter quantifies the uncertainty.
In the present work, we set M = 1, 000. Similarly for the other terms, e.g., f /b.
Results and discussion
In this section, we first employ the normalizing flows for posterior estimation in the neural functional prior using the example of a one-dimensional function approximation case. We then test two differential equation problems in conjunction with PINN as well as DeepONet. In each case, we demonstrate that NF supports the minibatch training in posterior estimation.
Pedagogical example: Function Approximation
We first consider to employ the NFP as well as the NF to quantify uncertainties in a one-dimensional regression problem. The target function is expressed as follows:
u = sin 3 (3x), x ∈ [−1, 1].(7)
We assume that we have 128 noisy measurements on this function which serves as the training data. Further, the training data are equidistantly distributed in
x ∈ [−0.8, −0.2] ∪ [0.2, 0.8].
The measurement error is assumed to be a Gaussian distribution with zero mean and 0.1 as the standard deviation. For this specific case, we assume that the historical data are from the following Gaussian process, i.e.,
u ∼ GP(0, K), K = exp − (x − x ) 2 2l 2 ,
x, x ∈ [−1, 1], l = 0.2.
We then randomly draw 10,000 samples from the aforementioned GP to train the GANs to obtain the prior in the functional space. In addition, 30 equidistant discrete points are employed to resolve each sample. The details for the architecture as well as the training of GANs are presented in Appendix B, which will not be presented here.
With the learned neural functional prior, we now employ the NF to estimate the posterior distribution of ξ in the neural functional prior given testing data on an unseen task. We first conduct the fullbatch training in NF, i.e., M = 128, and illustrate the results in Fig. 3(a). As shown, (1) the predicted uncertainty increases at the regions that do not have training data, and (2) the computational errors between the predicted mean and the reference solution are bounded by the predicted uncertainty. To further test the minibatch training in NF, we then train the NF with two different batch sizes, i.e., M = 32 and 64. The results are also illustrated in Fig. 3(a). It is observed that (1) both the predicted mean and uncertainty from NF with the two batch sizes are similar, and (2) the results in these two cases agree well with those in the first test case, i.e., NF with fullbatch training, suggesting the effectiveness of minibatch training in NF.
Finally, the results from the "gold rule", i.e., HMC, are utilized as the reference in Fig. 3(b). Note that the minibatch training is not supported in vanilla HMC, we therefore only conduct the HMC with fullbatch training in the present study. As shown, both the predicted mean and uncertainty of NF with full-and minibatch training are similar as those from HMC, suggesting that NF is able to achieve similar accuracy as HMC. The minibatch training in NF makes it a promising tool for quantifying uncertainty in problems with big data, which outperforms the HMC.
PINNs: 1D nonlinear diffusion-reaction problem
We now consider to employ the neural functional prior and the NF to quantify uncertainties in an inverse differential equation problem. Specifically, the PI-GAN is utilized to learn the functional prior from historical data as well as the equation, and the NF is used to obtain the posterior samples in the latent space of PI-GAN given testing data. Specially, we consider the same case as in [1], i.e., a nonlinear diffusion-reaction system, which is governed by:
D∂ 2 x u − k r u 3 = f, x ∈ [−1, 1], u(−1) = u(1) = 0,(9)
where u is the solute concentration, D = 0.01 is the diffusion coefficient, k r is the chemical reaction rate, and f is the source term. The exact solution for this system is assumed to be as follows:
u = (x 2 − 1) 4 i=1 [ω 2i−1 sin(iπx) + ω 2i cos(iπx)] ,(10)
where ω i are uniformly sampled from U([0, 1]), i = 1, 2, ..., 8. The chemical reaction rate is set as a nonlinear function of the solute concentration, i.e., k r = 0.4 exp(−u), and f can then be derived from Equation (9). Similar as in [1], we assume that we have 10,000 pairs of (k r , f ) as the historical data for learning the functional priors of k r and f . Measurements from 40 equidistant sensors are used to resolve each k r /f sample. For the inverse problem we consider here, we assume that we have partial noisy measurements on u and f for a new task, the objective is to infer k r with uncertainties given data on u and f . Specifically, we utilize 10 and 2 measurements for f and u as the training data at the posterior estimation stage, respectively. We note that the historical data as well as the setup, and the training data for the inverse problem are the same as in [1].
We first train a PI-GAN to obtain the neural functional priors for k r and f based on the historical data (Details for the architectures of PI-GAN can be found in Appendix B). With the neural functional priors, we then employ the NF with full-and mini-batch training to compute the posterior given noisy measurements on u/f in the new task. The results from full-and mini-batch training are illustrated in Figs. 4(a)-4(b), respectively. As shown, (1) the predictions for f , u, and k r are quite similar, (2) the predicted k r agree well with the reference solution even we do not have any measurements on it, which is attributed to the informative prior learned from historical data, and (3) the computational errors between the predicted means and the reference solutions for f , u and k r are bounded by the predicted uncertainties. Finally, we present the results from the HMC to serve as the reference solutions in Fig. 4(c). As observed, the results from NF with full-/mini-batch training are similar as those from HMC, which demonstrate that NF with full-/mini-batch training is able to achieve similar accuracy as HMC in posterior estimation.
DeepONet: 100-dimensional Darcy problem
We here employ the neural functional prior to quantify uncertainties in the predictions of DeepONet. Particularly, we utilize the test case in [20,23], which is a problem of two-dimensional steady flow through porous media. The governing equation for this problem is described by Darcy's law as follows:
∇ · (λ(x, y)∇u(x, y)) = f, 0 ≤ x, y ≤ 1,(11)
where x, y are the space coordinates, u(x, y) is the hydraulic head, f is a constant, i.e., f = 50, and λ(x, y) denotes the hydraulic conductivity field. The boundary conditions are expressed as follows:
u(x = 0, y; ξ) = 1, u(x = 1, y; ξ) = 0, (12a) ∂ n u(x, y = 0; ξ) = ∂ n u(x, y = 1; ξ) = 0, ∀ξ ∈ Ξ,
where n denotes the unit normal vector of the boundary. Generally, λ(x, y) is determined by the pore structure. We then employ a stochastic model for λ(x, y; ζ), to take in account of different porous media. In particular, λ(x, y; ζ) = exp(λ(x, y; ζ)), whereλ(x, y; ζ) is a truncated Karhunen-Loève expansion of a GP with zero mean and kernel given by In the following, we only keep the first 100 leading terms of the expansion and refer to the current problem as a 100-dimensional Darcy problem here.
kλ(x, y, x , y ) = exp(− (x − x ) 2 2l 2 − (y − y ) 2 2l 2 ),(13a)
To begin with, we assume that we have 9,900 different paired data (λ, u) as the historical data. For each sample ofλ/u, we utilize 20 × 20 uniform grid to resolve it. We then train a DeepONet to learn the mapping fromλ(x, y) to u using the paired data (λ, u). We further train a GAN to learn the functional prior forλ(x, y) based on the historical data onλ(x, y). We note that the historical data employed here are the same as in [20,23].
With the pretrained DeepONet and neural functional prior forλ, we assume that we have 20 and 10 noisy measurements for λ as well as u for an unseen task, which are displayed in Figs. 5-7. The objective is to reconstruct completeλ and u. Similarly, the measurements in the new task are the same as employed in [20,23]. We present the predictions forλ and u from NF with full-and mini-batch training in Figs. 5 and 6, respectively. It is observed that the (1) the results in these two cases show little discrepancy, and (2) the computational errors between the predicted means and the reference solutions forλ and u are mostly bounded by the predicted uncertainties in both cases. We also present the results from HMC for reference in Fig. 7. As shown, the NF with full-and mini-batch training are able achieve simialr accuracy as HMC in terms of the predicted mean and uncertainties.
Summary
In the present study, we utilize the generative adversarial networks (GANs) to learn the functional prior from historical data and available physics. In addition, two different scenarios for encoding the physics in GANs are considered, i.e., (1) the differential operators for defining the problems are known, we then encode the physical laws via automatic differentiation similar as in PINNs; and (2) the differential operators for defining the problems are unknown, we thus employ the DeepONet to learn the operators given data. We refer to the pre-trained GANs as neural functional prior. Further, we propose to employ the normalizing flows to compute the posterior in the latent space of neural functional prior in the context of variational inference as the alternative to the "gold rule" HMC. Specially, the NF is a unified framework for both full-and mini-batch training. We begin with a one-dimensional example to show that NF with full-and mini-batch training are able to achieve similar accuracy comparing to the "gold rule" HMC in posterior estimation. We further tested 1D and 2D differential equation problems using automatic differentiation and DeepONets to encode physics, respectively. In these two problems, we show that NF can provide accurate predicted means and reasonable uncertainty bounds, with relatively small number of sensors, which can be attributed to the informative functional priors that reflect our knowledge from historical data. Also, NF with full-and mini-batch training is capable of providing similar results as compared to HMC. The minibatch training in NF makes it a promising tool for quantifying uncertainties in high-dimensional parametric PDEs with big data. ., x m are the discrete points to resolve the input function, b 1 , b 2 , ..., b p and t 1 , t 2 , ..., t p are the outputs for the Branch Net and Trunk Net, respectively, u is the output target function. If the operator takes multiple functions as input (e.g., source term f , boundary condition b, etc.), then we just need to concatenate the multiple vectors that resolve these functions as the input of the Branch Net. Adapted from [1].
discrete function values at certain locations, i.e., x 1 , x 2 , ..., x m , and the output of BN is a vector [b 1 , b 2 , ..., b p ]. Further, TN takes x as input and outputs a vector [t 1 , t 2 , ..., t p ]. The output of the DeepONet is the inner product of these two vectors as u = p j=1 b j t j . The DeepONet is essentially a mapping between two function spaces, e.g., from the problem parameters λ to the solution of a PDE, i.e., u.
Similar as in [1], the uniform grids are used to discretize the input functions in the test cases of the present study. Also, we do not employ any constraint on the input x in the TN, we can thus evaluate the output u at any location. Both the BN and TN are trained simultaneously by minimizing the mean squared error (MSE) between the given and predicted u from DNNs using the Adam optimizer. More details on DeepONets can be found in [9,28].
Appendix B. Details for numerical computations
In this section, we present the details for data generation, architectures of employed NNs, as well as the training strategy.
For the training data, the historical data as well as the testing data used in Secs. 3.2 and 3.3 are the same as those in [1] and [20,23], respectively, which are available on github.com/Crunch-UQ4MI .
The architectures of the GANs for neural functional priors and NFs employed in the present work are displayed in Tables B.1 and B.2, respectively. Note that the architectures of GANs are the same as in [1]. We employ the Adam with a learning rate 10 −4 for both the training of neural functional priors as well as NFs. Furthermore, for the HMC, we employ the No-U-Turn [29], which can adaptively set path lengths in HMC in this study. In all cases, the initial step size is set as 1, the target acceptance rate is 0.6, and the number of burnin steps is 2,000. Particularly, the No-U-Turn is implemented using the Tensorflow Probability package [30].
G (g/h * ) D d
Algorithm 1
1Variational inference with normalizing flows Require: Pretrained neural functional prior. Require: An initialization for ρ.for k = 1, 2...N do Sample {z (j) } Nz j=1 independently from N (0, I d ξ ), Compute the loss, i.e., L(ρ), based on Equation(5) or Equation(6), Update ρ with gradient ∇ ρ L(ρ) using Adam optimizer.end for Sample {z (j) } M j=1 independently from N (0, I d ξ ), Obtain the posterior samples using {z (j) n } M j=1 = G ρ (z), Calculate {ũ(x, z (j) n )} M j=1 as posterior samples for u(x), similarly for other terms. With the posterior samples {ũ(x, z (j) n )} M j=1 for u in Algorithm 1, we can obtain the quantities of interest, i.e., the mean and standard deviation of {ũ(x, z
Figure 2 :
2Schematic of the normalizing flows.
Figure 3 :
31D function approximation with 128 training data: Predictions from (a) NF with full batch size (left most), batch size 64 (middle), and 32 (right most); and (b) HMC. 2σ: two standard deviations.
Figure 4 :
41D nonlinear diffusion-reaction problem: Predicted f , u and k r from (a) NF with full batch training, (b) NF with minibatch training in which M u = 1 and M f = 5, and (c) HMC.
Figure 5 :
5100D Darcy problem: Predictions from NF with full batch training. (a)λ. (b) u. Circle: testing data forλ, Cross: testing data for u.
Figure 6 :Figure 7 :
67100D Darcy problem: Predictions from NF with minibatch training, i.e., M f = 10, and M u = 5. (a)λ. (b) u. Circle: testing data forλ, Cross: testing data for u. 100D Darcy problem: Predictions from HMC. (a)λ. (b) u. Circle: testing data forλ, Cross: testing data for u.
Figure A. 8 :
8Schematic of DeepONet. λ is the input of Branch Net, x 1 , x 2 , .
ξ Training steps width × depth Activation width × depth Activation Sec. 3.1 64 × 2/64 × 2 tanh/tanh128 × 3
Leaky ReLu 10
500,000
Sec. 3.2 64 × 2/64 × 2 tanh/tanh
128 × 3
Leaky ReLu 40
500,000
Sec. 3.3 64 × 2/64 × 2 tanh/tanh
128 × 3
Leaky ReLu 100
500,000
Table B .
B1: Architecture and training steps of GANs in each case. The width and depth are for the hidden layers. d G is the dimension of ξ as well as the output dimension ofg andh * in G.Table B.2: Architecture and training steps of NF in each case. The width and depth are for the hidden layers in each block of NF.bolcks width × depth Activation Training steps
Secs. 3.1-3.3
4
256 × 2
tanh
200,000
Appendix A. Brief overview of DeepoNetsAs reported in[9], DeepONets can be used as a universal approximator to any continuous nonlinear operator. We present a schematic of DeepONets inFig. A.8. As shown, the DeepONet has two sub-networks, i.e., the Branch Net (BN) and the Trunk Net (TN). The input for BN is a function, which is represented by a set of
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A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data. L Lu, X Meng, S Cai, Z Mao, S Goswami, Z Zhang, G E Karniadakis, Computer Methods in Applied Mechanics and Engineering. 393114778L. Lu, X. Meng, S. Cai, Z. Mao, S. Goswami, Z. Zhang, G. E. Karniadakis, A comprehensive and fair comparison of two neural operators (with practical extensions) based on fair data, Computer Methods in Applied Mechanics and Engineering 393 (2022) 114778.
The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo. M D Hoffman, A Gelman, Journal of Machine Learning Research. 151M. D. Hoffman, A. Gelman, The No-U-Turn sampler: adaptively setting path lengths in Hamiltonian Monte Carlo, Journal of Machine Learning Research 15 (1) (2014) 1593-1623.
J Lao, C Suter, I Langmore, C Chimisov, A Saxena, P Sountsov, D Moore, R A Saurous, M D Hoffman, J V Dillon, tfp. mcmc: Modern Markov chain. J. Lao, C. Suter, I. Langmore, C. Chimisov, A. Saxena, P. Sountsov, D. Moore, R. A. Saurous, M. D. Hoffman, J. V. Dillon, tfp. mcmc: Modern Markov chain
Monte Carlo tools built for modern hardware. arXiv:2002.01184arXiv preprintMonte Carlo tools built for modern hardware, arXiv preprint arXiv:2002.01184.
| {'fraction_non_alphanumeric': 0.0591885883660615, 'fraction_numerical': 0.02227676917376806, 'mean_word_length': 4.323594674556213, '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': 'Physics-informed deep learning have recently emerged as an effective tool for leveraging both observational data and available physical laws. Physics-informed neural networks (PINNs) and deep operator networks (DeepONets) are two such models. The former encodes the physical laws via the automatic differentiation, while the latter learns the hidden physics from data. Generally, the noisy and limited observational data as well as the overparameterization in neural networks (NNs) result in uncertainty in predictions from deep learning models. In [1], a Bayesian framework based on the Generative Adversarial Networks (GAN) has been proposed as a unified model to quantify uncertainties in predictions of PINNs as well as DeepONets. Specifically, the proposed approach in [1] has two stages: (1) prior learning, and (2) posterior estimation. At the first stage, the GANs are employed to learn a functional prior either from a prescribed function distribution, e.g., Gaussian process, or from historical data and available physics. At the second stage, the Hamiltonian Monte Carlo (HMC) method is utilized to estimate the posterior in the latent space of GANs. However, the vanilla HMC does not support the mini-batch training, which limits its applications in problems with big data. In the present work, we propose to use the normalizing flow (NF) models in the context of variational inference, which naturally enables the minibatch training, as the alternative to HMC for posterior estimation in the latent space of GANs. A series of numerical experiments, including a nonlinear differential equation problem and a 100-dimensional Darcy problem, are conducted to demonstrate that NF with full-/mini-batch training are able to achieve similar accuracy as the "gold rule" HMC. Moreover, the minibatch training of NF makes it a promising tool for quantifying uncertainty in solving high-dimensional partial differential equation (PDE) problems with big data.', 'arxivid': '2302.10448', 'author': ['Xuhui Meng \nInstitute of Interdisciplinary Research for Mathematics and Applied Science\nSchool of Mathematics and Statistics\nHuazhong University of Science and Technology\n430074WuhanChina\n', 'Xuhui Meng \nInstitute of Interdisciplinary Research for Mathematics and Applied Science\nSchool of Mathematics and Statistics\nHuazhong University of Science and Technology\n430074WuhanChina\n'], 'authoraffiliation': ['Institute of Interdisciplinary Research for Mathematics and Applied Science\nSchool of Mathematics and Statistics\nHuazhong University of Science and Technology\n430074WuhanChina', 'Institute of Interdisciplinary Research for Mathematics and Applied Science\nSchool of Mathematics and Statistics\nHuazhong University of Science and Technology\n430074WuhanChina'], 'corpusid': 257050224, 'doi': '10.48550/arxiv.2302.10448', 'github_urls': [], 'n_tokens_mistral': 12713, 'n_tokens_neox': 11263, 'n_words': 7045, 'pdfsha': '33e9370b34073132f91eb8582bd3fe2aab8b031d', 'pdfurls': ['https://export.arxiv.org/pdf/2302.10448v1.pdf'], 'title': ['Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems', 'Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems', 'Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems', 'Variational inference in neural functional prior using normalizing flows: Application to differential equation and operator learning problems'], 'venue': []} |
arxiv |
MULTIPLIER IDEAL SHEAVES, COMPLEX SINGULARITY EXPONENTS, AND RESTRICTION FORMULA
25 Oct 2015
A N Qi'
Guan
MULTIPLIER IDEAL SHEAVES, COMPLEX SINGULARITY EXPONENTS, AND RESTRICTION FORMULA
25 Oct 2015
In this article, we obtain two sharp equality conditions in the restriction formula on complex singularity exponents: an equality between the codimension of the zero variety of related multiplier ideal sheaves and the relative codimension of the restriction of the variety on the submanifold (in the restriction formula); an equivalence between the transversality (between the variety and the submanifold) and the regularity of the restriction of the variety. As applications, we present sharp equality conditions in the fundamental subadditivity property on complex singularity exponents.
background, main results and applications
Complex singularity exponent (or log canonical threshold (lct) in algebraic geometry) is an important concept related to the multiplier ideal sheaves in complex geometry and complex algebraic geometry. Several important and fundamental related results have been established, e.g. the restriction formula and the subadditivity property on complex singularity exponents (see [11] [7]).
Recently, by giving lower bounds of the dimensions of the zero varieties of related multiplier ideal sheaves, sharp equality conditions in the restriction formula and the subadditivity property have been established [21]. In the present article, continuing the above work, we obtain two sharp equality conditions: an equality between the codimension of the zero variety and the codimension of the restriction of the variety on the submanifold (in the restriction formula); an equivalence between the transversality (between the zero variety and the submanifold) and the regularity of the restriction of the zero variety. As applications, we present sharp equality conditions in the fundamental subadditivity property.
Organization: We organize the present article as follows: in Section 1, we recall some related results of multiplier ideal sheaves and complex singularity exponents, and present the main results of the present article (Theorem 1.3, Theorem 1.4) and applications (Proposition 1.7 and Proposition 1.8); in Section 2, we recall some know results and present some preparatory results; in Section 3, we prove the main results and applications.
1.1. Multiplier ideal sheaves and complex singularity exponents. Let Ω be a domain in C n and o ∈ Ω. Let u be a plurisubharmonic function on Ω. Nadel [27] introduced the multiplier ideal sheaf I(u) which can be defined as the sheaf of germs of holomorphic functions f such that |f | 2 e −2u is locally integrable (background see e.g. [34,6,35,36,25,26,37,7]). Here u is regarded as the weight of I(u). As in [7], we denote the zero variety {z|I(u) z = O z , z ∈ Ω} of I(u) by V (I(u)).
Date: October 27, 2015. The author was partially supported by NSFC-11431013 and NSFC-11522101. 1 It is well-known that the multiplier ideal sheaf I(u) is coherent and integral closed, satisfies Nadel's vanishing theorem [27], and the strong openness property I(u) = ∪ ε>0 I((1 + ε)u) [17,18,19] i.e. the solution of Demailly's strong openness conjecture (the background and motivation of the conjecture could be referred to [6,7], the proof of dim ≤ 2 case could be referred to [22,23]).
The complex singularity exponent (see [38], see also [6,7], lct in algebraic geometry see [32,24]) is defined
c o (u) := sup{c ≥ 0 : exp (−2cu) is integrable near o}.
Berndtsson's solution ( [3]) of the openness conjecture posed by Demailly-Kollar, i.e. e −2co(u)u is not integrable near o, implies that {z|c z (u) ≤ c} = V (I(cu)) is an analytic set.
Let I ⊆ O o be a coherent ideal. The jumping number c I o (u) is defined (see e.g. [22,23]
) c I o (u) := sup{c ≥ 0 : |I| 2 exp (−2cu) is integrable near o}. Let F ⊆ O be a coherent ideal sheaf.
In [21], it has been presented that the strong openness property
I(u) = ∪ ε>0 I((1 + ε)u) implies that {z|c Fz z (u) ≤ p} = Supp(F /(F ∩ I(pu)))
is an analytic subset.
1.2.
Main results: Sharp equality conditions in the restriction formula on complex singularity exponents. Let u be a plurisubharmonic function near o ∈ C n . Let p : C n → C n−k satisfying p(z 1 , · · · , z n ) = (z k+1 , · · · , z n ), where (z 1 , · · · , z n ) and (z k+1 , · · · , z n ) are coordinates of C n and C n−k respectively. Let H = {z k+1 = · · · = z n = 0}. In [11], the following restriction formula (an "important monotonicity result" as in [11]) on complex singularity exponents has been presented by using Ohsawa-Takegoshi L 2 extension theorem.
Proposition 1.1. [11] c o (u| H ) ≤ c o (u) holds, where u| H ≡ −∞.
Let A = V (I(c o (u)u)). In [21], the following equality condition in Proposition 1.1 has been established
Theorem 1.2. [21] If c o (u| H ) = c o (u), then dim o A ≥ n − k.
For the case k = 1 and n = 2, Theorem 1.2 can be referred to [4] and [12]. For the case k = 1 and n > 2, Theorem 1.2 can be referred to [20] (a recent new proof by combining methods in [20] and [10] can be referred to [31]).
In the present article, combining recent results in [21] with some new ideas, we obtain the following sharp equality condition in Proposition 1.1 by giving the equality between n − dim o A and k − dim o (A ∩ H).
Theorem 1.3. If c o (u| H ) = c o (u), then dim o A = n − k + dim o (A ∩ H).
It is known that for the case k = 1 and any n, one can obtain the regularity of (A, o) (see [20] and [21]). Then it is natural to consider the regularity of (A, o) for general k. When n = 2, and u = log |z 2 − z 2 1 | (H = {z 2 = 0}), one can obtain that c o (u| H ) = 1/2 < 1 = c o (u). It is natural to consider the transversality between (A, o) and (H, o).
In the present article, we obtain the following sharp equality condition in Propo- (2) there exist coordinates (w 1 , · · · , w k , z k+1 , · · · , z n ) near o and l ∈ {1, · · · , k} such that (A, o) = (w 1 = · · · = w l = 0, o);
(3) there exist coordinates (w 1 , · · · , w k , z k+1 , · · · , z n ) near o and l ∈ {1, · · · , k} such that I(c o (u)u) o = (w 1 , · · · , w l ) o .
1.3. Applications: Sharp equality conditions in the fundamental subadditivity property of complex singularity exponents. Let u and v be plurisubharmonic functions near o ∈ C n . In [11,21], the fundamental subadditivity property of complex singularity exponents has been presented.
Theorem 1.5. [11, 21] c o (max{u, v}) ≤ c o (u) + c o (v). Let c = c o (u) + c o (v). Let A 1 = V (I(cu)
) and A 2 = V (I(cv)). In [21], the following sharp equality condition in Theorem 1.5 has been established.
Theorem 1.6. [21] If c o (max{u, v}) = c, then dim o A 1 + dim o A 2 ≥ n. Let B = {z|c z (u) + c z (v) ≤ c}, which is an analytic subset on A 1 ∩ A 2 (see subsection 3.3).
As an application of Theorem 1.3, we present the following more precise version of Theorem 1.6
Proposition 1.7. If c o (max{u, v}) = c, then dim o A 1 + dim o A 2 ≥ n + dim o B. Let n = 2, u = log |z 1 |, v = log |z 1 −z 2 2 |. As (|z 1 |+|z 2 2 |)/6 ≤ max{|z 1 |, |z 1 −z 2 2 |} ≤ 6(|z 1 | + |z 2 2 |), it is clear that c o (max{u, v}) = 1 + 1/2 < 2 = c, A 1 ∩ A 2 = {o}.
Then it is natural to consider the transversality between A 1 and A 2 .
As an application of Theorem 1.4, we present the following sharp equality condition in Theorem 1.5 by giving the regularity of A 1 and A 2 and the transversality between A 1 and A 2 .
Proposition 1.8. Assume that (A 1 , o) and (A 2 , o) are both irreducible such that (B, o) = (A 1 ∩ A 2 , o), which is regular. If c o (max{u, v}) = c, then both (A 1 , o) and (A 2 , o) are regular such that dim(T A1,o + T A2,o ) = n.
Some preparations
In this section, we recall some known results and present some preparatory results for the proofs of the main results and applications in the present article.
2.1. Ohsawa-Takegoshi L 2 extension theorem. We recall the famous Ohsawa-Takegoshi L 2 extension theorem as follows: [30], see also [28,34,1,5,35], etc.) Let D be a bounded pseudoconvex domain in C n . Let u be a plurisubharmonic function on D. Let H be an m-dimensional complex plane in C n . Then for any holomorphic function f on
Theorem 2.1. (H ∩ D satisfying H∩D |f | 2 e −2u dλ H < +∞,
there exists a holomorphic function F on D satisfying F | H∩D = f , and
D |F | 2 e −2u dλ n ≤ C D H∩D |f | 2 e −2u dλ H ,
where C D only depends on the diameter of D and m, and dλ H is the Lebesgue measure on H.
The optimal estimates of generalized versions of Theorem 2.1 could be referred to [14,15,16].
Following the symbols in Theorem 2.1, there is a local version of Theorem 2.1
Remark 2.2. (see [30], see also [11]) For any germ of holomorphic function f on o ∈ H ∩ D satisfying |f | 2 e −2u|H is locally integrable near o, there exists a germ of holomorphic function F on o ∈ D satisfying F | H∩D = f , and |F | 2 e −2u is locally integrable near o.
2.2.
Berndtsson's solution of the openness conjecture. Let (z 1 , · · · , z k ) be the coordinates of B k−l × B l ⊆ C k , and let p :
B k−l × B l → B k−l . Let H 1 := {z k−l+1 = · · · = z k = 0}.
In [21], by combining with Theorem 2.1, Berndtsson's solution of the openness conjecture [3] and Berndtsson's log-plurisubharmonicity of relative Bergman kernels [2], it has been presented
Remark 2.3. [21] Let u be a plurisubharmonic function on B k−l × B l ⊆ C k . Then there exists c ∈ (0, +∞], such that c za (u| La ) = c for almost every a = (a 1 , · · · , a k−l ) ∈ B k−l with respect to the Lebesgue measure on C k−l , where L a = {z 1 = a 1 , · · · , z k−l = a k−l } and z a ∈ L a ∩ H 1 .
We present a corollary of Remark 2.3 as follows (1) for any a = (a 1 , · · · , a k−l ) ∈ B k−l , A 3 ∩ L a = ∅, where L a = {z 1 = a 1 , · · · , z k−l = a k−l };
Corollary 2.4. Let u be a plurisubharmonic function on B k−l × B l . Assume that c z (u) ≤ 1 for any z ∈ H 1 and c o (u) = 1. Then for almost every a = (a 1 , · · · , a k−l ) ∈ B k−l with respect to the Lebesgue measure on C k−l , c za (u| La ) = 1 holds,
(2) there exists analytic subset A 4 ⊆ B k−l such that any z ∈ (A 3 ∩p −1 (B k−l \A 4 )) is the regular point in A 3 and the noncritical point of p| A3,reg .
Let u be a plurisubharmonic function on B k−l × B l . Assume that c z (u) ≤ 1 for any z ∈ A 3 and c o (u) = 1. Then for almost every a = (a 1 , · · · , a k−l ) ∈ B k−l with respect to the Lebesgue measure on C k−l , there exists z a ∈ A 3 ∩L a such that equality c za (u| La ) = 1.
Proof. By Remark 2.3, it follows that there exists c ∈ (0, ∞]. such that c z (u| L p(z) ) = c for almost every z ∈ (A 3 ∩p −1 (B k−l \A 4 )) with respect to the Lebesgue measure in (A 3 ∩ p −1 (B k−l \ A 4 )). By c o (u) = 1, it follows that c ≥ 1 (consider the integrability of e −2pu near o, where p < 1 near 1, and by contradiction).
By Proposition 1.1, it follows that 1 ≤ c ≤ c z (u| L p(z) ) ≤ c z (u) ≤ 1 holds, for almost every z ∈ (A 3 ∩p −1 (B k−l \A 4 )). Then one can find z 3 ∈ (A 3 ∩p −1 (B k−l \A 4 )) such that c z3 (u) = c z3 (u| L p(z 3 ) ) = 1.
By Corollary 2.4 (o ∼ z 3 ), it follows that for almost every a = (a 1 , · · · , a k−l ) ∈ (B k−l \ A 4 ) with respect to the Lebesgue measure on C k−l , there exists z a ∈ A 3 ∩L a such that equality c za (u| La ) = 1. As the Lebesgues measure of A 4 on C k−l is zero, then we obtain Remark 2.5.
2.3.
Hilbert's Nullstellensatz (complex situation). It is well-known that the complex situation of Hilbert's Nullstellensatz is as follows (see (4.22)
2.4.
A consequence of Demailly's strong openness conjecture. In [20], inspired by the proof of Demailly's equisingular approximation theorem (see Theorem 15.3 in [7]) and using the strong openness property I(u) = ∪ ε>0 I((1 + ε)u) [17,18,19], the following observation has been presented: We recall some direct consequences of Proposition 2.7 as follows Remark 2.8. ( [20], see also [21])ũ (as in Proposition 2.7) satisfies:
(1) for any z ∈ ({z|c z (u) ≤ 1}, o) = (V (I(u)), o), inequality c z (u) ≤ c z (ũ) ≤ 1 holds;
(2) if c z0 (u) = 1, then c z0 (ũ) = 1, where z 0 ∈ ({z|c z (u) ≤ 1}, o).
Let I ⊆ O o be a coherent ideal, and let u be a plurisubharmonic function near o. Using Proposition 2.7, we present the following result about the integrability of the ideals related to weight of jumping number one. After the present article has been written, Demailly kindly shared his manuscript [9] with the author addresses, which includes Proposition 2.9 (Lemma (4.2) in [9]). It suffices to prove that |I| 2 |J 0 | 2ε e −2 max{u,p0 log |J0|} is locally integrable near o for small enough ε > 0. We prove the above statement by contradiction: If not, then there exists ε 0 > 0, such that |I| 2 |J 0 | 2ε0 e −2 max{u,p0 log |J0|} is not locally integrable near o. Note that ε 0 log |J 0 | ≤ ε0 p0 max{u, p 0 log |J 0 |}, then it follows that |I| 2 e −2(1− ε 0 2.5. Product spaces. Let π i := Ω 1 × Ω 2 → Ω i i = 1, 2 be the projections, where Ω i ⊂ C n and containing the origin o ∈ C n for any i. Let ∆ be the diagonal of C n × C n . It is well-known that Let u and v be plurisubharmonic functions near o. In [11,21], the following statement has been presented, which was used to prove Theorem 1.5.
Proposition 2.12. [11,21]
c z×w (max{π * 1 (u), π * 2 (v)}) = c z (u) + c w (v). Let c = c o (u)+c o (v). Proposition 2.12 shows that c (o,o) (max{π * 1 (u), π * 2 (v)}) = c.
Proofs of main results and applications
In this section, we present the proofs of main results and applications.
Let l = k − dim o (A ∩ H). Let A 3 be a irreducible component of A ∩ H on B k−l × B l ⊂ H through o satisfying dim o A 3 = k − l.
By the parametrization of (A 3 , o) in H (see "Local parametrization theorem" (4.19) in [8]), it follows that one can find local coordinates (z 1 , · · · , z n ) of a neighborhood
U = B k−l × B l × B n−k of o satisfying H = {z k+1 = · · · = z n = 0} and dim(A ∩ U ) = dim o A such that (1) A 3 ∩ ((B k−l × B l ) ∩ H) is reduced and irreducible;
(2) for any a = (a 1 , · · · , a k−l ) ∈ B k−l , A 3 ∩ L a = ∅, where L a = {z 1 = a 1 , · · · , z k−l = a k−l };
(3) there exists analytic subset A 4 ⊆ B k−l such that any z ∈ (A 3 ∩ p −1 (B k−l \ A 4 )) is the regular point in A 3 and the noncritical point of p| A3,reg , where p : (z 1 , · · · , z k ) = (z 1 , · · · , z k−l ).
By (2), (3), c o (ũ) = c o (ũ| H ) = 1 (inequality 3.3), c z (ũ) ≤ 1 for any z ∈ A 3 (inequality 3.1), and Remark 2.5, it follows that for almost every a = (a 1 , · · · , a k−l ) ∈ B k−l with respect to the Lebesgue measure on C k−l , there exists z a ∈ A 3 ∩ L a such that equality c za (ũ| La ) = 1 (the set of a denoted by A ae ).
LetL a = {z 1 = a 1 , · · · , z k−l = a k−l }. By inequality 3.1 and Proposition 1.1, it follows that for any a ∈ A ae , 1 = c za (ũ| La ) ≤ c za (ũ|L a ) ≤ c za (ũ) = 1, which implies c za (ũ| La ) = c za (ũ|L a ) = 1. Using Theorem 1.2 (C n ∼L a , H ∼ H ∩L a = L a , o ∼ z a , u ∼ũ|L a ), one can obtain that for any a ∈ A ae , max
za∈p −1 (a) dim za {z|c z (ũ|L a ) ≤ 1} ≥ n − l − (k − l) = n − k. By the definition ofũ, it follows that (A ∩ U ) ∩L a ) ⊇ {c o (ũ|L a ) ≤ 1}, which implies dim((A∩U )∩L a ) ≥ max za∈p −1 (a) dim za {z|c z (ũ|L a ) ≤ 1}.
Then we obtain that the 2(n − k)-dimensional Hausdorff measure of dim((A ∩ U ) ∩L a ) is not zero for any a ∈ A ae .
Note that the 2(k − l)-dimensional Hausdorff measure of A ae is not zero, then it follows that the 2(n−k)+2(k−l) = 2(n−l)-dimensional Hausdorff measure of A near o is not zero (see Theorem 3.2.22 in [13]), which implies that In order to prove Theorem 1.4, by Theorem 1.3, it suffices to prove the following statement ((1) ⇒ (2)).
dim o A = dim(A∩U ) ≥ n − l. Note that l = k − dim o (A ∩ H) implies dim o A ≤ n − k + (k − l) = n − l,
Assume that (A ∩ H, o) is regular, and
k − dim o A ∩ H = n − dim o A. If c o (u) = c o (u| H )
, then there exist coordinates (w 1 , · · · , w k , z k+1 , · · · , z n ) near o and l ∈ {1, · · · , k}, such that (w 1 = · · · = w l = 0, o) = (A, o). H on (H, o)), it follows that there exist l ∈ {1, · · · , k} and holomorphic functions f 1 , · · · , f l near o ∈ H such that (a) df 1 | o , · · · , df l | o are linear independent; (b) ({f 1 = · · · = f l = 0}, o) = (A ∩ H, o) holds; (c) |f j | 2 e −2ũ|H are all locally integrable near o for j ∈ {1, · · · , l}.
Let J 0 = I(c o (u)u) o . By Remark 2.8 (u ∼ c o (u)u),
By Remark 2.2 and (c), it follows that there exist holomorphic functions F 1 , · · · , F l near o ∈ C n such that and |F j | 2 e −2ũ are integrable near o for any j ∈ {1, · · · , l}, which implies that {F 1 = · · · = F l = 0} ⊇ A. Combining F j = f j and (a), we obtain that dF 1 | o , · · · , dF l | o , dz k+1 | o , · · · , dz n | o are linear independent.
Note that {F 1 = · · · = F l = 0} is regular near o and n − dim o A = k − dim o A ∩ H = l, then it follows that {F 1 = · · · = F l = 0} = A near o. Choosing w j = F j for any j ∈ {1, · · · , l}, one can find holomorphic functions w l+1 , · · · , w k near o such that dw 1 | o , · · · , dw k | o , dz k+1 | o , · · · , dz n | o are linear independent. Then Theorem 1.4 has been proved.
3.3. Proof of Proposition 1.7. Let A 1 = V (I(cu)) and A 2 = V (I(cv)), and A = {(z, w)|c (z,w) (max{π * 1 (u), π * 2 (v)}) ≤ c}. By Proposition 1.5, it follows that , o), k ∼ n, n ∼ 2n), we obtain dim (o,o) A = dim (o,o) (A ∩ ∆) + n. By Proposition 2.12, it follows that
c (o,o) (max{π * 1 (u), π * 2 (v)}) = c o (u) + c o (v) = c = c o (max{u, v}) = c (z,w) (max{π * 1 (u), π * 2 (v)}| ∆ ). Using Theorem 1.3 (u ∼ max{π * 1 (u), π * 2 (v)}, H ∼ ∆, o ∼ (oA = {(z, w)|c z (u) + c w (v) ≤ c} ⊆ {(z, w)| max{c z (u), c w (v)} ≤ c} = A 1 × A 2 , which implies dim o A 1 + dim o A 2 = dim (o,o) (A 1 × A 2 ) ≥ dim (o,o) A. Note that B = {z|c z (u) + c z (v) ≤ c} is biholomophic to A ∩ ∆, then it follows that dim o A 1 + dim o A 2 ≥ dim (o,o) A = dim (o,
sition 1.1 by giving the equivalence between the transversality (between (A, o) and (H, o)) and the regularity of (A ∩ H, o).
Theorem 1 . 4 .
14If c o (u| H ) = c o (u), then the following statements are equivalent (1) (A ∩ H, o) is regular;
where L a and z a are as in Remark 2.3. Proof. By Remark 2.3 and c o (u) = 1, it follows that c ≥ 1 (consider the integrability of e −2pu near o, where p < 1 near 1, and by contradiction). By c z (u) ≤ 1 for any z ∈ H 1 and Proposition 1.1, it follows that c za (u| La ) ≤ c za (u) ≤ 1 for any z a ∈ L a ∩ H 1 . Combining c ≥ 1, we obtain Corollary 2.4. The following remark is the singular version of Corollary 2.4: Remark 2.5. Let A 3 be a reduced irreducible analytic subvariety on B k−l × B l through o satisfying dim o A 3 = k − l such that
. (see[8]) For every idealI ⊂ O o , J V (I),o = √ I, where √I is the radical of I, i.e. the set of germs f ∈ O o such that some power f k lies in I.
Proposition 2.7. ([20], see also[21]) Let D be a bounded domain in C n , and o ∈ D. Let u be a plurisubharmonic function on D. Let J ⊆ I(u) o . Then there exists p > 0 large enough, such that the plurisubharmonic functionũ = max{u, p log |J|} on a small enough neighborhood V o of o satisfying that e −2u − e −2ũ is integrable.
Proposition 2 . 9 .
29Let J ⊆ O o be a coherent ideal. Assume that c I o (u) = 1. If (V (I(u)), o) ⊆ (V (J), o), then |I| 2 |J| 2ε e −2u is locally integrable near o for any ε > 0.
Proof. (proof of Proposition 2.9) Let J 0 ⊆ O o be a coherent ideal satisfying (V (J 0 ), o) ⊇ (V (I(u)), o). By Theorem 2.6 (I ∼ I(u) o ), it follows that there exists large enough positive integer N such that J N 0 ⊆ I(u) o . By Proposition 2.7, it follows that exist p 0 > 0 large enough such that e −2u − e −2 max{u,p0 log |J0|} is locally integrable near o.
p 0 )
0max{u,p0 log |J0|} is not locally integrable near o. Note that u ≤ max{u, p 0 log |J 0 |}, then it follows that |I| 2 e −2(1− ε 0 p 0 )u is not locally integrable near o, which contradicts c I o (u) = 1. Then we prove Proposition 2.9. Let I = O o , ε = 1. Using Proposition 2.9, we obtain the following result.
Corollary 2 . 10 .
210Let J ⊆ O o be a coherent ideal. Assume that c o (u) = 1. Then the following two statements are equivalent (1) (V (J), o) ⊇ (V (I(u)), o); (2) |J| 2 e −2u is locally integrable near o, i.e. J ⊆ I(u) o .
Remark 2 . 11 .
211Let A 1 and A 2 be two varieties on Ω 1 and Ω 2 respectively through o. Assume that A 1 and A 2 are both regular at o. Then dim(T A1,o ∩ T A2,o ) = dim(T A1×A2,(o,o) ∩ T ∆,(o,o) ).
3. 1 .
1Proof of Theorem 1.3. By Remark 2.8 (u ∼ c o (u)u, J = I(c o (u)u) o ), it follows that c z (ũ) ≤ 1 (3.1) for any z ∈ (A, o) and c o (ũ) = 1 (⇐ c o (c o (u)u) = 1). By Proposition 1.1, it follows that c z (ũ| H ) ≤ 1 (3.2) for any z ∈ (A ∩ H, o). Using Proposition 1.1 and inequality 3.2, one can obtain that c o (ũ| H ) ≤ c o (ũ) ≤ 1. Combining with 1 = c o (u)/c o (u) = c o (u| H )/c o (u) = c o (c o (u)u| H ) ≤ c o (ũ| H ) (⇐ c o (u)u ≤ũ), one can obtain that c o (ũ| H ) = c o (ũ) = 1. (3.3)
Proof of Theorem 1.4. By Corollary 2.10, it follows that (2) ⇔ (3).
it follows that there exists p 0 > 0 large enough, such thatũ := max{c o (u)u, p 0 log |J 0 |} satisfies:(1) c o (ũ) = 1 (⇐ c o (c o (u)u) = 1); (2) ({z|c z (ũ) ≤ 1}, o) = (A, o). Byũ| H ≥ c o (u)u| H = c o (u| H )u| H , it follows that c o (ũ| H ) ≥ c o (c o (u)u| H ) = c o (c o (u| H )u| H ) = 1. Combining with the fact that c o (ũ| H ) ≤ c o (ũ) = 1, we obtain that c o (ũ| H ) = 1.(3.4) Note that c z (ũ| H ) ≤ c z (ũ) for any z ∈ A ∩ H, then by (2) (⇒ c z (ũ) ≤ 1) for any z ∈ A ∩ H, it follows that ({z|c z (ũ| H ) ≤ 1}, o) ⊇ (A ∩ H, o). Combining with the definition ofũ (⇒ ({z|c z (ũ| H ) < +∞}, o) ⊆ (V (J o ) ∩ H, o) = (A ∩ H, o)), we obtain (V (I(ũ| H )), o) = ({z|c z (ũ| H ) ≤ 1}, o) = (A ∩ H, o). (3.5) In the following part of the present section, we considerũ instead of u. By equality 3.5, it follows that (V (I(ũ| H )), o)(= (A∩H, o)) is regular. Combining with equality 3.4 and Corollary 2.10 (u ∼ũ|
o) (A ∩ ∆) + n = n + dim o B. Proposition 1.7 has thus been proved. 3.4. Proof of Proposition 1.8. Following the symbols in subsection 3.3, by Theorem 1.4(n ∼ 2n, k ∼ n, u ∼ max{π * 1 u, π * 2 v}, o ∼ (o, o) ∈ C n × C n , H ∼ ∆ the diagonal of C n × C n ), it follows that A is regular at ((o, o)) satisfying dim (o,o) A = dim (o,o) (A ∩ ∆) + n. As A 1 ∩ A 2 = B, it follows that (A 1 × A 2 ) ∩ ∆ = A ∩ ∆, which implies dim (o,o) A = dim (o,o) (A ∩ ∆) + n = dim (o,o) ((A 1 × A 2 ) ∩ ∆) + n. (3.6) Note that A 1 × A 2 ⊇ A and equality 3.6 holds, then it follows that dim (o,o) (A 1 × A 2 ) ≥ dim (o,o) A = dim (o,o) ((A 1 ×A 2 )∩∆)+n. As ∆ is regular, then it is clear that dim (o,o) (A 1 × A 2 ) ≤ dim (o,o) ((A 1 × A 2 ) ∩ ∆) + n, which implies dim (o,o) (A 1 × A 2 ) = dim (o,o) ((A 1 × A 2 ) ∩ ∆) + n = dim (o,o) A. Note that (A, (o, o)) is regular and A 1 × A 2 is irreducible at (o, o) (A 1 and A 2 are both irreducible at o), then we obtain A = A 1 × A 2 , which implies A 1 and A 2 are both regular. By the transversality between A 1 × A 2 = A and ∆ at (o, o) and Remark 2.11, it follows that 2n = dim(T A1×A2,(o,o) + T ∆,(o,o) ) = dim T A1×A2,(o,o) + dim T ∆,(o,o) − dim(T A1×A2,(o,o) ∩ T ∆,(o,o) ) = (dim T A1,o + dim T A2,o ) + n − dim(T A1,o ∩ T A2,o ) = dim(T A1,o + T A2,o ) + n. It is clear that dim(T A1,o + T A2,o ) = n, then we prove Proposition 1.8.
Acknowledgements. The author addresses would like to sincerely thank my advisor, Professor Xiangyu Zhou, for bringing me to the theory of multiplier ideal sheaves and for his valuable help to me in all ways.The author would also like to sincerely thank Professor Jean-Pierre Demailly for giving series of talks on related topics at CAS and PKU and sharing his related recent work, Professor Liyou Zhang for pointing out some typos, and Zhenqian Li for helpful comments.
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. Qi'an Guan, Beijing, 100871, ChinaSchool of Mathematical Sciences, and Beijing International Center for Mathematical Research, Peking UniversityE-mail address: [email protected]'an Guan: School of Mathematical Sciences, and Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China. E-mail address: [email protected]
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arxiv |
Faddeev-Jackiw Hamiltonian Reduction for Free and Gauged Rarita-Schwinger Theories
13 Sep 2016
Suat Dengiz
Center for Theoretical Physics
Massachusetts Institute of Technology
02139CambridgeMAUSA
Faddeev-Jackiw Hamiltonian Reduction for Free and Gauged Rarita-Schwinger Theories
13 Sep 2016(Dated: September 14, 2016)
We study the Faddeev-Jackiw symplectic Hamiltonian reduction for 3 + 1-dimensional free and Abelian gauged Rarita-Schwinger theories that comprise Grassmannian fermionic fields. We obtain the relevant fundamental brackets and find that they are in convenient forms for quantization. The brackets are independent of whether the theories contain mass or gauge fields, and the structure of constraints and symplectic potentials largely determine characteristic behaviors of the theories. We also note that, in contrast to the free massive theory, the Dirac field equations for free massless Rarita-Schwinger theory cannot be obtained in a covariant way.
I. INTRODUCTION
In 1941, Rarita and Schwinger constructed a theory of spin- 3 2 vector-spinor fields which has a local fermionic gauge-invariance [1]. However, this symmetry is lost when the vector-spinor field has mass or couples to the other lower spin fields. More precisely, in 1961, Johnson and Sudarshan studied massive Rarita-Schwinger field minimally coupled to an external electromagnetic field, and showed that the equal-time commutators and relativistic covariance of the theory are in conflict, which makes the quantization a rather subtle issue [2]. In 1969, Velo and Zwanziger found that the massive gauged extension of the theory also admits superluminal wave propagation. Thus, the causality principle is also violated in the theory [3]. Despite these persistent problems, the massless theory keeps its importance particularly in two aspects. First, the massless (Majorana) Rarita-Schwinger field plays a central role in the construction of covariantly interacting supergravity theory [4][5][6]. The theory describes a generalization of the Rarita-Schwinger fermionic gaugeinvariance and the vector-spinor fields are fermionic superpartner of gravitons, namely gravitinos of the supergravity. In this concept, Das and Freedman showed that the massless theory is free from the non-causal wave propagation and has a unitary propagator structure [7]. Secondly, the massless Rarita-Schwinger theory is valuable for the cancellation of SU(8) gauge anomalies. Unlike the generic anomaly cancellation mechanisms in which the anomalies are supposed to be canceled withing the lower spin fermionic fields, it was shown by Marcus [8] and later studied by Adler [9], that a complete SU (8) gauge theory can be constructed via Rarita-Schwinger fields. In this set-up, the vector-spinor field acquires a crucial role in canceling anomalies arising in the gauge theory. Thus, it is left to determine whether the gauged Rarita-Schwinger fields describe well-behaved, complete classical or quantum field theories. For this purpose, Adler has recently studied minimally gauged massless Rarita-Schwinger theories at both classical and quantum levels in detail [10]. He showed that, unlike the massive case, the massless gauged Rarita-Schwinger theory provides consistent classical and quantum theories with a generalized fermionic gauge-invariance.
Taking the above mentioned observations as inspiration points and noting the hard task of getting proper brackets of constrained systems providing viable quantization, we study the Faddeev-Jackiw (FJ) symplectic Hamiltonian reduction [11,12] for free and gauged Rarita-Schwinger theories. Unlike Dirac's approach for constrained systems [16], FJ symplectic first-order formalism does not require any classification of constraints 1 . In other words, the method avoids analyzing systems by evaluating all commutation relations among the constraints and classifying them accordingly. Apparently, the FJ approach supplies a rather economical way of quantizing constrained systems. In doing so, we find the fundamental brackets for the free and gauged Rarita-Schwinger theories for both massless and massive versions. Here, the brackets are in admissible structures to be quantized. We also observe that the brackets are identical for all kinds of the theories; the brackets are independent of whether the theory is massive or interacting with external electromagnetic field or not. The differences between the theories arise among the constraints they have. We also notice that, in contrast to the massive case, the Dirac field equations for free massless Rarita-Schwinger theory cannot be obtained in a covariant way.
The layout of the paper is as follows: In Sec. II, we recapitulate the fundamental properties of free massless Rarita-Schwinger theory and apply FJ Hamiltonian reduction to the theory. In Sec. III, we turn our attention to the FJ Hamiltonian reduction for free massive Rarita-Schwinger theory. Sec. IV and Sec. V are devoted to the first-order symplectic analysis for Abelian gauged extensions of massless and massive Rarita-Schwinger theories. In Sec. VI, we conclude our results. In the Appendix A, the derivation of the transverse and traceless decomposition of the fields in the free massless Rarita-Schwinger theory is given as a sample. In the Appendix B, we briefly review the FJ approach for constrained and unconstrained systems. We also give an example of the application of symplectic method to anti-commuting spin-1 2 Dirac theory.
II. FREE MASSLESS RARITA-SCHWINGER THEORY
The 3 + 1-dimensional free massless Rarita-Schwinger theory is described by the Lagrangian
L = −ǫ λµνρψ λ γ 5 γ µ ∂ ν ψ ρ ,(1)
where ψ µ andψ µ are vector-spinor fields with spinor indices suppressed. We work in the metric signature (+, −, −, −), γ 5 = iγ 0 γ 1 γ 2 γ 3 and {γ µ , γ ν } = 2η µν . We consider the fermionic fields as independent anti-commuting Grassmannian variables. Recall that, unlike the complex Dirac field, for the Grassmannian variables there is no such relation asψ µ = γ 0 ψ + µ . Instead, ψ µ andψ µ are independent generators in the Grassmann algebra. Thus, one can define the conjugation as follows:
ψ * µ =ψ ν (γ 0 ) ν µ , (ψ µ ) * = (γ 0 ) µ ν ψ ν .(2)
Notice that this does not mean that Eq.(2) produces a new element in the Grassmannian algebra. This is merely the conjugation of independent variables. Therefore, with the help of the conjugation of the Grassmannian variables (θ 1 θ 2 ) * = θ * 2 θ * 1 , one can show that the Lagrangian in Eq.(1) is selfadjoint up to a boundary term:
L * = L + ∂ ν (ǫ λµνρψ λ γ 5 γ µ ψ ρ ),(3)
such that the total derivative term naturally drops at the action level. Moreover, variations with respect to independent variables respectively yield
ǫ λµνρ γ 5 γ µ ∂ ν ψ ρ = 0, ǫ λµνρ ∂ νψλ γ 5 γ µ = 0,(4)
which are the corresponding field equations. From now on, we will work with the first of Eq.(4). But, by following the same steps, one could easily obtain the similar results for the second equation. Notice that by using the identity
ǫ λµνρ γ 5 γ µ = i(η λρ γ ν − η λν γ ρ − γ λ η ρν + γ λ γ ν γ ρ ),(5)
one can recast the field equation in Eq.(4) as follows
/ ∂ψ λ − ∂ λ (γ · ψ) − γ λ ∂ · ψ + γ λ / ∂(γ · ψ) = 0.(6)
Here / ∂ = γ µ ∂ µ and γ · ψ = γ µ ψ µ . Contracting Eq.(6) with γ λ gives
∂ · ψ − / ∂(γ · ψ) = 0.(7)
Finally, by plugging this result in Eq.(6), the field equation reduces to
/ ∂ψ λ − ∂ λ (γ · ψ) = 0.(8)
To obtain the real propagating degrees of freedom, let us now study gauge transformation and corresponding gauge conditions. For this purpose, let us recall that under the local Rarita-Schwinger fermionic gauge transformation
δψ ρ (x) = ∂ ρ ǫ(x),(9)
the Lagrangian in Eq.(1) transforms as
δL = ∂ λ (−ǫ λµνρǭ γ 5 γ µ ∂ ν ψ ρ ).(10)
Here ǫ is an arbitrary four-component spinor field. As is seen in Eq.(10), the free massless Rarita-Schwinger Lagrangian changes by a total derivative under the Rarita-Schwinger gauge transformation, which drops at the action level and thus we have a completely gauge-invariant theory. This means that the theory admits a gauge redundancy. To find the correct physical degrees of freedom of the theory, one needs to fix this gauge-freedom. For this purpose, let us assume the Coulomb-like gauge condition
γ i ψ i = 0,(11)
where i = 1, 2, 3. In fact, this is a reasonable gauge choice: Any initial data ψ ′ i (x, t) that does not satisfy Eq. (11) can be tuned to the desired form via 2
ǫ(x, t) = −γ i ∂ iˆd 3 y 4π|x − y| γ j ψ j (y, t).(12)
(See [7] and [17] for further discussions). For the sake of the self-completeness, one needs to examine the theory further to see whether Eq.(11) imposes any additional conditions or not. For this purpose, note that ψ 0 component does not have a time derivative, so it is a Lagrange multiplier. In other words, as in the electromagnetic case, the zeroth component of the vector-spinor field is a zero mode which is followed with a constraint. More precisely, the λ = 0 component of the field equation in Eq.(8) reads
γ i ∂ i ψ 0 − ∂ 0 (γ i ψ i ) = 0.(13)
One can also get a secondary constraint by contracting the field equation with ∂ λ . But since our primary aim is not analyzing the system by examining all the existing constraints, we leave it as a comment. As is seen in Eq.(13), gauge fixing condition γ i ψ i = 0 imposes γ i ∂ i ψ 0 = 0. Here, since the operator is not invertible, we are not allowed to get ψ 0 = 0 as a corollary of γ i ψ i = 0; yet we assume an additional condition of ψ 0 = 0. Furthermore, splitting the fully contracted equation in Eq.(7) into its space and time components yields
∂ i ψ i − γ 0 ∂ 0 (γ i ψ i ) − γ i ∂ i (γ 0 ψ 0 ) − γ i ∂ i (γ j ψ j ) = 0.(14)
In Eq. (14), one should notice that the gauge fixing condition γ i ψ i = 0 together with the assumed condition ψ 0 = 0 impose ∂ i ψ i = 0. As a consequence of this, we obtain the set of consistency conditions
γ i ψ i = 0 , ∂ i ψ i = 0 , ψ 0 = 0.(15)
Observe that Eq. (15) can also be written in covariant forms as follows
γ µ ψ µ = 0 , ∂ µ ψ µ = 0,(16)/ ∂ψ λ = 0.(17)
Symplectic Reduction for Free Massless Rarita-Schwinger Theory
In this section, we study the FJ Hamiltonian reduction for the free massless Rarita-Schwinger theory which will lead us to the fundamental brackets of the theory. For this purpose, let us recast the Lagrangian in Eq.(1) in a more symmetric form:
L = − 1 2 ǫ λµνρψ λ γ 5 γ µ ∂ ν ψ ρ + 1 2 ǫ λµνρ (∂ νψλ )γ 5 γ µ ψ ρ .(18)
To study the theory in the first-order symplectic formalism, one needs to convert Eq.(18) into the desired symplectic form. That is, one needs to split the Lagrangian into its space and time components. After a straightforward decomposition, one gets
L = A (k) 1ψ k + A (k) 2ψ k − H(ψ 0 ,ψ 0 , ψ k ,ψ k ),(19)
where the symplectic coefficients are
A (k) 1 = − 1 2 ǫ ijkψ i γ 5 γ j , A (k) 2 = 1 2 ǫ ijk γ 5 γ j ψ i ,(20)
and the corresponding symplectic potential reads
H(ψ 0 ,ψ 0 , ψ k ,ψ k ) = 1 2 ǫ ijkψ 0 γ 5 γ i ∂ j ψ k − 1 2 ǫ ijkψ i γ 5 γ 0 ∂ j ψ k − 1 2 ǫ ijkψ i γ 5 γ j ∂ k ψ 0 − 1 2 ǫ ijk (∂ jψ0 )γ 5 γ i ψ k + 1 2 ǫ ijk (∂ jψi )γ 5 γ 0 ψ k + 1 2 ǫ ijk (∂ kψi )γ 5 γ j ψ 0 .(21)
As expected, all the non-dynamical components have been relegated into the Hamiltonian part of the system. In analyzing the theory, one could also choose the conjugate momenta ofψ k as a dynamical variable. But in our analysis, we will not work with it. Instead, we consider ψ µ and ψ µ as the independent variables. Note that ψ 0 andψ 0 are not dynamical components, so they are Lagrange multipliers. Following [11,12], the elimination of constraints give the equations
ǫ ijk (∂ kψi )γ 5 γ j = 0, ǫ ijk γ 5 γ i ∂ j ψ k = 0.(22)
To solve the constraint equations, one can decompose the independent fields into its local transverse and γ-traceless parts as
ψ i = ψ T i +ψ iψi =ψ T i +ψ i ,(23)
where "T " and "ˆ" stand for the transverse and traceless parts, respectively. Here the γ-traceless parts areψ
i = ψ i − 1 3 γ i γ j ψ j ,ψ i =ψ i − 1 3ψ j γ j γ i ,(24)
such that γ iψ i = 0 and γ iψ i = 0. Then, by using the identity
ǫ ijk γ 5 γ k = −γ 0 σ ij where σ ij = i 2 [γ i , γ j ],(25)
as well as the constraints in Eq. (22), one can show that the transverse and traceless decomposition of the fields in Eq.(23) can actually be written as follows
ψ i = ψ T i + ∂ i ζ ∇ 2 ,ψ i =ψ T i + ∂ iζ ∇ 2 ,(26)where ζ = / ∂(γ · ψ T ) and ∇ 2 = ∂ i ∂ i .
As a side comment, one should note that as is done in [12], without addressing the transverse and γ-traceless parts (23), one could also directly start with the (26). Here, we further provide what the explicit form of the Longitudinal part is. (See Appendix A for the derivation of Eq.(26)). Accordingly, the constraint equations in Eq.(22) turn into completely transverse ones
ǫ ijk (∂ kψ T i )γ 5 γ j = 0, ǫ ijk γ 5 γ i ∂ j ψ T k = 0.(27)
Finally, by inserting Eq.(26) and Eq.(27) in the Eq. (19), up to a boundary term, one gets a completely transverse Lagrangian
L = A (k) T 1ψ T k + A (k) T 2ψ T k − H T (ψ T k ,ψ T k ).(28)
Here the transverse symplectic coefficients and potential are
A (k) T 1 = − 1 2 ǫ ijkψT i γ 5 γ j , A (k) T 2 = 1 2 ǫ ijk γ 5 γ j ψ T i , H T (ψ T k ,ψ T k ) = − 1 2 ǫ ijkψT i γ 5 γ 0 ∂ j ψ T k + 1 2 ǫ ijk (∂ jψ T i )γ 5 γ 0 ψ T k .(29)
Thus, by defining the symplectic variables as (ξ 1 , ξ 2 ) = (ψ T k ,ψ T k ), one gets the corresponding symplectic matrix
f αβ = 0 ǫ ijk γ 5 γ j −ǫ ijk γ 5 γ j 0 = ǫ αβ ǫ ijk γ 5 γ j ,
which is clearly non-singular. Notice that the minus sign in the sub-block is due to the antisymmetric ǫ tensor. Therefore, by taking care of the epsilons contraction in the current signature, one can easily show that the inverse symplectic matrix is
f −1 αβ = 0 − 1 2 ǫ imk γ 5 γ m 1 2 ǫ imk γ 5 γ m 0 = 1 2 ǫ βα ǫ imk γ 5 γ m .
Once the inverse symplectic matrix is found, one can evaluate the fundamental brackets. That is, by using the definition of the FJ equal-time brackets for the Grassmann variables
{ξ β , ξ α } F J = −f −1 αβ ,(30)
one gets the fundamental brackets for free massless Rarita-Schwinger theory as follows
{ψ T i (x),ψ T k (y)} F J = − 1 2 ǫ imk γ 5 γ m δ 3 (x − y), {ψ T i (x), ψ T k (y)} F J = 0, {ψ T i (x),ψ T k (y)} F J = 0.(31)
Note that, with the help of the identity in Eq.(25), the non-vanishing bracket can also be rewritten as
{ψ T i (x),ψ T k (y)} F J = i 2 γ k γ i γ 0 δ 3 (x − y),(32)
which is identical with the one found in [18].
III. FREE MASSIVE RARITA-SCHWINGER THEORY
The Lagrangian that describes the 3 + 1-dimensional free massive Rarita-Schwinger theory is
L = −ǫ λµνρψ λ γ 5 γ µ ∂ ν ψ ρ + imψ λ σ λρ ψ ρ ,(33)
where
σ λρ = i 2 [γ λ , γ ρ ] = i(η λρ − γ ρ γ λ ).
Recall that the fermionic fields are anti-commuting Grassmannian variables. Accordingly, the field equations of the independent variables respectively read
ǫ λµνρ γ 5 γ µ ∂ ν ψ ρ − imσ λρ ψ ρ = 0, ǫ λµνρ ∂ νψλ γ 5 γ µ + imψ λ σ λρ = 0.(34)
In dealing with the fundamental properties of the theory, as we did in the massless theory, we will work only with the first field equation in Eq.(34). Notice that by using the identity in Eq.(5), one can recast the field equation as follows
i[ / ∂ψ λ − ∂ λ (γ · ψ) − γ λ ∂ · ψ + γ λ / ∂(γ · ψ)] − imσ λρ ψ ρ = 0.(35)
Observe that the contraction of Eq.(35) with γ λ yields
2i[ / ∂(γ · ψ) − ∂ · ψ] + 3mγ · ψ = 0,(36)
and the contraction of Eq.
(35) with ∂ λ gives m[ / ∂(γ · ψ) − ∂ · ψ] = 0.(37)
Combining both contracted field equations Eq.(36) and Eq.(37), one obtains
γ · ψ = 0, ∂ · ψ = 0.(38)
With these gauge-fixing conditions, the equation in Eq.(35) turns into the Dirac field equation for massive spin-3 2 vector-spinor field
(i / ∂ + m)ψ λ = 0.(39)
Note that, unlike the massless theory, one obtains the Dirac field equation in Eq.(39) without addressing the space and time decompositions of the field equations. On the other hand, due to the mass term, the Rarita-Schwinger gauge-invariance is inevitably lost.
Symplectic Reduction for Free Massive Rarita-Schwinger Lagrangian
Let us now study the symplectic Hamiltonian reduction of the free massive Rarita-Schwinger theory. For this purpose, let us recall that the Lagrangian in Eq.(33), up to a boundary term, can be written as
L = − 1 2 ǫ λµνρψ λ γ 5 γ µ ∂ ν ψ ρ + 1 2 ǫ λµνρ ∂ νψλ γ 5 γ µ ψ ρ + imψ λ σ λρ ψ ρ .(40)
In order to proceed the FJ symplectic reduction of Eq.(40), one needs to separate the dynamical components from the non-dynamical ones so that the non-dynamical components can be relegated to Hamiltonian part of the Lagrangian. Therefore, by splitting the Lagrangian into its space and time components, one will obtain
L = A (k) 1ψ k + A (k) 2ψ k − H(ψ 0 ,ψ 0 , ψ k ,ψ k ),(41)
where the coefficient of the dynamical parts are
A (k) 1 = − 1 2 ǫ ijkψ i γ 5 γ j , A (k) 2 = 1 2 ǫ ijk γ 5 γ j ψ i ,(42)
and the explicit form of the symplectic potential is
H(ψ 0 ,ψ 0 , ψ k ,ψ k ) = 1 2 ǫ ijkψ 0 γ 5 γ i ∂ j ψ k − 1 2 ǫ ijkψ i γ 5 γ 0 ∂ j ψ k − 1 2 ǫ ijkψ i γ 5 γ j ∂ k ψ 0 − 1 2 ǫ ijk (∂ jψ0 )γ 5 γ i ψ k + 1 2 ǫ ijk (∂ jψi )γ 5 γ 0 ψ k + 1 2 ǫ ijk (∂ kψi )γ 5 γ j ψ 0 − imψ 0 σ 0i ψ i − imψ i σ i0 ψ 0 − imψ i σ ij ψ j .(43)
Like the free massless theory, ψ 0 andψ 0 are zero modes of the system whose eliminations give rise the constraints
ǫ ijk (∂ kψi )γ 5 γ j − imψ i σ i0 = 0, ǫ ijk γ 5 γ i ∂ j ψ k − imσ 0i ψ i = 0.(44)
As was done in the previous section, by decomposing the fields into the local transverse and γtraceless parts as in the Eq.(23), the constraints in Eq.(44) turn into completely transverse ones
ǫ ijk (∂ kψ T i )γ 5 γ j − imψ T i σ i0 = 0, ǫ ijk γ 5 γ i ∂ j ψ T k − imσ 0i ψ T i = 0.(45)
In this case, the longitudinal part reads ζ = ( / ∂ + im)γ · ψ T . Thus, by plugging the Eq.(23) and the transverse constraints Eq.(45) into the Eq.(41), up to a boundary term, the Lagrangian turns into
L = A (k) T 1ψ T k + A (k) T 2ψ T k + imψ T i σ i0ζ ∇ 2 + imζ ∇ 2 σ 0i ψ T i − H T (ψ T k ,ψ T k ),(46)
where the transverse symplectic coefficients and potential respectively are
A (k) T 1 = − 1 2 ǫ ijkψT i γ 5 γ j , A (k) T 2 = 1 2 ǫ ijk γ 5 γ j ψ i , H T (ψ T k ,ψ T k ) = − 1 2 ǫ ijkψT i γ 5 γ 0 ∂ j ψ T k + 1 2 ǫ ijk ∂ jψ T i γ 5 γ 0 ψ T k − imψ T i σ ij ψ T j .(47)
Observe that the middle two terms in Eq.(46) are not in the symplectic forms. Therefore, by assuming the Darboux transformation
ψ T k → ψ ′ T k = e 2i ζ ∇ 2 ψ T k ,(48)
with an additional assumption of
ǫ ijkψT i γ 5 γ j ψ T k = me −2iζ ∇ 2ψ T i σ i0 ,(49)
the undesired terms in Eq.(46) drop and thus we are left with a completely transverse Lagrangian
L = A (k) T 1ψ T k + A (k) T 2ψ T k − H T (ψ T k ,ψ T k ) − λ k φ k (ψ T k ,ψ T k ) −λ iφ i (ψ T k ,ψ T k ).(50)
Note that the extra condition Eq.(49) is enforced by the Darboux transformation and the constraint equations; otherwise, the coupled terms in the symplectic part could not be decoupled. In fact, it seems there is a lack in the physical interpretation of Eq.(49). Therefore, it will be particularly interesting if one can show that it has a relation with the real constraints or not. Here, as is mentioned in Eq.(117), the remaining variables (i.e., the longitudinal components) are denoted as the Lagrange multipliers
λ k = ∂ k ζ ∇ 2 ,λ i = ∂ iζ ∇ 2 ,(51)
such that
φ k (ψ T k ,ψ T k ) = iǫ ijkψT i γ 5 γ 0 ψ T j ,φ i (ψ T k ,ψ T k ) = −iǫ ijkψT j γ 5 ψ T k .(52)
As noted in [11,12], since the last two terms in the Eq.(50) cannot be dropped via elimination of constraints anymore, Eq.(52) corresponds to the true constraints of the system. Note also that the true constraints cannot be rewritten as linear combinations of the ones that are obtained during the eliminations of the constraints; otherwise, they would also drop when the eliminations of constraint was performed. These are the constraints that cannot be eliminated anymore. Therefore, setting φ k (ψ T k ,ψ T k ) andφ i (ψ T k ,ψ T k ) to zero provides an unconstrained fully traceless Lagrangian
L = A (k) T 1ψ T k + A (k) T 2ψ T k − H T (ψ T k ,ψ T k ).(53)
Thus, with the definition of the dynamical variables (ξ 1 , ξ 2 ) = (ψ T k ,ψ T k ), the non-vanishing equaltime FJ bracket for the free massive Rarita-Schwinger theory becomes
{ψ T i (x),ψ T k (y)} F J = i 2 γ k γ i γ 0 δ 3 (x − y).(54)
which is same as the one found in [19].
IV. GAUGED MASSLESS RARITA-SCHWINGER THEORY
In this section, we study the massless Rarita-Schwinger field minimally coupled to an external electromagnetic field which is described by the Lagrangian
L = −ǫ λµνρψ λ γ 5 γ µ → D ν ψ ρ .(55)
Here the gauge covariant derivative is D ν = ∂ ν + gA ν , where g is the relevant coupling constant and A µ is an Abelian gauge field. The field equations read
ǫ λµνρ γ 5 γ µ → D ν ψ ρ = 0, ǫ λµνρψ λ ← D ν γ 5 γ µ = 0.(56)
As in the free massless and massive theories, while deducing the some basic properties of the theory, we will only deal with the first of Eq.(56). Notice that with the help of the identity in Eq.(5), the Eq.(56) turns into
/ Dψ λ − D λ (γ · ψ) − γ λ D · ψ + γ λ / D(γ · ψ) = 0.(57)
Moreover, contracting the Eq.(57) with γ λ yields
/ D(γ · ψ) − D · ψ = 0.(58)
Finally, substituting the Eq.(58) in Eq. (57) gives
/ Dψ λ − D λ (γ · ψ) = 0.(59)
On the other side, contracting the Eq.(56) with D λ becomes
gǫ λµνρ γ 5 γ µ F λν ψ ρ = 0,(60)
which is a secondary constraint in the theory and does not provide any further simplification in the field equation in Eq.(59).
Symplectic Reduction for Gauged Massless Rarita-Schwinger Theory
Let us now apply the first-order symplectic formalism to the massless Rarita-Schwinger fields minimally coupled to an external electromagnetic field. For this purpose, let us note that the Lagrangian of the theory in Eq.(55) can be recast in a more symmetric form as follows
L = − 1 2 ǫ λµνρψ λ γ 5 γ µ → D ν ψ ρ + 1 2 ǫ λµνρψ λ ← D ν γ 5 γ µ ψ ρ .(61)
Similarly, by splitting the Lagrangian in Eq.(61) into its space and time components, one gets
L = A (k) 1ψ k + A (k) 2ψ k − H(ψ 0 ,ψ 0 , ψ k ,ψ k , A 0 , A k ),(62)
where the symplectic coefficients are
A (k) 1 = − 1 2 ǫ ijkψ i γ 5 γ j , A (k) 2 = 1 2 ǫ ijk γ 5 γ j ψ i ,(63)
and the related symplectic potential is
H(ψ 0 ,ψ 0 , ψ k ,ψ k , A 0 , A k ) = 1 2 ǫ ijkψ 0 γ 5 γ i ∂ j ψ k − 1 2 ǫ ijkψ i γ 5 γ 0 ∂ j ψ k − 1 2 ǫ ikjψ i γ 5 γ k ∂ j ψ 0 − 1 2 ǫ ijk ∂ jψ0 γ 5 γ i ψ k + 1 2 ǫ ijk ∂ jψi γ 5 γ 0 ψ k + 1 2 ǫ ikj ∂ jψi γ 5 γ k ψ 0 + gǫ ijkψ i γ 5 γ j A 0 ψ k + gǫ ijkψ 0 γ 5 γ i A j ψ k − gǫ ijkψ i γ 5 γ 0 A j ψ k − gǫ ikjψ i γ 5 γ k A j ψ 0 .(64)
Note that although the gauge fields are non-dynamical variables, due to being external potentials, one cannot vary and then impose these variations to be vanished. Otherwise, as in the Quantum Electromagnetic Dynamics with external potential, the gauge field current would be enforced to be zero which is not a desired situation. Hence, as in the free theories, here ψ 0 andψ 0 are the only zero modes of the theory: Therefore, variations with respect to ψ 0 andψ 0 respectively give the following constraint equations
ǫ ikj ∂ jψi γ 5 γ k − gǫ ikjψ i γ 5 γ k A j = 0, ǫ ijk γ 5 γ i ∂ j ψ k + gǫ ijk γ 5 γ i A j ψ k = 0.(65)
As was done in the free theories, by decomposing the fields into the local transverse and γ-traceless parts as in Eq.(23) 3 and using the constraints in Eq.(65) as well as by assuming the Darboux transformation (48), with an additional assumption of
iǫ ijkψT i γ 5 γ j ψ T k = ge −2iζ ∇ 2 ǫ ijkψT i γ 5 γ k A j ,(66)
the Lagrangian (62) turns into a completely transverse one
L = A (k) T 1ψ T k + A (k) T 2ψ T k − H T (ψ T k ,ψ T k ) − λ k φ k (ψ T k ,ψ T k ) −λ iφ i (ψ T k ,ψ T k ),(67)
where the transverse symplectic coefficients and potential read
A (k) T 1 = − 1 2 ǫ ijkψT i γ 5 γ j , A (k) T 2 = 1 2 ǫ ijk γ 5 γ j ψ T i , H T (ψ T k ,ψ T k ) = − 1 2 ǫ ijkψT i γ 5 γ 0 ∂ j ψ T k + 1 2 ǫ ijk ∂ jψ T i γ 5 γ 0 ψ T k + gǫ ijkψT i γ 5 γ j A 0 ψ T k − gǫ ijkψT i γ 5 γ 0 A j ψ T k . (68)
Note that the symplectic potential also contains gauge field parts. Furthermore, as is given in (117), the remaining variables (i.e., the longitudinal components) are denoted as the Lagrange multipliers
λ k = ∂ k ζ ∇ 2 ,λ k = ∂ iζ ∇ 2 ,(69)
such that
φ k (ψ T k ,ψ T k ) = iǫ ijkψT i γ 5 γ 0 ψ T j + gǫ ijkψT i γ 5 γ j A 0 + gǫ ijkλ i γ 5 γ j A 0 − gǫ ijkψT i γ 5 γ 0 A j φ i (ψ T k ,ψ T k ) = −iǫ ijkψT j γ 5 ψ T k + gǫ ijk γ 5 γ j A 0 ψ T k − gǫ ijk γ 5 γ 0 A j ψ T k − gǫ ijk γ 5 γ 0 A j λ k ,(70)
which cannot be dropped via elimination of constraints anymore so, according to [11,12], they are the true constraint of the system. Thus, by setting φ k (ψ T k ,ψ T k ) andφ i (ψ T k ,ψ T k ) to zero, one arrives at a completely transverse Lagrangian
L = A (k) T 1ψ T k + A (k) T 2ψ T k − H T (ψ T k ,ψ T k ).(71)
Finally, with the definition of the symplectic dynamical variables (ξ 1 , ξ 2 ) = (ψ T k ,ψ T k ), one obtains the non-vanishing equal-time FJ basic bracket for the gauged massless Rarita-Schwinger theory as follows
{ψ T i (x),ψ T k (y)} F J = i 2 γ k γ i γ 0 δ 3 (x − y),(72)
which is consistent with the Pauli-spin-part of the fundamental bracket obtained in [10] in which Adler studies the Dirac quantization of the non-Abelian gauged Rarita-Schwinger theory via the left-chiral component of the fermionic field. One should notice that such a difference is expected because in [10], the corresponding gauge fields are non-Abelian variables; however here the gauge fields are Abelian vector fields.
V. GAUGED MASSIVE RARITA-SCHWINGER
In this section, we study the massive Rarita-Schwinger theory minimally coupled to an external electromagnetic field which is described by the Lagrangian
L = −ǫ λµνρψ λ γ 5 γ µ → D ν ψ ρ + imψ λ σ λρ ψ ρ ,(73)
where the gauge-covariant derivative is D ν = ∂ ν + gA ν . Accordingly, the field equations for the independent anti-commuting fermionic fields are
ǫ λµνρ γ 5 γ µ → D ν ψ ρ − imσ λρ ψ ρ = 0, ǫ λµνρψ λ ← D ν γ 5 γ µ + imψ λ σ λρ = 0,(74)
which with the help of the identity in Eq.(5) turns into
i[ / Dψ λ − D λ (γ · ψ) − γ λ D · ψ + γ λ / D(γ · ψ)] − imσ λρ ψ ρ = 0.(75)
Moreover, contraction of the equation in Eq.(75) with γ λ gives
2i( / D(γ · ψ) − D · ψ) + 3mγ · ψ = 0.(76)
And contraction of field equation in Eq.(74) with D λ becomes
gǫ λµνρ γ 5 γ µ F λν ψ ρ + m[( / D(γ · ψ) − D · ψ] = 0,(77)
which with the additional redefinition
F d = F d µ ρ = ǫ µ ρλ ν F λ ν ,(78)
turns into
m[ / D(γ · ψ) − D · ψ] − gγ 5 γ · F d · ψ = 0.(79)
Combining Eq.(76) and Eq.(79), one gets the secondary constraint that determines the equation of motion of ψ 0 component as follows
γ · ψ = − 2 3 m −2 igγ 5 γ · F d · ψ.(80)
Observe that using Eq.(80) in Eq.(79) gives the relation
D · ψ = −( / D − 3im 2 ) 2 3 m −2 igγ 5 γ · F d · ψ.(81)
Finally, by plugging Eq.(80) and Eq.(81) into the field equation in Eq.(75), one obtains
(i / D − m)ψ λ + (iD λ + m 2 γ λ ) 2 3 m −2 igγ 5 γ · F d · ψ = 0,(82)
which is the equation that is used by Velo and Zwanziger in deducing the acausal wave propagation of the solution by finding the future-directed normals to the surfaces at each point [3].
Symplectic Reduction for Gauged Massive Rarita-Schwinger Theory
Finally, let us apply FJ symplectic Hamiltonian reduction to the massive Rarita-Schwinger field minimally coupled to an external electromagnetic field. In order to do so, let us rewrite the Lagrangian in Eq.(73) in a more symmetric form:
L = − 1 2 ǫ λµνρψ λ γ 5 γ µ → D ν ψ ρ + 1 2 ǫ λµνρψ λ ← D ν γ 5 γ µ ψ ρ + imψ λ σ λρ ψ ρ .(83)
Subsequently, by splitting Lagrangian in Eq.(83) into its space and time components, one gets
L = A (k) 1ψ k + A (k) 2ψ k − H(ψ 0 ,ψ 0 , ψ k ,ψ k , A 0 , A k ),(84)
where the symplectic coefficients are
A (k) 1ψ k = − 1 2 ǫ ijkψ i γ 5 γ j , A (k) 2 = 1 2 ǫ ijk γ 5 γ j ψ i ,(85)
and the relevant Hamiltonian
H(ψ 0 ,ψ 0 , ψ k ,ψ k , A 0 , A k ) is H(ψ 0 ,ψ 0 , ψ k ,ψ k , A 0 , A k ) = 1 2 ǫ ijkψ 0 γ 5 γ i ∂ j ψ k − 1 2 ǫ ijkψ i γ 5 γ 0 ∂ j ψ k − 1 2 ǫ ikjψ i γ 5 γ k ∂ j ψ 0 − 1 2 ǫ ijk ∂ jψ0 γ 5 γ i ψ k + 1 2 ǫ ijk ∂ jψi γ 5 γ 0 ψ k + 1 2 ǫ ikj ∂ jψi γ 5 γ k ψ 0 − imψ 0 σ 0i ψ i − imψ i σ i0 ψ 0 − imψ i σ ij ψ j + gǫ ijkψ i γ 5 γ j A 0 ψ k + gǫ ijkψ 0 γ 5 γ i A j ψ k − gǫ ijkψ i γ 5 γ 0 A j ψ k − gǫ ikjψ i γ 5 γ k A j ψ 0 .(86)
Note that as is emphasized in the massless gauged part, since the gauge fields are external potentials, one is not allowed to set their variation to zero. Hence, here ψ 0 ,ψ 0 are the only Lagrange multipliers that induce constraints on the system. Therefore, eliminations of constraint yield
ǫ ikj ∂ jψi γ 5 γ k − imψ i σ i0 − gǫ ikjψ i γ 5 γ k A j = 0, ǫ ijk γ 5 γ i ∂ j ψ k − imσ 0i ψ i + gǫ ijk γ 5 γ i A j ψ k = 0. (87)
Like the free massive theory, by decomposing the dynamical components into the local transverse and traceless parts as in Eq.(23) 4 as well as using constraints in Eq.(87) and the Darboux transformation (48), with an additional assumption of
iǫ ijkψT i γ 5 γ j ψ T k = e −2iζ ∇ 2 (imψ i σ i0 + gǫ ijkψT i γ 5 γ k A j ),(89)
the Lagrangian, up to a boundary term, turns into
L = A (k) T 1ψ T k + A (k) T 2ψ T k − H T (ψ T k ,ψ T k ) − λ k φ k (ψ T k ,ψ T k ) −λ iφ i (ψ T k ,ψ T k ).(90)
Here the transverse symplectic coefficients and potential are
A (k) T 1 = − 1 2 ǫ ijkψT i γ 5 γ j , A (k) T 2 = 1 2 ǫ ijk γ 5 γ j ψ T i , H T (ψ T k ,ψ T k ) = − 1 2 ǫ ijkψT i γ 5 γ 0 ∂ j ψ T k + 1 2 ǫ ijk ∂ jψ T i γ 5 γ 0 ψ T k + gǫ ijkψT i γ 5 γ j A 0 ψ T k − gǫ ijkψT i γ 5 γ 0 A j ψ T k − imψ T i σ ij ψ T j .
(91) 4 In this case, from the constraint equation, one finds
ζ = ( / ∂ + im + g γ · A)γ · ψ T − gA · ψ T .(88)
Notice that, different from the free cases, the symplectic potential involves mass and gauge potentials. Here in the Eq.(90), as in the previous sections, the Lagrange multipliers are the Longitudinal parts of the vector-spinor field and the corresponding constraints read
φ k (ψ T k ,ψ T k ) = iǫ ijkψT i γ 5 γ 0 ψ T j + gǫ ijkψT i γ 5 γ j A 0 + gǫ ijkλ i γ 5 γ j A 0 − gǫ ijkψT i γ 5 γ 0 A j φ i (ψ T k ,ψ T k ) = −iǫ ijkψT j γ 5 ψ T k + gǫ ijk γ 5 γ j A 0 ψ T k − gǫ ijk γ 5 γ 0 A j ψ T k − gǫ ijk γ 5 γ 0 A j λ k ,(92)
which are same as Eq.(70). Similarly, by setting Eq.(92) to zero [11,12], one arrives at a completely transverse Lagrangian
L = A (k) T 1ψ T k + A (k) T 2ψ T k − H T (ψ T k ,ψ T k ),(93)
whose symplectic part is same as the ones found so far. Thus, with the definition of the dynamical variables (ξ 1 , ξ 2 ) = (ψ T k ,ψ T k ), the non-vanishing equal-time bracket for the gauged massive Rarita-Schwinger theory becomes
{ψ T i (x),ψ T k (y)} F J = i 2 γ k γ i γ 0 δ 3 (x − y),(94)
which is identical to the one found in [20].
VI. CONCLUSIONS
In this work, we studied 3 + 1-dimensional free and Abelian gauged Grassmannian Rarita-Schwinger theories for their massless and massive extensions in the context of Faddeev-Jackiw first-order symplectic formalism. We have obtained the fundamental brackets of theories which are consistent with the some results that we found in the literature but obtained in a more simpler way. The brackets are independent of whether the theories contain mass or gauge field or not, and thus the structure of constraints and symplectic potentials determine characteristic behaviors of the theories. It will be particularly interesting to find proper transformations that will relate the constraints obtained via the Faddeev-Jackiw symplectic method with the ones that are obtained via Dirac method. But since the constraints obtained in both methods are rather complicated, in this paper, we restrict ourselves only to the Faddeev-Jackiw analysis of Rarita-Schwinger theories and leave this as a future work. With the comparison with the literature, we concluded that the Faddeev-Jackiw symplectic approach provides a more economical way in deriving the fundamental brackets for the Rarita-Schwinger theories. In addition to these, we notice that, in contrast to the massive theory, the Dirac field equations for free massless Rarita-Schwinger theory cannot be covariantly deduced.
VII.
VIII. APPENDIX A: TRANSVERSE AND TRACELESS DECOMPOSITION OF FIELDS
In this section, let us give the derivations of (26) and (27): To solve the constraint equations, one can decompose the independent fields into its local transverse and γ-traceless parts as
ψ i = ψ T i +ψ iψi =ψ T i +ψ i ,(95)
where "T " and "ˆ" stand for the transverse and traceless parts, respectively. Here the γ-traceless parts areψ
i = ψ i − 1 3 γ i γ j ψ j ,ψ i =ψ i − 1 3ψ j γ j γ i .(96)
Therefore, we have
∂ i ψ T i = ∂ iψT i = 0 and γ iψ i = γ iψ i = 0.(97)
To find how the constraint equations in Eq.(22) decomposes under Eq.(95), let us focus on the following constraint equation
ǫ ijk γ 5 γ i ∂ j ψ k = 0.(98)
Note that with the identity ǫ ijk γ 5 γ k = −γ 0 σ ij and Eq.(95), the Eq.(98) turns into
iγ 0 2 [γ k , γ j ]∂ j (ψ T k +ψ k ) = 0.(99)
Furthermore, by using the relation
[γ k , γ j ] = {γ k , γ j } − 2γ j γ k = 2(η kj − γ j γ k ),(100)
and the transverse and traceless properties of the fields in Eq.(97), one gets
iγ 0 ∂ kψ k − γ j γ k ∂ j ψ T k = 0.(101)
Notice that after contraction with iγ 0 and relabeling of the dummy indices, it becomes
∂ iψ i − γ m γ n ∂ m ψ T n = 0,(102)which yieldsψ i = ∂ i ζ ∇ 2 where ζ = γ m γ n ∂ m ψ T n .(103)
This structure is also valid for the other theories. The only difference arises in the definition of ζ which we give its explicit form in each section. Finally, by substituting this result into the constraint in Eq.(98), it turns into
ǫ ijk γ 5 γ i ∂ j ψ T k + ǫ ijk γ 5 γ i ∂ j ∂ k ζ ∇ 2 = 0.(104)
Because of the symmetric and anti-symmetric contraction in "j, k" indices, the second term drops, and we are left with the transverse constraint equation
ǫ ijk γ 5 γ i ∂ j ψ T k = 0.(105)
IX. APPENDIX B: FADDEEV-JACKIW HAMILTONIAN REDUCTION FOR CONSTRAINED AND UNCONSTRAINED SYSTEMS
In this section, we review the Faddeev-Jackiw symplectic first-order formalism which was introduced particularly to quantize the constrained systems [11,12]. The method works on the first-order Lagrangian and does not require any classification of constraints. To better understand how the method works, let us consider
L = p αq α − H(p, q), α = 1, . . . n.(106)
With the definition of 2n-component phase-space coordinates ξ α = p α , α = 1, · · · , n and ξ β = q β , β = n + 1, · · · , 2n,
Eq.(106) can be rewritten as a Lagrangian one-form
Ldt = 1 2 ξ α f 0 αβ dξ β − V (ξ)dt.(108)
Here the symplectic 2n × 2n matrix is [11,12]. But, in general, the symplectic twoform does not have to be constant. Therefore, let us now consider the following generic Lagrangian
f 0 αβ = 0 I −I 0 αβ , where I is the identity matrix; A 0 ≡ 1 2 ξ α f 0 αβ dξ β is the canonical one-form; f 0 ≡ dA 0 ≡ 1 2 f 0 αβ dξ α dξ β is the symplectic two-form. Note that f 0 is constantLdt = A α dξ α − H(ξ)dt, α = 1, · · · , 2n,(109)
where A α is an arbitrary one-form. The variation of Eq.(109) with respect to ξ yields
f βαξα = ∂H ∂ξ β where f βα = ∂A α ∂ξ β − ∂A β ∂ξ α .(110)
In the case of when the symplectic matrix is nonsingular, Eq.(110) becomeṡ
ξ α = f −1 αβ ∂H(ξ) ∂ξ β .(111)
Thus, by using Eq.(111) and the Poisson brackets for the bosonic variables, one obtains the FJ fundamental brackets as follows
{ξ β , ξ α } F J = f −1 αβ .(112)
Note that, in the case of the Grassmannian variables, using the anti-commutation property of the variables as well as the Poisson brackets for the Grassmannian variables [21], one haṡ
ξ α = ∂H(ξ) ∂ξ β (f −1 ) αβ ,(113)
and the corresponding fundamental brackets become
{ξ β , ξ α } F J = −(f −1 ) αβ .(114)
On the other side, when there are constraints in the system which are induced by the existence of the zero-modes, then the symplectic matrix cannot be inverted. In that case, according to the Darboux's theorem which states that for any given one-form A = A α dξ α where α = 1, · · · , N , one can always do the following changes in the variables ξ α → (p β , q γ z ρ ), β, γ = 1, · · · , n, ρ = 1, · · · , N − 2n,
so that A turns into A = A α dq α . As is seen above, when there is no constraint, Eq.(115) diagonalizes f αβ . However when there are constraints, only a 2n × 2n sub-block of f αβ diagonalizes and the remaining N − 2n degrees of freedom (corresponding the zero-modes z ρ ) will not be in the symplectic form [11,12]; yet they occur in the rest of the Lagrangian:
L = p α dq α − Φ(p, q, z)dt.(116)
The equations ∂Φ ∂z α = 0 can be used to eliminate the zero-modes of z's only if ∂ 2 Φ ∂z ρ ∂z β is nonsingular. In the generic case, after diagonalization and elimination of z's as many as possible, one ultimately arrives at
L = p αq α − H(p, q) − λ ρ φ ρ (p, q),(117)
where the remaining z's are denoted by λ ρ (namely, Lagrange multipliers) and the φ ρ are the only true constraints in the system
φ ρ = 0.(118)
A. Symplectic Reduction for Dirac Theory of spin- 1 2 fields In this section, to see how the method works, we provide FJ Hamiltonian reduction for the Dirac theory for the spin-1 2 theory as an example. For this purpose, let us note that the Lagrangian can be written:
L = − i 2ψ ← / ∂ ψ + i 2ψ → / ∂ ψ − mψψ.(119)
As mentioned above, we assume that the independent dynamical variables are anti-commuting Grassmann variables. In order to pass to the symplectic analysis of the system, one needs to separate the dynamical components from the non-physical ones by splitting the Lagrangian (119) into its time and space components. In doing so, one arrives at
L = i 2 γ 0 ψψ + i 2 γ 0ψψ − i 2 ∂ iψ γ i ψ − i 2ψ γ i ∂ i ψ + mψψ ,(120)
whose variation, up to a boundary term, yields
δL = δψ iγ 0ψ + δψ iγ 0ψ − δψ(−iγ i ∂ i ψ + mψ) + δψ(−iγ i ∂ iψ − mψ) ,(121)
from which one gets the Dirac field equations as follows
iγ 0ψ = −iγ i ∂ iψ − mψ, iγ 0ψ = −iγ i ∂ i ψ + mψ.(122)
As is seen from Eq.(121) and Eq.(122), the symplectic matrix for the Dirac theory and its inverse are
f αβ = 0 iγ 0 iγ 0 0 , f −1 αβ = 0 −iγ 0 −iγ 0 0 = −f αβ .
One should observe that, in contrast to the bosonic case, the symplectic matrix for the Grassmannian variables is symmetric and the fundamental brackets are defined as follows
{ξ β , ξ α } F J = −(f −1 ) αβ ,(123)
from which one gets the basic bracket for the Dirac theory
{ψ,ψ} F J = iγ 0 .(124)
This is also valid for the massless theory. Note that since the theory does not have any gauge redundancy, one does not need to assume any gauge-fixing.
For the quantization of the constrained system, see for example[13][14][15].
Since the gauge choice ∂ i ψi = 0 on the initial data will also arise due to the self-consistency, one should also be able to regulate the gauge parameter via ǫ = − 1 ∇ 2 ∂iψ i . But since we start with the (11), we have to give(12).
Notice that, in this case, the longitudinal part becomes ζ = ( / ∂ + g γ · A)γ · ψ T − gA · ψ T .
ACKNOWLEDGMENTSWe would like to thank Roman Jackiw for suggesting the problem and several useful discussions. We would also like to thank Bayram Tekin for useful suggestions and Markus Schulze, Gilly Elor and Ibrahim Burak Ilhan for critical readings of the paper. S.D. is supported by the TUBITAK 2219 Scholarship.
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arxiv |
Multinomial Distribution Learning for Effective Neural Architecture Search
Xiawu Zheng
Department of Cognitive Science
School of Information Science and Engineering
Fujian Key Laboratory of Sensing and Computing for Smart City
Xiamen University
XiamenChina
Rongrong Ji [email protected]
Department of Cognitive Science
School of Information Science and Engineering
Fujian Key Laboratory of Sensing and Computing for Smart City
Xiamen University
XiamenChina
Peng Cheng Laboratory
ShenzhenChina
Lang Tang [email protected]
Department of Cognitive Science
School of Information Science and Engineering
Fujian Key Laboratory of Sensing and Computing for Smart City
Xiamen University
XiamenChina
Baochang Zhang [email protected]
Beihang University
China
Jianzhuang Liu
Huawei Noahs Ark Lab
Qi Tian [email protected]
Huawei Noahs Ark Lab
Multinomial Distribution Learning for Effective Neural Architecture Search
Architectures obtained by Neural Architecture Search (NAS) have achieved highly competitive performance in various computer vision tasks. However, the prohibitive computation demand of forward-backward propagation in deep neural networks and searching algorithms makes it difficult to apply NAS in practice. In this paper, we propose a Multinomial Distribution Learning for extremely effective NAS, which considers the search space as a joint multinomial distribution, i.e., the operation between two nodes is sampled from this distribution, and the optimal network structure is obtained by the operations with the most likely probability in this distribution. Therefore, NAS can be transformed to a multinomial distribution learning problem, i.e., the distribution is optimized to have high expectation of the performance. Besides, a hypothesis that the performance ranking is consistent in every training epoch is proposed and demonstrated to further accelerate the learning process. Experiments on CIFAR-10 and ImageNet demonstrate the effectiveness of our method. On CIFAR-10, the structure searched by our method achieves 2.4% test error, while being 6.0× (only 4 GPU hours on GTX1080Ti) faster compared with state-of-the-art NAS algorithms. On ImageNet, our model achieves 75.2% top-1 accuracy under MobileNet settings (MobileNet V1/V2), while being 1.2× faster with measured GPU latency. Test code is available at
Introduction
Given a dataset, Neural architecture search (NAS) aims to discover high-performance convolution architectures with a searching algorithm in a tremendous search space. NAS has achieved much success in automated architecture engineering for various deep learning tasks, such as image * Corresponding Author. classification [18,31], language modeling [19,30] and semantic segmentation [17,6]. As mentioned in [9], NAS methods consist of three parts: search space, search strategy, and performance estimation. A conventional NAS algorithm samples a specific convolutional architecture by a search strategy and estimates the performance, which can be regarded as an objective to update the search strategy. Despite the remarkable progress, conventional NAS methods are prohibited by intensive computation and memory costs. For example, the reinforcement learning (RL) method in [31] trains and evaluates more than 20,000 neural networks across 500 GPUs over 4 days. Recent work in [19] improves the scalability by formulating the task in a differentiable manner where the search space is relaxed to a continuous space, so that the architecture can be optimized with the performance on a validation set by gradient descent. However, differentiable NAS still suffers from the issued of high GPU memory consumption, which grows linearly with the size of the candidate search set.
Indeed, most NAS methods [31,17] perform the performance estimation using standard training and validation over each searched architecture, typically, the architecture has to be trained to converge to get the final evaluation on validation set, which is computationally expensive and limits the search exploration. However, if the evaluation of different architectures can be ranked within a few epochs, why do we need to estimate the performance after the neural network converges? Consider an example in Fig. 1, we randomly sample different architectures (LeNet [16], AlexNet [15], ResNet-18 [10] and DenseNet [13]) with different layers, the performance ranking in the training and testing is consistent (i.e, the performance ranking is ResNet-18 > DenseNet-BC > AlexNet > LeNet on different networks and training epochs). Based on this observation, we state the following hypothesis for performance ranking: Performance Ranking Hypothesis. If Cell A has higher validation performance than Cell B on a specific network and a training epoch, Cell A tends to be better than Cell Figure 1. We randomly choose widely used LeNet [16], AlexNet [14], ResNet-18 [10] and DenseNet-BC(k = 40) [13] to illustrate the proposed Performance Ranking Hypothesis. The training and testing are conducted on CIFAR-10. We report the top1 error and loss learning curves on both training and testing set. As we can see in the figure, the ranking of the test loss and accuracy keeps consistent in every training epoch, i.e., a good architecture tends to have better performance in the whole training process.
B on different networks after the trainings of these netwoks converge.
Here, a cell is a fully convolutional directed acyclic graph (DAG) that maps an input tensor to an output tensor, and the final network is obtained through stacking different numbers of cells, the details of which are described in Sec. 3.
The hypothesis illustrates a simple yet important rule in neural architecture search. The comparison of different architectures can be finished at early stages, as the ranking of different architectures is sufficient, whereas the final results are unnecessary and time-consuming. Based on this hypothesis, we propose a simple yet effective solution to neural architecture search, termed as Multinomial distribution for efficient Neural Architecture Search (MdeNAS), which directly formulates NAS as a distribution learning process. Specifically, the probabilities of operation candidates between two nodes are initialized equally, which can be considered as a multinomial distribution. In the learning procedure, the parameters of the distribution are updated through the current performance in every epoch, such that the probability of a bad operation is transferred to better operations. With this search strategy, MdeNAS is able to fast and effectively discover high-performance architectures with complex graph topologies within a rich search space.
In our experiments, the convolutional cells designed by MdeNAS achieve strong quantitative results. The searched model reaches 2.4% test error on CIFAR-10 with less parameters. On ImageNet, our model achieves 75.2% top-1 accuracy under MobileNet settings (MobileNet V1/V2 [11,25]), while being 1.2× faster with measured GPU latency. The contributions of this paper are summarized as follows:
• We introduce a novel algorithm for network architecture search, which is applicable to various large-scale datasets as the memory and computation costs are similar to common neural network training.
• We propose a performance ranking hypothesis, which can be incorporated into the existing NAS algorithms to speed up its search.
• The proposed method achieves remarkable search efficiency, e.g., 2.4% test error on CIFAR-10 in 4 hours with 1 GTX1080Ti (6.0× faster compared with stateof-the-art algorithms), which is attributed to using our distribution learning that is entirely different from RL-based [2,31] methods and differentiable methods [19,28].
Related Work
As first proposed in [30,31], automatic neural network search in a predefined architecture space has received significant attention in the last few years. To this end, many search algorithms have been proposed to find optimal architectures using specific search strategies. Since most handcrafted CNNs are built by stacked reduction (i.e., the spatial dimension of the input is reduced) and norm (i.e. the spatial dimensionality of the input is preserved) cells [13,10,12], the works in [30,31] proposed to search networks under the same setting to reduce the search space. The works in [30,31,2] use reinforcement learning as a meta-controller, to explore the architecture search space. The works in [30,31] employ a recurrent neural network (RNN) as the policy to sequentially sample a string encoding a specific neural architecture. The policy network can be trained with the policy gradient algorithm or the proximal policy optimization. The works in [3,4,18] regard the architecture search space as a tree structure for network transformation, i.e., the network is generated by a farther network with some predefined operations, which reduces the search space and speeds up the search. An alternative to RL-based methods is the evolutionary approach, which optimizes the neural architecture by evolutionary algorithms [27,23].
However, the above architecture search algorithms are still computation-intensive. Therefore some recent works are proposed to accelerate NAS by one-shot setting, where the network is sampled by a hyper representation graph, and the search process can be accelerated by parameter sharing [22]. For instance, DARTS [19] optimizes the weights within two node in the hyper-graph jointly with a continuous relaxation. Therefore, the parameters can be updated via standard gradient descend. However, one-shot methods suffer from the issue of large GPU memory consumption. To solve this problem, ProxylessNAS [5] explores the search space without a specific agent with path binarization [7]. However, since the search procedure of ProxylessNAS is still within the framework of one-shot methods, it may have the same complexity, i.e., the benefit gained in Prox-ylessNAS is a trade-off between exploration and exploitation. That is to say, more epochs are needed in the search procedure. Moreover, the search algorithm in [5] is similar to previous work, either differential or RL based methods [19,31]. Different from the previous methods, we encode the path/operation selection as a distribution sampling, and achieve the optimization of the controller/proxy via distribution learning. Our learning process further integrates the proposed hypothesis to estimate the merit of each operation/path, which achieves an extremely efficient NAS search.
Architecture Search Space
In this section, we describe the architecture search space and the method to build the network. We follow the same settings as in previous NAS works [19,18,31] to keep the consistency. As illustrated in Fig. 2, the network is defined in different scales: network, cell, and node.
Node
Nodes are the fundamental elements that compose cells. Each node x i is a specific tensor (e.g., a feature map in convolutional neural networks) and each directed edge (i, j) denotes an operation o (i,j) sampled from the operation search space to transform node x i to another node x j , as illustrated in Fig. 2 Following [19] set of possible operations, denoted as O, consists of the following 8 operations:
(1) 3 × 3 max pool- ing. (2) no connection (zero). (3) 3 × 3 average pooling. (4) skip connection (identity). (5) 3 × 3 dilated convolution with rate 2. (6) 5 × 5 dilated convolution with rate 2. (7) 3×3 depth-wise separable convolution. (8) 5×5 depth-wise separable convolution.
We simply employ element-wise addition at the input of a node with multiple operations (edges). For example, in Fig. 2(b), B 2 has three operations, the results of which are added element-wise and then considered as B 2 .
Cell
A cell is defined as a tiny convolutional network mapping an H × W × F tensor to another H × W × F . There are two types of cells, norm cell and reduction cell. A norm cell uses the operations with stride 1, and therefore H = H and W = W . A reduction cell uses the operations with stride 2, so H = H/2 and W = W/2. For the numbers of filters F and F , a common heuristic in most human designed convolutional neural networks [10,13,15,26] is to double F whenever the spatial feature map is halved. Therefore, F = F for stride 1, and F = 2F for stride 2.
As illustrated in Fig. 2(b), the cell is represented by a DAG with 7 nodes (two input nodes I 1 and I 2 , four intermediate nodes B 1 , B 2 , B 3 , B 4 that apply sampled operations on the input and upper nodes, and an output node that concatenates the intermediate nodes). The edge between two nodes denote a possible operation according to a multinomial distribution p (node1,node2) in the search space. In training, the input of an intermediate node is obtained by element-wise addition when it has multiple edges (operations). In testing, we select the top K probabilities to generate the final cells. Therefore, the size of the whole search space is 2 × 8 |E N | , where E N is the set of possible edges with N intermediate nodes. In our case with N = 4, the total number of cell structures is 2 × 8 2+3+4+5 = 2 × 8 14 , which is an extremely large space to search, and thus requires efficient optimization methods.
Network
As illustrated in Fig. 2(a), a network consists of a predefined number of stacked cells, which can be either norm cells or reduction cells each taking the output of two previous cells as input. At the top of the network, global average pooling followed by a softmax layer is used for final output. Based on the Performance Ranking Hypothesis, we train a small (e.g., 6 layers) stacked model on the relevant dataset to search for norm and reduction cells, and then generate a deeper network (e.g., 20 layers) for evaluation. The overall CNN construction process and the search space are identical to [19]. But note that our search algorithm is different.
Methodology
In this section, our NAS method is presented. We first describe how to sample the network mentioned in Sec. 3 to reduce GPU memory consumption during training. Then, we present a multinomial distribution learning to effectively optimize the distribution parameters using the proposed hypothesis.
Sampling
As mentioned in Sec. 3.1, the diversity of network structures is generated by different selections of M possible paths (in this work, M = 8) for every two nodes. Here we initialize the probabilities of these paths as p i = 1 M in the beginning for exploration. In the sampling stage, we follow the work in [5] and transform the M real-valued probabilities
{p i } with binary gates {g i }: g = [1, 0, ..., 0] M with probability p 1 ... [0, 0, ..., 1] M with probability p M(1)
The final operation between nodes i and j is obtained by:
o (i,j) = o (i,j) * g = o 1 with probability p 1 ... o M with probability p M .(2)
As illustrated in the previous equations, we sample only one operation at run-time, which effectively reduces the memory cost compared with [19].
Multinomial Distribution Learning
Previous NAS methods are time and memory consuming. The use of reinforcement learning further prohibits the methods with the delay reward in network training, i.e., the evaluation of a structure is usually finished after the network training converges. On the other hand, as mentioned in Sec. 1, according to the Performance Ranking Hypothesis, we can perform the evaluation of a cell when training the network. As illustrated in Fig. 3, the training epochs and accuracy for every operation in the search space are recorded. Operations A is better than B, if operation A has fewer training epochs and higher accuracy.
Formally, for a specific edge between two nodes, we define the operation probability as p, the training epoch as H e , and the accuracy as H a , each of which is a real-valued column vector of length M = 8. To clearly illustrate our learning method, we further define the differential of epoch as:
∆H e = ( 1 × H e 1 − H e ) T ... ( 1 × H e M − H e ) T ,(3)
and the differential of accuracy as:
∆H a = ( 1 × H a 1 − H a ) T ... ( 1 × H a M − H a ) T ,(4)p i ← p i + α * ( j 1(∆H e i,j < 0, ∆H a i,j > 0)− j 1(∆H e i,j > 0, ∆H a i,j < 0)),(5)
where α is a hyper-parameter, and 1 denotes as the indicator function that equals to one if its condition is true.
As we can see in Eq. 5, the probability of a specific operation i is enhanced with fewer epochs (∆H e i,j < 0) and higher performance (∆H a i,j > 0). At the same time, the probability is reduced with more epochs (∆H e i,j > 0) and lower performance (∆H a i,j < 0). Since Eq. 5 is applied after every training epoch, the probability in the search space can be effectively converge and stabilize after a few epochs. Together with the proposed performance ranking hypothesis (demonstrated latter in Section 5), our multinomial distribution learning algorithm for NAS is extremely efficient, and achieves a better performance compared with other state-of-the-art methods under the same settings. Considering the performance ranking is consisted of different layers according to the hypothesis, to further improve the search efficiency, we replace the search network in [19] with another shallower one (only 6 layers), which takes only 4 GPU hours of searching on CIFAR-10.
To generate the final network, we first select the operations with highest probabilities in all edges. For nodes with multi-input, we employ element-wise addition with top K probabilities. The final network consists of a predefined number of stacked cells, using either norm or reduction cells. Our multinomial distribution learning algorithm is presented in Alg. 1.
Experiment
In this section, we first conduct some experiments on the CIFAR-10 to demonstrate the proposed hypothesis. Then, we compare our method with state-of-the-art methods on both search effectiveness and efficiency on two widely-used classification datasets including CIFAR-10 and ImageNet.
Datasets
We follow most NAS works [19,4,31,18] in their experiment datasets and evaluation metrics. In particular, we conduct most experiments on CIFAR-10 [14] which has 50, 000 training images and 10, 000 testing images. In architecture search, we randomly select 5, 000 images in the training set as the validation set to evaluate the architecture. The color image size is 32 × 32 with 10 classes. All the color intensities of the images are normalized to [−1, +1]. To further evaluate the generalization, after discovering a good cell on CIFAR-10, the architecture is transferred into a deeper network, and therefore we also conduct classification on ILSVRC 2012 ImageNet [24]. This dataset consists of 1, 000 classes, which has 1.28 million training images and 50, 000 validation images. Here we consider the mobile setting where the input image size is 224 × 224 and the number of multiply-add operations in the model is restricted to be less than 600M.
Implementation Details
In the search process, according to the hypothesis, the layer number is irrelevant to the evaluation of a cell structures. We therefore consider in total L = 6 cells in the network, where the reduction cells are inserted in the second and third layers, and 4 nodes for a cell. The network is trained for 100 epoches, with a batch size as 512 (due to the shallow network and few operation sampling), and the initial number of channels as 16. We use SGD with momentum to optimize the network weights w, with an initial learning rate of 0.025 (annealed down to zero following a cosine schedule), a momentum of 0.9, and a weight decay of 3 × 10 −4 . The learning rate of the multinomial parameters is set to 0.01. The search takes only 4 GPU hours with only one NVIDIA GTX 1080Ti on CIFAR-10.
In the architecture evaluation step, the experimental setting is similar to [19,31,22]. A large network of 20 cells is trained for 600 epochs with a batch size of 96, with additional regularization such as cutout [8], and path dropout of probability of 0.3 [19]. All the experiments and models of our implementation are in PyTorch [21].
On ImageNet, we keep the same search hyperparameters as on CIFAR-10. In the training procedure, The network is trained for 120 epochs with a batch size of 1024, a weight decay of 4×10 −5 , and an initial SGD learning rate of 0.4 (annealed down to zero following a cosine schedule).
Baselines
We compare our method with both human designed networks and other NAS networks. The manually designed networks include ResNet [10], DenseNet [13] and SENet [12]. For NAS networks, we classify them according to different search methods, such as RL (NASNet [31], ENAS [22] and Path-level NAS [4]), evolutional algorithms (AmoebaNet [23]), Sequential Model Based Optimization (SMBO) (PNAS [18]), and gradient-based (DARTS [19]). We further compare our method under the mobile setting on ImageNet to demonstrate the generalization. The best architecture generated by our algorithm on CIFAR-10 is transferred to ImageNet, which follows the same experimental setting as the works mentioned above. Since our algorithm takes less time and memory, we also directly search on Im-ageNet, and compare it with another similar baseline (low computation consumption) of proxy-less NAS [5].
Evaluation of the Hypothesis
We first conduct experiments to verify the correctness of the proposed performance ranking hypothesis. To get some intuitive sense of the hypothesis, we introduce the Kendall rank correlation coefficient, a.k.a. Kendall's τ [1]. Given two different ranks of m items, the Kendall's τ is computed as follows:
τ = P − Q P + Q ,(6)
where P is the number of pairs that are concordant (in the same order in both rankings) and Q denotes the number of pairs that are discordant (in the reverse order). τ ∈ [−1, 1], with 1 meaning the rankings are identical and -1 meaning a rank is in reverse of another. The probability of a pair in two ranks being consistent is p τ = τ +1 2 . Therefore, a τ = 0 means that 50% of the pairs are concordant.
We randomly sample different network architectures in the search space, and report the loss, accuracy and Kendall's τ of different epochs on the testing set. The performance ranking in every epoch is compared with the final performance ranking of different network architectures. As illustrated in Fig. 4, the accuracy and loss are hardly distinguished due to the homogeneity of the sampled networks, i.e., all the networks are generated from the same space. On the other hand, the Kendall coefficient keeps a high value (τ > 0, p τ > 0.5) in most epochs, generally approaching 1 as the number of epochs increases. It indicates that the architecture evaluation ranking has highly convincing probabilities in every epoch and generally becomes more close to the final ranking. Note that, the mean value of Kendall's τ for each epoch is 0.474. Therefore, the hypothesis holds with a probability of 0.74. Moreover, we discover that the combination of the hypothesis with the multinomial distribution learning can enhance each other. The hypothesis guarantees the high expectation when selecting a good architecture, and the distribution learning decreases the probability of sampling a bad architecture.
Results on CIFAR-10
We start by finding the optimal cell architecture using the proposed method. In particular, we first search neural architectures on an over-parameterized network, and then we evaluate the best architecture with a deeper network. To eliminate the random factor, the algorithm is run for several times. We find that the architecture performance is only slightly different with different times, as well as compar- ing to the final performance in the deeper network (<0.2), which indicates the stability of the proposed method. The best architecture is illustrated in Fig. 5.
The summarized results for convolutional architectures on CIFAR-10 are presented in Tab. 1. It is worth noting that the proposed method outperforms the state-of-theart [31,19], while with extremely less computation consumption (only 0.16 GPU days << 1,800 in [31]). Since the performance highly depends on different regularization methods (e.g., cutout [8]) and layers, the network architectures are selected to compare equally under the same settings. Moreover, other works search the networks using either differential-based or black-box optimization. We attribute our superior results based on our novel way to solve the problem with distribution learning, was well as the fast learning procedure: The network architecture can be directly obtained from the distribution when the distribution converges. On the contrary, previous methods [31] evaluate architectures only when the training process is done, which is highly inefficient. Another notable phenomena observed in Tab. 1 is that, even with randomly sampling in the search space, the test error rate in [19] is only 3.49%, which is comparable with the previous methods in the same search space. We can therefore reasonable conclude that, the high performance in the previous methods is partially due to the good search space. At the same time, the proposed method quickly explores the search space and generates a better architecture. We also report the results of hand-crafted networks in Tab. 1. Clearly, our method shows a notable enhancement, which indicates its superiority in both resource consumption and test accuracy.
Results on ImageNet
We also run our algorithm on the ImageNet dataset [24]. Following existing works, we conduct two experiments with different search datasets, and test on the same dataset. As reported in Tab. 1, the previous works are time consuming on CIFAR-10, which is impractical to search on ImageNet. Therefore, we first consider a transferable experiment on ImageNet, i.e., the best architecture found on CIFAR-10 is directly transferred to ImageNet, using two initial convolution layers of stride 2 before stacking 14 cells with scale reduction (reduction cells) at 1, 2, 6 and 10. The total number of flops is decided by choosing the initial number of channels. We follow the existing NAS works to compare the performance under the mobile setting, where the input image size is 224 × 224 and the model is constrained to less than 600M FLOPS. We set the other hyper-parameters by following [19,31], as mentioned in Sec. 5.1.2. The results in Tab. 2 show that the best cell architecture on CIFAR-10 is transferable to ImageNet. Note that, the proposed method achieves comparable accuracy with state-of-the-art methods, while using much less computation resource. The extremely minimal time and GPU memory consumption makes our algorithm on ImageNet feasible. Therefore, we further conduct a search experiment on Im-ageNet. We follow [5] to design network setting and the search space. In particular, we allow a set of mobile convolution layers with various kernels {3, 5, 7} and expanding ratios {1, 3, 6}. To further accelerate the search, we directly use the network with the CPU and GPU structure obtained in [5]. In this way, the zero and identity layer in the search space is abandoned, and we only search the hyper-parameters related to the convolutional layers. The results are reported in Tab. 3, where we have found that our MdeNAS achieves superior performance compared to both human-designed and automatic architecture search methods, with less computation consumption. The best architecture is illustrated in Fig. 6.
Conclusion
In this paper, we have presented MdeNAS, the first distribution learning-based architecture search algorithm for convolutional networks. Our algorithm is deployed based on a novel performance rank hypothesis that is able to further reduce the search time which compares the architecture performance in the early training process. Benefiting from our hypothesis, MdeNAS can drastically reduce the computation consumption while achieving excellent model accuracies on CIFAR-10 and ImageNet. Furthermore, Mde-NAS can directly search on ImageNet, which outperforms the human-designed networks and other NAS methods. 32×56×56 32×56×56 32×56×56 32×56×56 48×28×28 48×28×28 48×28×28 48×28×28 88×14×14 88×14×14 104×14×14 104×14×14 104×14×14 104×14×14 216×7×7 216×7×7 216×7×7 216×7×7 360×7×7 32×56×56 56×28×28 56×28×28 112×14×14 112×14×14 128×14×14 128×14×14 256×7×7 128×14×14 256×7×7 256×7×7 256×7×7 432×7×7 Figure 6. Optimal CPU and GPU structures found by MdeNAS with various kernel sizes K = {3, 5, 7} and expansion ratios E = {1, 3, 6}.
We found that the structures with different layer numbers show different preferences. GPU structures (shallower) tend to select wide kernel sizes and high expansion ratios, and at the same time, CPU structures (deeper) prefer to chose small kernel sizes and low expansion ratios.
propagations. In Advances in neural information processing systems, pages 3123-3131, 2015. 3
Figure 2 .
2Searching networks with different scales. (a) A network consists of stacked cells, and each cell takes the output of two previous cells as input. (b) A cell contains 7 nodes, two input nodes I1 and I2, four intermediate nodes B1, B2, B3, B4 that apply sampled operations on the input nodes and upper nodes, and an output node that concatenates the outputs of the four intermediate nodes. (c) The edge between two nodes denotes a possible operation according to a multinomial distribution in the search space.
(c). There are three types of nodes in a cell: input node x I , intermediate node x B , and output node x O . Each cell takes the previous output tensor as an input node, and generates the intermediate nodes x i B by applying sampled operations o (i,j) to the previous nodes (x I and x j B , j ∈ [1, i)). The concatenation of all intermediate nodes is regarded as the final output node.
Figure 3 .
3The overall search algorithm: (1) Sample one operation in the search space according to the corresponding multinomial distribution with parameters θ. (2) Train the generated network with one forward and backward propagation. (3) Test the network on the validation set and record the feedback (epoch and accuracy). (4) Update the distribution parameters according to the proposed distribution learning algorithm. In the right table, the epoch number of operation 1 is 10, which means that this operation is selected 10 times among all the epochs.
Algorithm 1 :
1Multinomial Distribution Learning Input: Training data: D t ; Validation data: D v ; CNN model: F 1 . Output: Cell operation probabilities: P 2 . for t= 1,...
Figure 4 .
4The test error (left), top 1 accuracy (middle), and Kendall's τ (right) of different architectures. The error and accuracy curves are entangled, since they are sampled from the same search space defined in Section 3. Therefore, we further calculate the Kendall's τ between every epoch and the final result. Note that the Kendall's τ > 0 can be considered as a high value, which means more than half of the rankings are consistent.
Figure 5 .
5Detailed structure of the best cells discovered on CIFAR-10. The definition of the operations on the edges is in Section 3.1. In the reduction cell (up) the stride of operations on 2 input nodes is 2, and in the norm cell (down), the stride is 1.
variables H e , H a , ∆H e and ∆H a are calculated by the evaluation results. The parameters of the multinomial distribution can be updated through:where 1 is a column vector with length 8 and all its elements
being 1, ∆H e and ∆H a are 8 × 8 matrices, where ∆H e
i,j =
H e
i − H e
j , ∆H a
i,j = H a
i − H a
j . After one epoch training,
the corresponding
Table 2 .
2Comparison with state-of-the-art image classification methods on ImageNet with the mobile setting. All the NAS networks are searched on CIFAR-10, and then directly transferred to ImageNet.Architecture
Test Error
Params
Search Cost
Search
(%)
(M)
(GPU days)
Method
ResNet-18 [10]
3.53
11.1
-
manual
DenseNet [13]
4.77
1.0
-
manual
SENet [12]
4.05
11.2
-
manual
NASNet-A [31]
2.65
3.3
1800
RL
AmoebaNet-A [23]
3.34
3.2
3150
evolution
AmoebaNet-B [23]
2.55
2.8
3150
evolution
PNAS [18]
3.41
3.2
225
SMBO
ENAS [22]
2.89
4.6
0.5
RL
Path-level NAS [4]
2.49
5.7
8.3
RL
DARTS(first order) [19]
2.94
3.1
1.5
gradient-based
DARTS(second order) [19]
2.83
3.4
4
gradient-based
Random Sample [19]
3.49
3.1
-
-
MdeNAS (Ours)
2.40
4.06
0.16
MDL
Table 1. Test error rates of our discovered architecture, human-designed network and other NAS architectures on CIFAR-10. To be fair, we
select the architectures and results with similar parameters (< 10M) and training conditions (same epochs and regularization).
Architecture
Accuracy (%)
Params
Search Cost
Search
Top1
Top5
(M)
(GPU days)
Method
MobileNetV1 [11]
70.6
89.5
6.6
-
manual
MobileNetV2 [25]
72.0
91.0
3.4
-
manual
ShuffleNetV1 2x (V1) [29]
70.9
90.8
∼5
-
manual
ShuffleNetV2 2x (V2) [20]
73.7
-
∼5
-
manual
NASNet-A [31]
74.0
91.6
5.3
1800
RL
AmoebaNet-A [23]
74.5
92.0
5.1
3150
evolution
AmoebaNet-C [23]
75.7
92.4
6.4
3150
evolution
PNAS [18]
74.2
91.9
5.1
225
SMBO
DARTS [19]
73.1
91.0
4.9
4
gradient-based
MdeNAS (Ours)
74.5
92.1
6.1
0.16
MDL
Table 3 .
3Comparison with state-of-the-art image classification on ImageNet with the mobile setting. The networks are directly searched on ImageNet with the MobileNetV2[25] backbone.Model
Top-1
Search time GPU latency
GPU days
MobileNetV2
72.0
-
6.1ms
ShuffleNetV2
72.6
-
7.3ms
Proxyless (GPU) [5] 74.8
4
5.1ms
Proxyless (CPU) [5] 74.1
4
7.4ms
MdeNAS (GPU)
75.2
2
6.2ms
MdeNAS (CPU)
73.8
2
4.8ms
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arxiv |
A SIMPLE SVD ALGORITHM FOR FINDING HIDDEN PARTITIONS
15 Apr 2014
Van Vu
A SIMPLE SVD ALGORITHM FOR FINDING HIDDEN PARTITIONS
15 Apr 2014arXiv:1404.3918v1 [math.CO]
Finding a hidden partition in a random environment is a general and important problem, which contains as subproblems many famous questions, such as finding a hidden clique, finding a hidden coloring, finding a hidden bipartition etc.In this paper, we provide a simple SVD algorithm for this purpose, answering a question of McSherry. This algorithm is very easy to implement and works for sparse graphs with optimal density.
The problem and a new algorithm
The hidden partition problem is the following: let X be a set of n vertices with a partition X = ∪ k i=1 X i ; for all 1 ≤ i ≤ j ≤ n and any x ∈ X i , y ∈ X j , we put a random edge between x and y with probability p ij . Given one such random graph, the goal is to recover the sets X i . This problem is of importance in computer science and statistics and contains as special cases several well-studied problems such as the hidden clique, hidden bisection, hidden coloring, clustering etc (see, for instance, [1,2,3,6,7,8,9,11,12,15,16,18,13,21,19] and the references therein). In what follows, we refer to X i as clusters.
In an influential paper [24], Mc Sherry provided a (randomized) polynomial time algorithm that solves the general hidden partition problem for a large range of parameters. As corollary, he derived several earlier results obtained for special cases.
The general idea [24] (and in many earlier works on clustering) is to find a good geometric representation of the vertices. We say that a representation is perfect if there is a number r > 0 such that • Vertices in the same cluster have distance at most r from each other.
• Vertices from different clusters have distance at least 4r from each other.
Once a perfect representation is obtained, it is easy to find the clusters. If r is known, then the solution is obvious. If r is not known, then there are several simple algorithms. For instance, one can create a minimal spanning tree (with respect to the distances) on the vertices and then remove the largest k − 1 edges. In what follows, we put all these simple algorithms under a subroutine called Clustering by Distances and the reader can choose his/her favorite to implement. Our main goal is to present a simple way to obtain a perfect representation.
1991 Mathematics Subject Classification. 26C10, 30C15. V. Vu is supported by research grants from NSF and Airforce.
In the rest of the paper, let s u := |X i | if u ∈ X i and s := min u∈X s u = min i |X i |. We assume that n is sufficiently large, whenever needed. Asymptotic notation are used under the assumption n → ∞. All explicit constants (such as the 4 above) are adhoc and we make no attempt to optimize them.
A popular way to find a perfect representation is to project the points of X (seen as vectors in R n ) onto a properly chosen low-dimensional subspace H. The main technical part of Mc Sherry's algorithm is a subroutine called CP roj (Combinatorial Projection), which creates H in a combinatorial way. The inputs in this subroutine are a matrixÂ, parameters k, s, and a properly chosen threshold τ . Let P be the probability matrix (p ij ) 1≤i,j≤n . For a vertex u ∈ X, u denotes the corresponding column in P . Define
∆ := min u − v ,
where the minimum is taken over all pairs u, v belonging to different clusters. Mc Sherry proved [24] Theorem 1. Assume that σ 2 ≫ log 6 n/n is an upper bound on the variances of the entries. There is a constant C > 0 such that if
(1) ∆ ≥ Cσk 1/2 ( n s + log n ǫ ),
the above algorithm (with a proper choice of the threshold τ ) recovers the partition with probability 1 − ǫ with respect to the random graph and k −1 with respect to the auxiliary random bits.
The main open question raised by Mc Sherry in [24] is to find a more natural and simpler algorithm, which does not involve the subroutine CPROJ (see [24,Section 4.4]). The goal of this paper is to answer this question.
To this end, M k denotes the subspace spanned by the first k left singular vectors of a matrix M . LetP be our input, namely the adjacency matrix of a random graph generated by P . Arguably, the most natural choice for H would beP k (SVD), which leads to the algorithm below While SVD I could well win the contest for being the simplest algorithm, it is not easy to analyze in the general case. In what follows, we analyze a slightly more technical alternative, SVD II, which is a variant of an algorithm proposed in [24, Section 1].
Algorithm 4: SVD II
(0) Randomly partition X into two subsets Y and Z. Let B be the adjacency matrix of the bipartite graph between Y and Z. Let Y 1 be a random subset of Y by selecting each element with probability 1/2 independently and let be the submatrix of B formed by the columns indexed by Y 1 .
(1) Project the columns of B indexed by Y 2 := Y \Y 1 on k .
(2) Run Clustering by Distances on the projected points.
Compared to SVD I, the extra steps in SVD II are the random partitions in Step (0) done in order to reduce the correlation. (A careful reading of [24] reveals that one also need an extra partition in Algorithm 2 to make the analysis go through.)
Notice that SVD II gives a partition of Y 2 , not X. There are many ways to extend it to a partition of X. For instance, we can run the algorithm l times (for some small l) and find partitions of Y 1 2 , . . . , Y l 2 , where Y i 2 are random subsets of X with density 1/4 (the input graph is the same, only the random partitions are different). If a cluster C in Y i 2 and a cluster C ′ in Y i ′ 2 intersect, then they must belong to the same cluster in X and we can merge them. If we choose l = 3 log n, say, then with probability 1 − o(n −1 ), all vertices of X must belong to some Y i 2 and we recover the clusters X 1 , . . . , X k at the end. We can also first find the partitions of Y 1 , Y 2 and Z 1 , Z 2 by reversing the role of Y 1 and Y 2 and Y and Z and find which four clusters must belong to an original cluster by looking at the edge densities; we omit the details.
Beside being simple, SVD II is also very convenient to implement, as its main step, the computation of the projection onto k (given as input) is a routine operation (SVD) which appears in most standard mathematical packages.
Let us now analyze SVD II. For convenience, we assume that P has rank k. The general case when P can have a smaller rank is discussed later. Let λ be the least non-trivial singular value of P . Theorem 2. There is a constant C > 0 such that the following holds. Assume that σ 2 ≥ C log n n and s ≥ C log n, k = o((n/ log n) 1/2 ). Then SVD II clusters Y 2 correctly with probability 1 − o(n −1 ) if one of the following two conditions is satisfied
• Condition 1. ∆ ≥ C(σ n s + √ log n). • Condition 2. ∆ ≥ C(σ n s + σ √ k log n + σ √ nk λ )
If we omit the assumption s ≥ C log n, the statement still holds but with probability
1 − o(n −1 ) − c k i=1 e −|Xi|/c for some constant c.
Remark 3. We would like to point out a few remarks
• The lower bound σ 2 ≥ C log n/n is optimal, up to the value of C. If σ 2 < log n/n, then there are many isolated points, which can be assigned to any cluster. • We can reduce the failure probability O(n −1 ) to O(n −K ) for any constant K at the cost of increasing the constant C.
Let us now consider the performance of SVD II on various subproblems. We allow the value of C to be flexible in order to omit smaller order terms for convenience. It is instructive to compare the corollaries below with Corollaries 1,2,3 from [24].
Hidden clique. In this problem, k = 2, s is the size of the clique, and ∆ = (1 − p) √ s, where p is the density of the random graph. Condition 1 becomes
(1 − p)s 1/2 ≥ C(p 1/2 n s + log n) which is satisfied if s ≥ C( √ np + √ log n). As np = Θ(σ 2 n) = Θ(log n), this simplifies to s ≥ C √ np.
Corollary 4. There is a constant C such that for any p ≥ C log n n and s ≥ C √ np, SVD II finds the hidden clique of size s with probability 1 − o(1).
Hidden Coloring.
Here k is the number of color classes, each has size n/k; ∆ = p 2n/k; s = n/k; σ 2 = p(1 − p). The singular values of P are k−1 k n, 1 k n, . . . , 1 k n. If p ≥ 1/k, Condition 1 is p n/k ≥ C(p 1/2 √ k + log n) which is satisfied for k = o((n/ log n)) 1/3 ).
If p < 1/k, then the bound λ ≥ σ √ ns holds, and the ∆ bound in Condition 2 is
p n/k ≥ C( pk log n + k log n √ n ) which is satisfied if p ≥ C k 3/2 log n n .
Corollary 5. There is a constant C such that the following holds. For any k = o((n/ log n) 1/3 and edge density .99 > p ≥ C k 3/2 log n n , SVD II finds the hidden k-coloring with probability 1 − O(n −1 ).
Hidden Bipartition. Let the two densities be .99 ≥ p > q > 0. We have k = 2, ∆ = |p − q|n 1/2 , s = n/2, σ 2 = Θ(p). The two singular values of P are (p + q)n and (p − q)n. Condition 2 requires
p−q p 1/4 ≥ C log n n .
Corollary 6. There is a constant C such that the following holds Let .99 > p > q ≥ C log n/n be edge densities such that p−q p 1/4 ≥ C log n n then SVD II finds the hidden bipartition with probability 1 − o(n −1 ).
One can replace p 1/4 in the denominator by a better term p 1/2 by considering an approximate algorithm; see Corollary 11.
The rest of the paper is organized as follows. In the next section, we present a few technical lemmas and prove Theorem 2 and Theorem 10 in Section 3. In Section 4, we discuss variants of SVD II, including an approximate version which works under weaker assumptions.
Technical lemmas
Lemma 7 (Projection of a Random Vector). There are constants C 1 , C 2 such that the following holds. Let ξ = (ξ 1 , . . . , ξ n ) be a random vector in R n whose coordinates ξ i are independent random variables with mean 0 and variance at most σ 2 ≤ 1. Let H be a subspace of dimension d and Π H ξ be the length of the orthogonal projection of ξ onto H. Then
P(Π H X ≥ σ √ d + C 1 log n) ≤ n −3 . Furthermore, if H has an orthornormal bases v 1 , . . . , v d such that max 1≤i≤d v i ∞ ≤ α, then P(Π H X ≥ C 2 √ d(σ + α log n)) ≤ n −3 .
We prove this lemma in the appendix.
Lemma 8 (Norm of a random matrix). There is a constant C 0 > 0 such that the following holds. Let E be a symmetric matrix whose upper diagonal entries e ij are independent random variables where e ij = 1 − p ij or −p ij with probabilities p ij and 1 − p ij , respectively, where 0 ≤ p ij ≤ 1. Let σ 2 := max ij p ij (1 − p ij . If σ 2 ≥ C 0 log n/n, then
P( E ≥ C 0 σn 1/2 ) ≤ n −3 .
If σ 2 ≥ log 4 n n , the statement is a corollary of [26,Theorem 1.4]. For smaller σ, one can prove this lemma using the ǫ-net approach by Kahn and Szemeredi [20]. We omit the details, which is very similar to the proof of This lemma is a well known result in numerical linear algebra, known as Davis-Kahan-Wedin theorem; see [5,10,28,17].
Proof of Theorems 2
Let A be the probability matrix p ij corresponding toÂ. As A is a large random submatrix of P , it is not hard to show that λ k (A) = Θ(λ k (P )) with high probability (we provide a verification of this fact at the end of the proof). In the rest of this proof, we assume
(2) λ k (A) ≥ c 0 σ √ ns,
for some constant c 0 > 0.
We view the adjacency matrix (between Y 1 and Z) as a random perturbation of A, := A+E, where the entries e ij of E are independent and e ij = 1 − p ij with probability p ij and −p ij with probability 1 − p ij . We denote byû, u, e u the columns corresponding to a vertex u inÂ, A, E, respectively. All matrices are of size approximately n/2 × n/4 by the definitions of Y, Z and Y 1 , Y 2 .
Our leading idea is that the random perturbation E does not change A k too much, thus hopefully the projections onto k and A k differ by only a small amount. The heart of the matter, of course, is to bound this error term. While inviting, a straightforward application of Lemma 9 is too crude in the general case (it does lead to some simple solution for some subproblems in certain range of parameters). We will still make use of this lemma, but for a quite different purpose.
For simplicity, we assume in the rest of the proof that s ≥ C log n. For a sufficiently large C, this implies that with probability 1 − o(n −1 ), each cluster X i intersects Z in at least |X i |/3 elements. Thus, the distance between two columns (belonging to different clusters) in A is at least ∆/3. We aim to show that with high probability PÂ kû − u < ∆/12 for all u ∈ Y 2 ; this will provide a perfect geometric representation. If there is no lower bound on s, then the probability that the random partition has this property is at least 1 − c k i=1 e −|Xi|/c for some constant c > 0.
For a fixed u, by the triangle inequality
PÂ kû − u ≤ PÂ k (û − u) + (PÂ k − I)u = PÂ k e u + (PÂ k − I)u .
To bound the second term, we follow an argument from [24] and consider
(PÂ k − I)A = (PÂ k − I)Â − (PÂ k − I)E.
The spectral norm of the first term is λ k+1 (A k ) ≤ λ k+1 (A) + E = E , as A has rank at most k. The spectral norm of the second term is also at most E . Thus, by Lemma 8, by probability at least 1 − n −3
(PÂ k − I)A ≤ 2 E ≤ C 0 σn 1/2 , for some constant C 0 .
Let χ u be the unit vector s −1/2 u I u where I u is the indicator vector for the cluster containing u, we have
(PÂ k − I)A ≥ (PÂ k − I)Aχ u = s 1/2 u (PÂ k − I)u .
Combining the last two inequalities and using the union bound, we conclude that with probability at least 1 − n −2
(PÂ k − I)u ≤ C 0 σ n s u ,
for all u ∈ X.
Now we tend to the first term, whose analysis is more involved. By the first part of Lemma 7, PÂ k e u ≤ σk 1/2 + C 1 log n with probability 1 − o(n −2 ), for a properly chosen constant C 1 . As sk ≤ n, the term σk 1/2 is at most σ n/s and can be omitted. This yields that if ∆ ≥ C 0 σ n/s + C 1 log n then the algorithm succeeds with probability at least 1 − o(n −1 ). This proves the first part of the theorem concerning Condition 1.
To prove the second part of the theorem, let us reconsider the distance PÂ k e u . Notice that if s ≤ 10k log n, then Condition 2 implies Condition 1 (with some modification on the value of C).
Thus, in what follows, we can assume s ≥ 10k log n.
Rewrite =
A + E and let v be a singular vector of A. Recall that |X i ∩ Z| ≥ 1 3 |X i | = s i /3 for all i By symmetry, each coordinate in v is repeated at least s/3 times, thus v ∞ ≤ 2s −1/2 . Furthermore, by Lemma 9 and Lemma 8, we have with probability 1 − o(n −2 ) that
sin(A k , k ) ≤ C 0 σ √ n λ which implies that for any unit vector v ∈ k , v ∞ ≤ 2s −1/2 + C 0 σ √ n λ ≤ C ′ 0 s −1/2
by the condition on λ, with some properly chosen constant C ′ 0 . Using the second part of Lemma 7, we conclude that with probability 1 − o(n −2 ), PÂ k e u ≤ C(σk 1/2 + s −1/2 log n) for all u and some properly chosen constant C, concluding the proof.
For the sake of completeness, let us show that with high probability, the least (non-trivial) singular value of P and A are essentially the same, up to a constant factor. We first compare the singular values of P with the singular values ofP , the probability matrix of the bipartite graph spanned by Y and X. Using Chernoff's bound, one can easily show that with probability at least 1 − n −2
(3) ||X i ∩ Y | − |X i |/2| ≤ 5 |X i | log n for all 1 ≤ i ≤ k.
We use the fact that for any matrix M of rank k λ k (M ) = inf rank(M ′ )=k−1 M − M ′ F . For simplicity, let us assume for a moment that |X i ∩ Y | = |X i |/2. LetP ′ be the matrix that define λ k (P ). We define P ′ , a rank (k − 1) approximation of P , by extendingP ′ as follows. For the block indexed by X i \Y , simply copy the block ofP ′ corresponding to X i ∩ Y . It is trivial that P ′ has rank k − 1 and
P − P ′ 2 F = 2 P −P ′ 2 F which implies λ k ≤ √ 2λ k (P ).
With the same argument, we can compare λ k (P ) with λ k (B) and the later with λ k (A), each time losing a factor of √ 2. At the end it would give λ k (P ) ≤ 2 3/2 λ k (A).
To make the argument precise, we need to remove the assumption |X i ∩ Y | = |X i |/2. Using (3), we can create a matrix P ′ such that
P − P ′ 2 F ≤ 2 P −P ′ 2 F + 5 k i=1 |X i | log nσ 4 .
On the other hand, the extra term 5 k i=1
|X i | log nσ 4 is less than 1 4 λ k (P ) 2 by the assumption of the theorem. Thus, we can use the above estimate to get a slightly weaker bound λ k (P ) ≤ 2λ k (P ), completing the proof.
Variants
Dimension and Density.
In the case rank A = l < k, it makes more sense to project ontô A l rather than onto k ; the rest of the algorithm and analysis remains the same.
If we do not know either k or l in advance. we can modify SVD II slightly as follows. The idea is to define the essential rank of to be the largest index l such that λ l (Â) ≥ C 3 σn 1/2 , for a properly chosen C 3 , and replace the projection onto k by the projection onto l . After redefining λ := λ l P , the content of Theorem 2 remains the same. Its proof also remains the same, expect few nominal changes. The error term caused by smaller singular values is not going to effect the final conclusion.
The only information we need in SVD II (using essential dimension) is the value of σ. Even if this information is not known, we can still solve the problem by considering a sequence of O(log n) trials with σ 1 = log n √ n , σ i = 2σ i−1 and run SVD II in each case. Each trial will output a clustering and it is easy to decide which one is correct by considering the degree densities of the vertices from one cluster to the others.
An approximate Solution.
In practice, one is often satisfied with an approximate solution. We say that a partition
X = ∪ k i=1 X ′ i is ǫ-correct if |X i \X ′ i | ≤ ǫ|X i |.
Similarly, we say that a geometric representation of X is ǫ-perfect if there are points x 1 , . . . , x k with distance at least 4r from each other so that at least (1 − ǫ)|X i | points from X i has distance at most r to x i . One can use an ǫ-perfect representation to find an ǫ-correct partition. It is worth mentioning that in various situations, an ǫ-correct partition can be upgraded to a fully correct one by a simple "correction" procedure, as shown in the following example:
Hidden bipartition. Assume p > q. Let X = X ′ 1 ∪ X ′ 2 be an ǫ-correct partition, for some small ǫ (say ǫ = .1). Then both X ′ i have size at most 1 2 (1 − ǫ)n. Assume |X i \X ′ i | ≤ ǫn/2; it follows that |X ′ 1 \X 1 | ≤ ǫn. With probability 1 − n −2 , the following holds. For any u ∈ X, let d u be the number of its neighbors in X ′ 1 . If u ∈ X 1 , then
d u ≥ |X 1 ∩ X ′ 1 |p + |X ′ 1 \X 1 |q − 10 np log n = D 1 .
On the other hand, if u ∈ X 2 , then
d u ≤ |X 1 ∩ X ′ 1 |q + |X ′ 1 \X 1 |p + 10 np log n = D 2 .
It is clear that if (p − q) ≥ 30 p log n n , then D 1 > D 2 . Thus, one can correct the partition by defining X 1 be the set of n/2 vertices u with largest d u .
Corollary 11. There is a constant C such that the following holds Let .99 > p > q ≥ C log n/n be edge densities such that p−q p 1/2 ≥ C log n n then the approximation algorithm with correction finds the hidden bipartition with probability 1 − o(n −1 ).
Proof of Theorem 10. We first bound PÂ k e u . Recall that
E PÂ k e u 2 ≤ σ 2 k.
By Markov's inequality, it follows that P( PÂ k e u ≥ Kσk 1/2 ) ≤ K −2 . We call a vertex u good if PÂ k e u ≤ Kσk 1/2 . For a sufficiently large C (depending on K), all good vertices will be clustered correctly. Moreover, choosing K ≥ 2ǫ −1/2 , the probability for u being good is at least 1 − ǫ/4, thus the expectation of the number of good elements in X i is at least |X i |(1 − ǫ/4). As the good events are independent, Chernoff's bound implies that with probability 1 − n −2 , at least |X i |(1 − ǫ) points from X i are good. This completes the proof.
Appendix A. Proof of Lemma 7
Notice that the function Π H (X) is 1-Lipschitz and convex, thus by Talagrand's inequality [25] for any t > 0 P(Π H X ≥ µ + t) ≤ 2 exp(−t 2 /4) where µ is the mean of Π H (X). We do not know µ; however, we can bound from above. Slightly abusing the notation, let Π := (π ij ) denote the projection matrix onto H, then
E|Π H X| 2 = EX T ΠX = n i=1 π ii Eξ 2 i ≤ σ 2 n i=1 π ii = dσ 2 .
Combining this with the concentration inequality, it is not hard to show that µ ≤ σd 1/2 + O(1), concluding the proof of the first part of the lemma.
The second part follows immediately from Claim 12. Let (a 1 , . . . , a n ) be real numbers such that i a 2 i = 1 and |a i | ≤ α for all i. Let ξ i be independent random variables with mean 0 and E|ξ i | k ≤ σ 2 for all k ≥ 2. Let S := n i=1 a i ξ i . Then P(|S| ≥ 4(σ log n + α log n) ≤ 2n −3 .
To prove Claim 12, notice that for any 0 < t ≤ α −1 we have
E exp(tS) = i E exp(ta i ξ i ) = i (1 + σ 2 a 2 i t 2 2! + t 3 a 3 i Eξ 3 i 6! + . . . )
Since Eξ k i ≤ σ 2 for all k ≥ 2 and t|a i | ≤ 1, the right most formula is
≤ i (1 + σ 2 t 2 a 2 i ) ≤ exp(σ 2 t 2 ).
Markov's inequality yields P(S ≥ T ) ≤ exp(−tT + t 2 σ 2 ).
To optimize the RHS, let us consider two cases Case 1. σ ≥ α √ log n. Take T = 4σ √ log n and t = √ log n σ ≤ α −1 . With this setting −tT +t 2 σ 2 = −3 log n.
Case 2. σ < α √ log n. Take T = 4α log n and t = α −1 . In this setting, −tT + t 2 σ 2 ≤ −4 log n + log n = −3 log n.
One can bound P(−S ≤ T ) the same way.
Algorithm 1 :
1Combinatorial Projection (CProj) (1) While there are at least s/2 unclassified nodes, choose an unclassified node v i randomly and define T i := {u| PÂ T (Â T vi −Â T u ) ≤ τ }. Mark each u ∈ T i as classified. (2) Assign each remaining node to the T i with the closest projected v i . (3) Letĉ i be the characteristic vector of T i . (4) Return Pĉ, the projection onto the span of theĉ i . Algorithm 2: Mc Sherry's algorithm (1) Randomly partition the set {1, . . . , n} into two parts A and B. LetÂ,B be the submatrices of the adjacency matrix formed by columns from A and B. (2) Let P 1 = CP roj(B), P 2 = CP roj(Â) and computeH = [P 1 (Â)|P 2 (B)]. (3) Run Clustering by Distances on the projected points.
the columns ofP ontoP k .(2) Run Clustering by distances on the projected points.
Feige and Ofek for [14, Theorem 1.1]. Lemma 9 (Perturbation bound). Let M, N be matrices where δ := λ k (M ) − λ k+1 M > 0. Then sin ∠(M k , (M + N ) k ) ≤ δ −1 N .
Theorem 10 .
10Given ǫ > 0, there is a constant C > 0 such that the following holds. If σ 2 ≥ probability 1 − o(n −1 ) the projection in SVD II produce an (1 − ǫ)-perfect representation of the point sin Y 2 .
Acknowledgement. The author would like to thank NSF and AFORS for their support and K. Luh for his careful proof reading.
A spectral technique for coloring random 3-colorable graphs. N Alon, N Kahale, SIAM Journal on Computing. 266N. Alon, N. Kahale. A spectral technique for coloring random 3-colorable graphs. SIAM Journal on Comput- ing, 26(6):1733-1748, 1997.
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Spectral analysis of large dimensional random matrices. Z Bai, J W Silverstein, SpringerZ. Bai, J.W. Silverstein. Spectral analysis of large dimensional random matrices. Springer, 2009.
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Eigenvalues and graph bisection: An average-case analysis. R B Boppana, Proceedings of the 28th Annual Symposium on Foundations of Computer Science. the 28th Annual Symposium on Foundations of Computer ScienceR.B. Boppana. Eigenvalues and graph bisection: An average-case analysis. Proceedings of the 28th Annual Symposium on Foundations of Computer Science, 280-285, 1987.
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Finding hidden cliques of size N/e in nearly linear time. Y Deshpande, A Montanari, arXiv:1304.7047Available atmath.PRY. Deshpande, A. Montanari. Finding hidden cliques of size N/e in nearly linear time. Available at arXiv:1304.7047 [math.PR].
Fast solution of some random NP-hard problems. M E Dyer, A M Frieze, 27th Annual Symposium on Foundations of Computer Science. M.E. Dyer, A.M. Frieze. Fast solution of some random NP-hard problems. 27th Annual Symposium on Foundations of Computer Science, 221-336, 1986.
Spectral techniques applied to sparse random graphs. U Feige, E Ofek, Random Structures Algorithms. 272251275U. Feige and E. Ofek, Spectral techniques applied to sparse random graphs, Random Structures Algorithms 27 (2005), no. 2, 251275.
Finding and certifying a large hidden clique in a semirandom graph. U Feige, R Krauthgamer, Random Structures and Algorithms. 162U. Feige, R. Krauthgamer. Finding and certifying a large hidden clique in a semirandom graph. Random Structures and Algorithms, 16 (2): 195-208, 2000.
Finding hidden cliques in linear time. U Feige, D Ron, DMTCS proc. AM. U. Feige, D. Ron. Finding hidden cliques in linear time. DMTCS proc. AM, 189-204, 2010.
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A new look at independence. M Talagrand, Ann. Probab. 241M. Talagrand. A new look at independence. Ann. Probab., 24 no. 1, 1-34, 1996.
Spectral norm of random matrices. V Vu, Combinatorica. 276V. Vu. Spectral norm of random matrices. Combinatorica, 27(6):721-736, 2007.
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Perturbation bounds in connection with singular value decomposition. P.-Å Wedin, BIT Numerical Mathematics. 121P.-Å. Wedin. Perturbation bounds in connection with singular value decomposition. BIT Numerical Mathe- matics, 12(1):99-111, 1972.
| {'fraction_non_alphanumeric': 0.06788781951774811, 'fraction_numerical': 0.03049432543378523, 'mean_word_length': 3.301186943620178, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 19, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Finding a hidden partition in a random environment is a general and important problem, which contains as subproblems many famous questions, such as finding a hidden clique, finding a hidden coloring, finding a hidden bipartition etc.In this paper, we provide a simple SVD algorithm for this purpose, answering a question of McSherry. This algorithm is very easy to implement and works for sparse graphs with optimal density.', 'arxivid': '1404.3918', 'author': ['Van Vu '], 'authoraffiliation': [], 'corpusid': 8561244, 'doi': '10.1017/s0963548317000463', 'github_urls': [], 'n_tokens_mistral': 9829, 'n_tokens_neox': 8656, 'n_words': 5660, 'pdfsha': 'c0748fca237e7a3061fce1180314d73ba9aa397c', 'pdfurls': ['https://arxiv.org/pdf/1404.3918v1.pdf'], 'title': ['A SIMPLE SVD ALGORITHM FOR FINDING HIDDEN PARTITIONS', 'A SIMPLE SVD ALGORITHM FOR FINDING HIDDEN PARTITIONS'], 'venue': []} |
arxiv |
A Probe of New Resonant Structures with Four Fermion Processes at a 1 TeV e + e − Collider
September 94
J Layssac
F M Renard
C Verzegnassi
Dipartimento di Fisica Teorica
Physique Mathématique et Théorique
CNRS-URA 768
Université Montpellier II
F-34095, Cedex 5Montpellier
Università di Trieste Strada Costiera
11 MiramareI-34014Trieste
Sezione di Trieste
Italy
A Probe of New Resonant Structures with Four Fermion Processes at a 1 TeV e + e − Collider
September 94arXiv:hep-ph/9409429v2 29 Sep 1994
Oblique contributions from vector particles that are strongly coupled to the known gauge bosons are calculated for the case of various observables at a 1 TeV e + e − collider. Constraints provided by LEP 1 results are taken into account in a model-independent way through a dispersion relation technique. Depending on the assumed theoretical properties, mass limits of several TeV should be observable.
In this paper we investigate the potential of a 1 TeV e + e − collider for the indirect search of Technicolour-like vector bosons. High precision tests at LEP 1 have already set rather stringent bounds on Technicolour models, following the original proposal of Peskin and Takeuchi [1]. They are based on the effect to the quantity S defined in [1], that appear through the so-called [2] oblique parts of the one-loop contributions. We here apply a general formalism established in recent publications [3], [4], [5], which allows to calculate the relevant oblique contributions to a number of processes in future higher energies e + e − experiments.
The main idea is that of expressing the various effects in the form of a once-subtracted dispersion integral, and of fixing the necessary subtraction constants by suitable modelindependent LEP 1 results. In this way, we are led to a compact "representation" of several observables which present two main advantages. The first one is that it allows to express the New Physics contributions through convergent integrals. The second one is that LEP 1 constraints are automatically incorporated in the expressions of the observables. For example, the cross section for muon production at cm energy √ q 2 , σ µ (q 2 ), at one loop level takes the form
σ SE µ (q 2 ) = 4πq 2 3 [ α(M 2 Z ) q 2 ] 2 [1 + 2D γ (q 2 )] + 1 (q 2 − M 2 Z ) 2 + M 2 Z Γ 2 Z [ 3Γ l M Z ] 2 [1 − 2D Z (q 2 ) − 16s 2 1 v 1 1 − v 2 1 D γZ(q 2 ) ](1)
Here Γ l is the leptonic Z width, α(M 2 Z ) = [1 ± 0.001]/128.87 and
D γ (q 2 ) ≡ ∆α(q 2 ) − ∆α(M 2 Z ) = − q 2 − M 2 Z π P ∞ 0 ds Im F γ (s) (s − q 2 )(s − M 2 Z ) (2) D Z (q 2 ) ≡ Re [I Z (q 2 ) − I Z (M 2 Z )] = q 2 − M 2 Z π P ∞ 0 ds s Im F ZZ (s) (s − q 2 )(s − M 2 Z ) 2 (3) D γZ (q 2 ) ≡ Re [∆κ ′ (q 2 ) − ∆κ ′ (M 2 Z )] = q 2 − M 2 Z π P ∞ 0 ds Im F κ ′ (s) (s − q 2 )(s − M 2 Z ) (4) (F ′ κ = c 1 /s 1 F Zγ , s 2 1 c 2 1 = πα √ 2G µ M 2 Z , s 2 1 = 1 − c 2 1 ≃ 0.217 , v 1 = 1 − 4s 2 1 ).
The Imaginary parts which appear in these expressions are constructed from the selfenergies. For Technicolour models, they are separately gauge-invariant. Similar representations were established for several other observables like forwardbackward asymmetries, polarization asymmetries and ratios of hadron to muon production. For each observable we finally obtain an expression that include the full effect of the oblique correction at one-loop in the form:
O(q 2 ) = c 0 [1 + c γ D γ (q 2 ) + c Z D Z (q 2 ) + c γZ D γZ (q 2 )](5)
where the analytic expressions of the various coefficients have been derived for each observable and computed numerically in [4].
We have presently used this formalism to calculate the possible effects of a pair of vector (V) and axial vector (A) resonances strongly coupled to the photon and to the Z. The parameters which enter the expressions of the imaginary parts of the various spectral functions are the couplings F V,A and the masses M V,A (assumed to be larger than √ q 2 ). We have treated two different theoretical models.
In model(I) we consider a Technicolour-like framework in which we exploited the validity of the two Weinberg sum rules [6]. We only retain their very general consequence, i.e. the positivity of S. In a zero-width approximation (in the actual computation of the effects on the observables we have used a finite width description of the V,A resonances) one has:
S = 4π[ F 2 V M 2 V − F 2 A M 2 A ] = 4πF 2 π M 2 V [1 + M 2 V M 2 A ](6)
The present experimental constraint on S is [7]:
− 0.9 ≤ S ≤ 0.4
In this first model only the positive upper bound is effective.
In model (II) we release the constraints due to the Weinberg sum rules. This choice has the consequence of introducing one more degree of freedom. It eliminates the theoretical relation between F V and F A . As a consequence S can now take negative values, and in addition the strength of the ratios F V /M V and F A /M A is no more bounded. We decided to consider the limiting case consisting in a "strongly interacting regime" for which the value of the ratio F V /M V is equal to twice the QCD value
F V M V = 2 f ρ m ρ = 1 √ 2π (8)
Then , for every choice of F 2 V /M 2 V , F 2 A /M 2 A is allowed to saturate both limits imposed by the experimental bounds on S.
Assuming a certain accuracy for the measurement of each observable, one accordingly obtains the observability limit of the self-energy effect that is translated in an upper bound on the masses M V,A . For 1 T eV e + e − collider the assumed accuracies are of a relative one percent for σ µ , A F B,µ , A LR,h ,R (5) , two percent for R b,µ and five percent for A τ .
Results are shown in Fig. 1,2 for both models respectively, the different curves corresponding to the various observables, and the shaded area to the combined overall mass bounds. In model (I) the resulting bounds on M V,A are located in the 2 T eV range, and rather strongly correlated. The only hadronic observable which contributes appreciably to the bound is A LR,h , that allows to improve the pure leptonic result by about 200 GeV . In model (II) the effect of releasing the validity of the Weinberg sun rule is roughly that of increasing the bounds on (M V , M A ) from the 2 T eV region to the 4 T eV region.
Compared to the results obtained in [3], [4], an improvement by a factor two is found as compared to the case of a 500 GeV collider and by a factor 6 to 8 as compared to the LEP 2 case. http://arxiv.org/ps/hep-ph/9409429v2
The mass range of M V and M A which is explored in this indirect way should be able to give a definite hint of the existence of Technicolour-like resonances or of any other strongly coupled vector boson. ;h (dot-dashed) and A (dashed), using the Weinberg sum rules and the experimental information on S. The lighter shaded domain represents the result of combining quadratically the two leptonic limits. The darker one corresponds to the domain allowed by the leptonic and the hadronic limits. The two full lines correspond to M releasing the Weinberg sum rules but imposing the limitation on F V =M V , from (vertical,dotted), A (vertical, dashed), A LR;h (dotdashed), R b; (short dashed), R (5) (dotted), A FB; (long dashed). The shaded domains have the same meaning as in Fig.1. The two full lines now corre-
. M E Peskin, T Takeuchi, Phys. Rev. 46381M.E. Peskin and T. Takeuchi, Phys. Rev. D46 (1991) 381 .
Physics at LEP. B W Lynn, M E Peskin, R Stuart In, J. Ellis and R. Peccei eds., CERN1B.W. Lynn, M.E. Peskin and R. Stuart in "Physics at LEP", J. Ellis and R. Peccei eds., CERN 86-02 (1986), Vol 1.
. J Layssac, F M Renard, C Verzegnassi, Phys. Rev. 484037J. Layssac, F.M. Renard and C. Verzegnassi, Phys. Rev. D48 (1993) 4037 .
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. E Fahri, L Susskind, Phys. Rev. 203404E. Fahri and L. Susskind, Phys. Rev. D20 (1979) 3404 .
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arxiv |
Dissipative time crystals originating from parity-time symmetry
Yuma Nakanishi
Department of Physics
Tokyo Institute of Technology
2-12-1 Ookayama Meguro-ku152-8551TokyoJAPAN
Tomohiro Sasamoto
Department of Physics
Tokyo Institute of Technology
2-12-1 Ookayama Meguro-ku152-8551TokyoJAPAN
Dissipative time crystals originating from parity-time symmetry
(Dated: January 11, 2023)
This study aims to provide evidence regarding the emergence of a class of dissipative time crystals when PT symmetry of the systems is restored in collective spin systems with Lindblad dynamics. First, we show that a standard model of boundary time crystals (BTCs) satisfies the Liouvillian PT symmetry, and prove that BTC exists only when the stationary state is PT symmetric in the large-spin limit. Also, a similar statement is confirmed numerically for another BTC model. In addition, the mechanism of the appearance of BTCs is discussed through the development of a perturbation theory for a class of the one-spin models under weak dissipations. Consequently, we show that BTCs appear in the first-order correction when the total gain and loss are balanced. These results strongly suggest that BTCs are time crystals originating from PT symmetry. arXiv:2203.06672v2 [quant-ph]
Introduction. Crystals are ubiquitous many-body systems wherein continuous space-translation symmetry is spontaneously broken. Similarly, dynamic manybody states that spontaneously break continuous timetranslation symmetry, namely (continuous) time crystals, were proposed by Wilczek in 2012 [1]. However, it has been proven that time crystals do not exist in ground and equilibrium states, at least for long-range interacting systems [2,3]. In non-equilibrium systems such as Floquet systems [4][5][6][7][8][9] and dissipative systems [10][11][12][13][14][15][16][17][18][19][20][21][22][23][24], (discrete or dissipative) time crystals have been observed theoretically and experimentally.
Dissipative time crystals are non-trivial states characterized by persistent periodic oscillations at late times induced by coupling with the external environment [20]. In particular, a kind of dissipative time crystal, called boundary time crystal (BTC), has been often studied recently [12][13][14][15][16][17][18]. BTCs were first introduced using a collective spin model with Lindblad dynamics [12], which describes a collection of spin-1/2 with all-to-all couplings interacting collectively with external Markovian baths. This model could be derived by tracing out the bulk (environment) degrees of freedom while leaving the boundary (system) degrees of freedom. Further, it has been confirmed that persistent oscillatory phenomena at late times emerge only in the thermodynamic limit. Even though such phenomena were already noted 40 years earlier as cooperative resonance fluorescence [25], it should be emphasized that there are various novel aspects in recent studies of BTCs. In particular, the importance of Liouvillian eigenvalues has been realized [12,13,18] because the dynamics can be fully understood in terms of their eigenvalues and eigenmodes. In addition, recent developments of the spectral theory of dissipative phase transitions [26,27] and exact solutions of the Liouvillian spectrum [28][29][30][31][32][33][34] have also increased the interest in investigating Liouvillian eigenvalues. * [email protected]
The dynamical properties of BTCs are often investigated via numerical calculations of Liouvillian eigenvalues [12,13], mean-field approximation method [12][13][14]25], and quantum trajectory method [35]. In particular, BTCs must satisfy two conditions for the Liouvillian spectrum, which characterize non-stationary periodic oscillations at late times: (i) there exist pure imaginary eigenvalues iλ j and (ii) the quotient of each pure imaginary eigenvalue is a rational number λ j /λ k ∈ Q for all j, k. Moreover, BTCs are characterized by static properties such as the existence of a highly mixed and low-entangled eigenmode with a zero eigenvalue [13,17,[36][37][38]. Also, the necessity of Hamiltonians' Z 2 symmetry has been argued recently [13]. However, the physical origin of the emergence of BTCs has not yet been elucidated, and most studies on Liouvillian eigenvalues for BTCs have been numerical.
Phase transitions accompanied by parity-time (PT ) symmetry breaking, namely PT phase transitions [39,40], are also phenomena wherein persistent oscillations emerge at late times in non-equilibrium systems. These are well-known phenomena in the context of non-Hermitian Hamiltonians (NHHs) [41] with exactly balanced gain and loss, and have been widely investigated in a variety of physical experimental systems, such as mechanics [42], photonics [43], plasmonics [44], electronics [45], and open quantum systems without quantum jumps [46]. Mathematically, the Hamiltonian H is considered to be PT symmetric if it holds that [H, PT ] = 0, where P is a parity operator and T is a time reversal operator [39,40]. In addition, PT phase transitions in systems with Lindblad dynamics, hereafter referred to as Liouvillian PT phase transitions, have been recently discussed, and their understanding has been progressing [47][48][49][50][51][52][53].
This study attempts to demonstrate the emergence of a class of dissipative time crystals when PT symmetry is realized. We focus on a specific class of systems, collective spin systems with Lindblad dynamics, and provide the results about the Liouvillian eigenvalues and stationary state for some specific examples. First, we show that the PT symmetric phase of an open two-spin model with Liouvillian PT symmetry [47][48][49] is a BTC. Here, an n-spin model is a system with n-collective spin operators in the interaction term. Second, we show that an open collective spin model with interaction owing to a transverse magnetic field and excitation decay (hereafter referred to as the one-spin BTC model) satisfies the proposed definition of the Liouvillian PT symmetry [48,49] if the parity transformation is appropriately chosen. In addition, we prove that the PT symmetry breaking of the stationary state occurs at the BTC phase transition point in the large-spin limit. Next, we confirm that the generalized one-spin BTC model studied in Ref. [13] also has Liouvillian PT symmetry. Further, we numerically show that the stationary state exhibits PT symmetry in the BTC phase. Finally, we perform a perturbative analysis of a class of one-spin models, including the one-spin BTC model under weak dissipation. Consequently, we show that BTCs appear in the first-order correction owing to the balanced total gain and loss. These results strongly suggest that BTCs in collective spin systems are time crystals originating from PT symmetry.
Liouvillian spectrum and Liouvillian PT symmetry.
In open quantum systems where the evolution of states is completely positive and trace-preserving (CPTP) Markovian, the time evolution of the density matrix ρ(t) is described by the Lindblad master equation (GKSL equation) [54][55][56][57] as follows
dρ dt = −i[H, ρ(t)] + i D[L i ]ρ,(1)
where H is a Hamiltonian, L i is the Lindblad operator, and the dissipation superoperators D[L i ] are defined as
D[L i ]ρ = 2L i ρL † i − L † i L i ρ − ρL † i L i .
Here, index i labels the Lindblad operators.
The Lindblad master equation (1) is linear in ρ, thus, it can be rewritten with a superoperator, which is a linear operator acting on a vector space of linear operators, as follows:
dρ(t) dt =Lρ(t).(2)
Here,L is referred to as the Liouvillian superoperator. The eigenvalues λ i and eigenmodes ρ i of the Liouvillian can be obtained by solving the following equation:
Lρ i = λ i ρ i .(3)
It is generally known that Re [56,57]. Here, we assume the existence of a unique steady state and set the eigenvalues as 0 = |Re[λ 0 ]| < |Re[λ 1 ]| ≤ |Re[λ 2 ]| ≤ · · · . The steady state is then written as ρ ss = ρ 0 /Tr[ρ 0 ]. In addition, the absolute value of the real part of the second maximal eigenvalue is one-spin PT model. These models satisfy Liouvillian PT symmetry (4) when the parity operator is (a) the exchange of two spins, (b) the reflection of the basis of S z , (c) the identity operator or the reflection of the basis of S z . referred to as the Liouvillian gap [26,27] and determines the slowest relaxation rate. Closing the Liouvillian gap is necessary for dissipative phase transitions in steady state [26,27].
[λ i ]≤ 0, ∀i; ifLρ i = λ i ρ i , thenLρ † i = λ * i ρ † i
It should be noted that imaginary eigenvalues emerge only in the thermodynamic limit for Liouvillian PT phases and BTCs. Therefore, the thermodynamic limit and the long-time limit are not commutative, that is, lim S →∞ lim t→∞ ρ(t) lim t→∞ lim S →∞ ρ(t) [58]. In the former case, the steady state is static without oscillation. We refer to the state lim t→∞ ρ(t) as the "stationary state", whereas, in the latter case, the state at late times includes oscillating non-decay modes.
Many studies on Liouvillian PT symmetry have been conducted recently [47][48][49][50][51][52][53]; however, the definition of Liouvillian PT symmetry has not been uniquely determined yet. In our arguments, we adopted (a slightly modified version of ) the definition proposed in Ref. [49] because similar properties to those of NHH PT phase transitions have been confirmed in a specific two-spin model that satisfies this definition, as mentioned below. A Liouvillian associated with the Lindblad equation (1) is considered to be PT symmetric if the following relation holds.
L[PT(H); PT (L µ ), µ = 1, 2, · · · ] =L[H; L µ , µ = 1, 2, · · · ],(4)
where
PT(H) = PT H(PT ) −1 = PHP −1 , PT (L µ ) = PL † µ P −1 .
Here, P and T are parity and time reversal operators, andH means the complex conjugation of H.
Further, PT(H) denotes the conventional PT transformation of Hamiltonian. However, in the PT transformation, the time-reversal transformation of dissipations represents an exchange of creation and annihilation operators.
Two-spin Liouvillian PT symmetric model is BTC.
To investigate the usefulness of the definition (4), an open two-spin-S model with exactly balanced gain and loss [ Fig.1 (a)] was actively investigated [47][48][49]. Here, S denotes the total spin. Denoting the subsystems as A and B, the Lindblad equation is expressed as
∂ ∂t ρ = −ig[H, ρ] + Γ 2S D[S +,A ]ρ + Γ 2S D[S −,B ]ρ,(5)
where H = (S +,A S −,B + H.c.)/2S , S ± = S x ± iS y , and g and Γ are the strengths of the interaction and dissipation, respectively. Note that it decays to a unique steady state for a finite S as spin ladder operators S ± exist as one of the dissipation operators [60]. This model satisfies the definition (4) when the parity transformation is the exchange of two spins, and a dissipative phase transition occurs at Γ = g when S = ∞. Moreover, the symmetry parameter of a stationary state [59], which provides a measure of the parity symmetry of the density operator, changes from 0 to a finite value at the transition point when S = ∞ (see Eq.(A.1) in the Supplemental Material [61]). This suggests the occurrence of the PT symmetry breaking of the stationary state [49].
In addition, the eigenvalue structure and dynamics were obtained when S = ∞ [47]. In particular, in the PT phase, when S = ∞, the eigenvalues are expressed as
λ = in g 2 − Γ 2 + r,(6)
where n ∈ N and r ∈ R − 0 , and information on degeneracy is omitted. Here, R − 0 is a set comprising real negative numbers including zero. Equation (6) shows that commensurable pure imaginary eigenvalues exist. Therefore, some physical quantities such as magnetization oscillate periodically at late times. However, in the PT broken phase, all eigenvalues are real (see Eq.(A.3) in the Supplemental Material [61]); thus, the state decays toward a steady state without oscillation, which is the same as the stationary state. This behavior corresponds to one of the NHH PT phase transitions.
Moreover, it can be easily confirmed that the PT phase is a boundary time crystal because the eigenvalue structure satisfies the two conditions of nonstationary periodic dynamics only in the thermodynamic limit. (Detailed explanation of this model is provided in the Supplemental Material [61].) In addition, if a model satisfies the definition (4), it has been shown that there exists a stationary state that approaches the identity eigenmode ρ ∝ 1l in the limit of zero dissipation rate [48].
Boundary time crystals. Among various BTC models, we first focused on the one-spin BTC model investigated in Refs. [12,25,[36][37][38]. The Lindblad equation is expressed as
d dt ρ = −2ig[S x , ρ] + κ S D[S − ]ρ,(7)
where g and κ are the strengths of interaction and dissipation, respectively [ Fig.1 (b)]. In this model, the stationary state is solved exactly for finite S [36,37] as
ρ ss = 1 D 2S n,n =0 i κ g S − S n −i κ g S + S n ,(8)
where D is the normalization constant. In this model, it was found that various physical quantities, such as magnetization and purity, clearly change; namely, the dissipative phase transition occurs at κ/g = 1 when S → ∞ [36][37][38].
Moreover, the eigenvalue structure and dynamics were numerically investigated above and below the transition point [12]. For the BTC phase (κ/g < 1), the real parts of many eigenvalues approach zero for an enormous S , and the imaginary parts are plotted at regular intervals for any S . This suggests that pure imaginary eigenvalues exist when S → ∞ and that these eigenvalues are commensurable. In other words, the imaginary part of the eigenvalues can be written as Im[λ] = −icq when S → ∞, where q ∈ N is the sector and c is a real number dependent on κ/g. Here, the imaginary part is invariant within the same sector. However, for the BTC broken phase (κ/g > 1), there are no eigenvalues with a non-zero imaginary part that approaches the imaginary axis as S increases.
BTCs are PT symmetric phases. Here, we first show that the one-spin BTC model (7) has the Liouvillian PT symmetry (4). We choose the parity operator to reflect the basis of S z , which acts on each spin operator as follows:
PS z P −1 = −S z , PS ± P −1 = S ∓ .(9)
We can verify that PT(S x ) = PS x P −1 = S x and PT (S − ) = P(S − ) † P −1 = S − hold, that is, the model has Liouvillian PT symmetry (4).
Then, we analytically show that PT symmetry breaking of the stationary state (8) occurs at the BTC phase transition point. Using (9), the conventional PT transformation of the stationary state can be written as:
PT ρ ss PT = 1 D 2S n,n =0 −i κ g S + S n i κ g S − S n .(10)
In the limit S → ∞, we show that −i κ g S + S n and i κ g S − S n are commutative only when κ/g < 1.
This implies that the stationary state (8) of the one-spin BTC model is PT symmetric in the BTC phase but not in the BTC broken phase. The details of the proof are provided in Sec. I.A of the Supplemental Material [61].
Next, we consider the open one-spin model studied in Ref. [13] whose Lindblad equation is given by
d dt ρ = −i[H, ρ] + κ − S D[S − ]ρ + κ + S D[S + ]ρ,(11)
where H = S (g z s p z z + g x s p x x ), and p z , p x ∈ N. and s z , s x are normalized spin operators S z /S , S x /S . This model has not been solved analytically; however, it has been numerically observed that BTCs appear when p z is even [13]. By choosing the parity operator as a reflection of the basis of S z as before, the model can have Liouvillian PT symmetry (4) if p z is even.
To investigate the PT symmetry breaking of a stationary state, we introduced the PT -symmetry parameter Q PT ,
Q PT (ρ) := 1 Z i, j |(ρ − PT ρPT ) i j |,(12)
where Z :
= i, j |(ρ) i, j | + |(PT ρPT ) i j | is a normalization constant, and thus 0 ≤ Q PT ≤ 1. Further, (ρ) i, j is the (i, j) element of matrix ρ. If Q PT is zero, ρ has PT symmetry. Figures 2 (a) and (b)
show the purity and PT -symmetry parameter Q PT of the stationary state for p z = 2, p x = 1. In the BTC phase (i.e., the phase with almost zero purity), Q PT is close to 0. Further, Fig.2 (c) shows that Q PT decreases in the BTC phase with increase in S . Therefore, these results suggest that the PT symmetry of the stationary state is unbroken in the thermodynamic limit, whereas it is broken in the BTC broken phase. In addition, all elements of |ρ ss − PT ρ ss PT | are close to 0 in the BTC phase [ Fig.2 (c)], whereas certain elements have finite values in the broken BTC phase ( Fig.2 (d)). Here, |ρ| denotes the matrix that accepts the absolute value of each matrix element ρ. These results indicate that the stationary state exhibits PT symmetry only in the BTC phase. In Fig C.2 in the Supplemental Material [61], we numerically investigated the time evolution and quantum trajectory of the normalized magnetization and normalized magnetization of the stationary state.
We also consider the one-spin model studied in Refs. [31,63,64] whose Liouvillian is expressed aŝ
Lρ = −ig[S x , ρ] + κ(1 + p) S D[S + x ]ρ + κ(1 − p) S D[S − x ]ρ,(13)
where −1 ≤ p ≤ 1 and S ± x := S y ± iS z . The model is BTC only when p = 0. When p = 0 the model is solvable for any S [31], the eigenvalues λ l,q are expressed as
λ l,q = igq − 2κ S [|q| + l(1 + l + 2|q|)],(14)
where q = {−S , −S + 1, ..., S } is the sector and l = {0, 1, ..., 2S − |q|}, which satisfies the two conditions for the emergence of non-stationary oscillating dynamics only in the thermodynamic limit. We can also discuss the relationship between the BTCs and PT symmetry for this model. Upon choosing the parity operator as the identity operator or reflection of (11) for
p z = 2, p x = 1, κ + /g z = 0, S =23. These are (a) purity, (b) PT symmetry parameter Q PT , (c) S -dependence of Q PT for κ − /g z = 0.5, g x /g z = 3 (orange circle), (d) |ρ ss − PT ρ ss PT | for κ − /g z = 0.5, g x /g z = 3 (orange circle), (e) |ρ ss − PT ρ ss PT | for κ − /g z = 2, g x /g z = 3 (black triangle),
where |ρ| implies the matrix taking the absolute value for each element of the matrix ρ. Here, elements are computed on the basis of the z-magnetization. These results indicate that the stationary state has PT symmetry only in the BTC phase. Here, we have used QuTip [62] to obtain the stationary state numerically. the basis of S z , this model (13) satisfies Liouvillian PT symmetry only when p = 0. This implies that this model is BTC only when it has Liouvillian PT symmetry. In addition, the stationary state ρ ss ∝ 1l has PT symmetry for p = 0. In the following, we refer to this model with p = 0 as the one-spin PT model [ Fig.1 (c)].
Understanding the mechanism of appearance of BTCs. Let us focus the eigenvalues with the smallest real part for the one-spin PT model, namely λ l=0,q in Eq. (14). The corresponding eigenmodes ρ 0,q are proportional to
(S + x ) |q| for q < 0 and (S − x ) q for q > 0 [63]
. These exact specific eigenmodes facilitate an understanding of the BTC's mechanism. For example, for
q = 1 we calculated −ig[S x , S − x ] = igS − x in the coherent part, and κ S (D[S + x ] + D[S − x ])S − x = κ S ([S + x , S − x ]S − x + S − x [S − x , S + x ]) = 2κ S (S x S − x − S − x S x ) = 2κ S [S x , S − x ] = − 2κ S S − x ,(15)
in dissipative parts. [The calculation for any q is provided in the Supplemental Material [61]; Eqs. (A.6), and (A.7).] Moreover, it has a 1/S -dependence in dissipative parts owing to the cancellation of several terms and the use of commutation relations. Thus, purely imaginary eigenvalues emerge when p = 0 and S → ∞, implying that they are BTC. However, when p 0, such cancellations of terms are generally unexpected. Indeed, the Liouvillian gap is not closed, even for S → ∞ [31] and no time crystals emerge.
Next, we investigated a class of the one-spin models using perturbation theory [65,66] under weak dissipations, whose Liouvillian is expressed aŝ
Lρ = −ig[S x , ρ] + κ S µ D[L µ ]ρ,(16)L µ = α µ S + x + β µ S − x + γ µ S x ,(17)
where α µ , β µ , γ µ ∈ C. This class includes the one-spin BTC model and the model (13). We can show that BTCs appear in first-order perturbation under a weak dissipation rate κ if and only if it holds that
µ |α µ | 2 = µ |β µ | 2 .(18)
Here, this condition can be regarded as exactly balanced total gain and loss on the x-basis. Note that BTCs basically appear under weak dissipations. Therefore, if it is not BTC in the first-order correction under weak dissipations, it may not be BTC for all dissipation regimes. First, we apply the degenerated perturbation theory to the model (13). For the prescription of the n-degenerate case, the first-order eigenvalue correctionλ (1) n,i can be obtained by solving the following equation:
L (n) ψ n,i =λ (1) n,i ψ n,i (i = 1, 2, ..., n),(19)
where L (n) and ψ n,i are the nsquare matrix and n coefficient vector, respectively. (see Eq.(D.10) in the Supplemental Material [61]). Choosing the non-perturbative LiouvillianL 0 as a coherent part of the model (13), namelyL 0 · = −ig[S x , ·], the non-perturbative eigenvalues are (2S + 1 − |q|)degenerated for each sector q. Then, L (2S +1−|q|) is a tridiagonal matrix with real-number elements. In particular, for p = 0, it becomes a symmetric matrix because of the balanced gain and loss (see Eq.(D.20) in the Supplemental Material [61]). In addition, all high-order corrections are zero because sector q is invariant for the model (13). Thus, perturbative analysis up to the first-order correction yields exact solutions.
Considering these aspects, we also performed perturbation theory on a class of the one-spin models (16). The non-perturbative LiouvillianL 0 was again chosen to be a coherent part, and consequently, the same tridiagonal real matrix L (2S +1−|q|) was obtained except for a constant multiplication and a sum of scalar multiplications. Therefore, its eigenvalues have the same properties, and the BTCs emerge for the symmetric case, but not for the non-symmetric case within the first-order perturbation scheme. In the Supplemental Material [61], we provide details of our proof and numerically indicate that certain properties of the BTC phase transition can be caught up to the second-order corrections for the one-spin BTC model.
Choosing the parity operator as a reflection of the basis of S z , the Liouvillian PT symmetry (4) guarantees the condition (18). Therefore, model (16) is BTC when it exhibits Liouvillian PT symmetry within the first-order perturbation scheme. Note that conservation is not always true, that is, the condition (18) does not necessarily imply Liouvillian PT symmetry. This suggests that the definition of Liouvillian PT symmetry leaves room for further improvements.
Finally, we provide an example. When (16) that satisfies the condition (18), the normalized magnetization S z /S oscillates and the relaxation time increases with increase in S [ Fig.3 (a)]. Furthermore, Figs.3 (c) and 3 (d) show that the real parts of the eigenvalues decrease to zero with an increase in S and that the imaginary parts are invariant even as S increases. These results imply that BTC emerges in the thermodynamic limit.
L 1 = −i(S + x − S − x )/2 = S z in the model
Summary and discussion. In this study, we provided evidence that dissipative time crystals originating from PT symmetry exist in collective spin systems. In partic-ular, we showed that BTCs are only such examples. In addition, we performed perturbation analysis for a class of one-spin models and showed that BTCs appear in the first-order correction because of the exactly balanced total gain and loss.
Finally, we discuss the robustness of our results. For a class of the one-spin models (16), the BTCs were stable for perturbations of the dissipations satisfying Eq. (18). It was also stable in case of perturbations of Hamiltonian terms that do not break the Liouvillian PT symmetry (4), such as S 2n z or S n x (n ∈ N) [12,13]. However, the rigidity of the periodic time, which is a property of discrete time crystals, did not appear because the periodic time is generally the variant for the dissipation and interaction strength.
As a natural extension, investigation of the relation- . Then we call it the two-spin PT model. Using the HP transformation [67] and the third quantization [28,29], the physical quantities and eigenvalue structure can be obtained when S = ∞. In particular, for Γ g = Γ l = Γ, symmetry parameter ∆ and purity, µ := Tr[ρ 2 ] in the stationary state are given by
∆ = | S +,A S −,A − S +,B S −,B | S +,A S −,A + S +,B S −,B =1 − g Γ 2 , (A.2) µ = 1 − g Γ 2 , (A.3)
for Γ/g > 1, and ∆ = µ = 0 for Γ/g < 1 when S = ∞. Also, the Liouvillian eigenvalues λ for each phase are given by
λ = in g 2 − Γ 2 + r, (Γ/g < 1), −2(m + β AF M + + m − β AF M − ), (Γ/g > 1), (A.4)
where r ∈ R − 0 , n ∈ N, m ± ∈ Z + and β AF M ± = (Γ ± g)/2 [47]. Note that the information on degeneracy is omitted in Eq.(A.4). For Γ/g < 1, namely in the PT phase, we can easily confirm that (i) there exist pure imaginary eigenvalues iλ j and (ii) the quotient of each pure imaginary eigenvalue is a rational number only when S = ∞. Therefore, we can find that the PT phase of the two-spin PT symmetric model is a boundary time crystal.
Let's mention here the properties of the symmetry parameter ∆ and the PT symmetry parameter Q PT . If Q PT (ρ) = 0, the density matrix ρ is exactly PT symmetric. However, even if ∆(ρ) = 0, the density matrix ρ is not always PT symmetric. Rather, it shows parity symmetry of the diagonal terms in the S z basis representation. If ρ is Hermitian, the diagonal terms are real, and then ∆(ρ) = 0 is the necessary condition for the PT symmetry of the stationary state. Fig.A.1 (a), (b) show the phase diagram when S = ∞ [47] and the numerical calculation of the PT symmetry parameter Q PT in the stationary state for S = 7. This shows that Q PT is less than 1 in the PT phase, while it is almost 1 in the PT broken phase. Fig.A.1 (c) shows the S -dependence of Q PT in the stationary state for some dissipation strength. The symmetry parameter Q PT decreases as S increases for each dissipation strength. These imply that the PT symmetry breaking in the stationary state occurs in the thermodynamic limit.
B. One-spin PT model
We consider the one-spin PT model whose Liouvillian is given bŷ
Lρ = −ig[S x , ρ] + κ S D[S + x ]ρ + κ S D[S − x ]ρ. (A.5)
This model is equivalent to the model (13) when p = 0. It has been exactly solved for any S [31], and the eigenvalues are given as Eq. (14). In particular, for l = 0, the eigenmodes ρ 0,q are proportional to (S + x ) |q| for q < 0 and (S − x ) q for q > 0 [63]. Indeed, substituting (S − x ) q for ρ in the model (A.5),
L(S − x ) q = −ig[S x , (S − x ) q ] + κ S D[S + x ](S − x ) q + κ S D[S − x ](S − x ) q = iqg(S − x ) q + κ S (2S + x (S − x ) q S − x − S − x S + x (S − x ) q − (S − x ) q S − x S + x ) + κ S (2S − x (S − x ) q S + x − S + x S − x (S − x ) q − (S − x ) q S + x S − x ) = iqg(S − x ) q + κ S (S + x (S − x ) q S − x − S − x S + x (S − x ) q + S − x (S − x ) q S + x − (S − x ) q S + x S − x ) (A.6)
Here, the several terms of dissipations are canceled out in Eq.(A.6). Further, it can be calculated aŝ
L(S − x ) q = iqg(S − x ) q + κ S ([S + x , S − x ](S − x ) q + (S − x ) q [S − x , S + x ]) = iqg(S − x ) q + κ S (2S x (S − x ) q − 2(S − x ) q S x ) = iqg(S − x ) q + 2κ S [S x , (S − x ) q ] = iqg(S − x ) q − 2κq S (S − x ) q = q(ig − 2κ S )(S − x ) q , (A.7)
where we use the commutation relations Furthermore, choosing the parity operator to be the identity operator or the reflection of the basis of S z , this model (13) satisfies the Liouvillian PT symmetry. Also, the stationary state ρ ss ∝ 1l has PT symmetry.
[S ± x , S ∓ x ] = ±2S x and [(S ± x ) n , S x ] = ∓n(S ± x ) n .
II. PROOF OF THE PT SYMMETRY BREAKING IN THE STATIONARY STATE FOR THE ONE-SPIN BTC MODEL
We show that the stationary state ρ ss (8) has PT symmetry for κ/g < 1, namely it satisfies ρ ss = PT ρ ss PT , while it is not PT symmetric for κ/g > 1. Firstly, we consider the case where κ/g < 1. Let us start from a well-known commutation relation of S n + and S − [37],
S n + S − = S − S n + + n(n − 1)S n−1 + + 2nS n−1 + S z . (B.1)
By using Eq.(B.1), we can write down the commutation relation of S n + and S n − as
S n + S n − = S n − S n + + n(n − 1) n k=1 S k−1 − S n−1 + S n −k − + 2n n k=1 S k−1 − S n−1 + S z S n −k − . (B.2)
Therefore, the commutation relation of −i κ
−i κ g S + S n , i κ g S − S n = +(−1) n n(n − 1) S 2 i κ g n+n n k=1 S − S k−1 S + S n−1 S − S n −k + (−1) n 2n S i κ g n+n n k=1 S − S k−1 S + S n−1 S z S S − S n −k . (B.
3)
The first term on the right-hand side in Eq.(B.3) approaches 0 when S → ∞ since it holds that κ g n+n n(n − 1)
S 2 n k=1 S − S k−1 S + S n−1 S − S n −k ≤ κ g n+n n(n − 1) S 2 n e → 0 (S → ∞), (B.4)
where we use that normalized spin operators are less than or equal to Next, we consider the case where κ/g > 1. We transform the variable κ/g as
κ g = 1 1 − p 2 , (B.5)
where p can take a value from 0 to 1. Let us compare the elements pS | ρ ss | pS and − pS | ρ ss | − pS , where the symbol means the floor function. We can calculate these elements as
pS | κ g 2n S − S S + S n | pS n l=0 κ g 2 1 − pS + l S 2 = n l=0 1 − pS +l S 2 1 − p 2 (B.6) and −pS | κ g 2n S − S S + S n | −pS n l=0 κ g 2 1 − − pS + l S 2 = n l=0 1 − − pS +l S 2 1 − p 2 , (B.7)
where we use the following relation, S ± S |m = 1 ∓ m S 1 ± m S + 1 S |m ± 1 1 − m S 2 |m ± 1 for a large S . We find that Eq.(B.6) is less than or equal to Eq.(B.7) since ( pS + l) 2 ≥ (− pS + l) 2 . In particular, for n = 2 pS , Eq.(B.6) is less than Eq.(B.7). As a result, it holds that pS | ρ ss | pS < − pS | ρ ss | − pS for 0 < p < 1, and then it can be seen that the PT symmetry of the steady state ρ ss (8) is broken for κ/g > 1 when S → ∞.
III. NUMERICAL CALCULATION FOR THE BTC MODELS
A. Numerical calculation for the one-spin BTC model
We numerically investigate the PT symmetry breaking of the stationary state. Fig.C.1 shows eigenvalue structures (top) and |ρ ss | (medium) and |ρ ss − PT ρ ss PT | (bottom) for S = 10. Here, |ρ| means the matrix takes the absolute value for each matrix element ρ. The top figures show that there exist near pure imaginary numbers in the BTC phase, while eigenvalues with the slow decay eigenmodes are real in the BTC broken phase. Medium figures imply that the stationary state ρ ss is likely to be PT symmetric in the BTC phase, while it is broken in the BTC broken phase. Also, the bottom figures indicate that all the elements of |ρ ss − PT ρ ss PT | are close to 0 in the BTC phase, while some elements are finite values in the BTC broken phase. These results indicate that the stationary state ρ ss is PT symmetric in the BTC phase, while it is not in the BTC broken phase. These results show that the stationary state ρ ss is PT symmetric in the BTC phase, while it is not in the BTC broken phase.
B. Numerical calculation of the model (11)
BTC and Liouvillian PT phases can be determined not only by dynamical properties such as dynamics [12-14, 25, 47, 49] and quantum trajectory [35,47] but also by static properties such as magnetization [12,[36][37][38][47][48][49], purity [13,38,48], PT symmetry parameter Q PT of the stationary state. Here we numerically investigate the normalized magnetization of the stationary state, the time evolution, and the quantum trajectory of the normalized magnetization for the model (11) with p z = 2, p x = 1. Fig.C.2 (a) shows the magnetization of the stationary state. Magnetization can usually be regarded as the order parameter even in dissipative systems. Comparing Fig.2 (a), (b), we can see that the magnetization is also zero in the region where the purity and Q PT are zero. Next, we investigate the time evolution with fixed g x /g z in Fig.C.2 (b). These results show that in the BTC phase, the magnetization periodically oscillates, and the relaxation time increases with increasing S . On the other hand, in the BTC broken phase, the magnetization decays without oscillation, and the behavior of dynamics little changes with increasing S . Next, we examine the quantum trajectory at the same point in Fig.C.2 (c). In the BTC (Liouvillian PT ) phase, quantum fluctuations are known to be large since the contribution from quantum jumps is dominant due to exactly balanced dissipation [35,47]. Indeed, these results show that the fluctuations are large in the BTC phase and small in the BTC phase. , (c-4) κ − /g z = 2 (black triangle). These results imply that the BTC phase can be determined by the dynamics, the quantum fluctuations, or the properties of the stationary state, such as purity, magnetization, and PT symmetry parameter Q PT . Here, we have used QuTip [62] to numerically obtain the stationary state and the quantum trajectory.
IV. PERTURBATION THEORY FOR THE ONE-SPIN MODELS
A. Degenerate perturbation theory of Liouvillians
We consider the perturbation theory of Liouvillians [65,66]. Firstly, a LiouvillianL is divided into the nonperturbative partL 0 and the perturbative partL 1 ,L (α) =L 0 + αL 1 .
(D.1)
Suppose that the eigenmodes of the non-perturbative partL 0 are u (0) n with eigenvalue λ (0) n , namely it holds that L 0 u (0) n = λ (0) n u (0) n , and the eigenmodes of the LiouvillianL(α) are u n (α) with eigenvalue λ n (α), namely it holds that L(α)u n (α) = λ n (α)u n (α). The Hilbert-Schmidt inner product is introduced as ρ, σ := Tr[ρ † σ], and the Hermitian adjoint of the LiouvillianL † is also defined as L ρ, σ = ρ,L † σ .
(D.2)
Then it holds thatL † ω n = (λ n ) * ω n , ω m , u n = δ mn .
(D.3)
Next, the eigenvalues and eigenmodes are expanded as
u n (α) = ∞ k=0 α k u (k) n , λ n (α) = ∞ k=0 α k λ (k) n , (D.4)
where we assume that eigenvalues do not degenerate. Also, using the conventional perturbation prescription, the following equation can be obtained,
λ (1) n = ω (0) n ,L 1 u (0) n , (D.5) u (1) n = k n ω (0) k ,L 1 u (0) n λ (0) n − λ (0) k u (0) k , (D.6) λ (2) n = k n ω (0) n ,L 1 u (0) k ω (0) k ,L 1 u (0) n λ (0) n − λ (0) k . (D.7)
Next, we consider the degenerate case. In this case, new non-perturbative eigenmodes are constructed as where u (0) n, j is a non-perturbative eigenmode with eigenvalue λ (0) n . Then, the coefficient c ji and the first-order eigenvalue correctionλ (1) n can be given by solving the following secular equation,
1,L 1 1 1,L 1 2 · · · · · · 2,L 1 1 2,L 1 2 · · · · · · . . . . . . · · · · · · . . . . . .
· · · · · · c 1i c 2i . . . . . . =λ (1) n,i c 1i c 2i . . . . . . , (D.10)
where i,L 1 j := ω (0) n,i ,L 1 u (0) n, j . Also, the first-order eigenmodes correction and the second-order eigenvalues correction are given byũ (1) n,i = k n j
ω (0) k, j ,L 1ũ (0) n,i λ (0) n − λ (0) kũ (0) k, j , (D.11) λ (2) n = k n j ω (0) n,i ,L 1ũ (0) k, j ω (0) k, j ,L 1ũ (0) n,i λ (0) n − λ (0) k . (D.12)
We analyze the one-spin model (13) using the (degenerate) perturbation theory. Now, we choose that the nonperturbative part is the coherent part in Eq.(13), namelyL 0 [·] = −i[H, ·], and the perturbative parts are dissipation parts in Eq.(13), namelyL 1
=L + (1 + p)/S +L − (1 − p)/S , whereL + = D[S + x ] andL − = D[S − x ].
Here, κ is the perturbation parameter. In this case, u (0) and ω (0) and eigenvalues are written as
u (0) = ω (0) = ρ n,q = |n x n − q| x , λ (0) n,q = −iq, (D.13)
where the subscript x of the bra and ket means the eigenbasis of the operator S x . It can be easily found that the eigenvalues −iq are (2S + 1 − |q|)-order degenerated. Also, the Hilbert-Schmidt inner products ρ m,q ,L + ρ n,q and ρ m,q ,L − ρ n,q can be calculated as ρ m,q ,L + ρ n,q = 2 (S − n)(S + n + 1)(S − n + q)(S + n − q + 1)δ m,n+1 δ q ,q − {(S − n)(S + n + 1) + (S − n + q)(S + n − q + 1)}δ m,n δ q ,q , (D.14)
ρ m,q ,L − ρ n,q = 2 (S + n)(S − n + 1)(S + n − q)(S − n + q + 1)δ m,n−1 δ q ,q − {(S + n)(S − n + 1) + (S + n − q)(S − n + q + 1)}δ m,n δ q ,q . (D.15) Therefore, the matrices in Eq.(D.10) for the sector q can be written as the (2S + 1 − |q|) real tridiagonal matrices on the basis of the x-magnetization with the elements
ρ n,q ,L 1 ρ n,q = − 1 + p S {(S − n)(S + n + 1) + (S − n + q)(S + n − q + 1)} − 1 − p S {(S + n)(S − n + 1) + (S + n − q)(S − n + q + 1)}, (D.16) ρ n+1,q ,L 1 ρ n,q = 2(1 + p) S (S − n)(S + n + 1)(S − n + q)(S + n − q + 1), (D.17) ρ n−1,q ,L 1 ρ n,q = 2(1 − p) S (S + n)(S − n + 1)(S + n − q)(S − n + q + 1),(4 −1 1 0 · · · · · · 1 −3 + 1 S 2 − 1 S · · · · · · 0 2 − 1 S −5 + 4 S · · · · · · . , (D.20)
where we use the following relations, , it can be found that many first-order eigenvalue corrections are closer to 0 as S increases, and all the eigenvalues are real. This means that many eigenvalues are zero when S → ∞.
In this model, all the high-order perturbation terms are 0 since ρ m,q ,L 1ρn,q = 0, for (m, q ) (n, q) since the sector q is invariant whenL 1 acts onρ n,q as shown in Eq.(D.14), (D.15). So the perturbative analysis up to first-order correction gives the exact result for the one-spin model (13).
C. Perturbation theory for a class of the one-spin model
Next, we apply the degenerate perturbation theory to a class of the one-spin model (16). Now, we choose that the non-perturbative part is a coherent part in Eq. (16), namelyL 0 [·] = −i[H, ·], and the perturbative part is the dissipation part in Eq.(16), namelyL 1 = µLµ /S = µ D[L µ ]/S . Here, κ is the perturbation parameter. In this case, u (0) and ω (0) and eigenvalues are also written as Eq.(D.13). Now, we calculate the Hilbert-Schmidt inner product ρ m,q ,L µ ρ n,q , ρ m,q ,L µ ρ n,q = 2|α µ | 2 (S − n)(S + n + 1)(S − n + q)(S + n − q + 1)δ m,n+1
+ 2|β µ | 2 (S + n)(S − n + 1)(S + n − q)(S − n + q + 1)δ m,n−1 − |α µ | 2 {(S − n)(S + n + 1) + (S − n + q)(S + n − q + 1)}δ m,n − |β µ | 2 {(S + n)(S − n + 1) + (S + n − q)(S − n + q + 1)}δ m,n − |γ µ | 2 q 2 δ m,n .
(D.24)
Note that there exists non-zero Hilbert-Schmidt inner product ρ m,q ,L µ ρ n,q for q q in general, so the high-order perturbation terms are not 0 in this class. However, these terms do not contribute to the first-order correction. Similarly to the model (13), the matrices in Eq.(D.10) for the sector q can be written as the (2S + 1 − |q|) real tridiagonal matrices on the basis of the x-magnetization with the elements (13) except for a constant multiplication and a sum of scalar multiplication −|γ µ | 2 q 2 /S of the identity matrix. Therefore, these eigenvalues properties are the same as those in the model (13) for µ |α µ | 2 , µ |β µ | 2 , µ |γ µ | 2 q 2 S . Also, it can be found that the case for µ |α µ | 2 = µ |β µ | 2 corresponds to the case for p = 0 in the model (13). On the other hand, the case for µ |α µ | 2 µ |β µ | 2 corresponds to the case for p 0 in the model (13). Therefore, for µ |α µ | 2 = µ |β µ | 2 , the perturbative analysis up to the first-order corrections shows that the real parts of many eigenvalues approach to 0 and the commensurability holds. This means that the BTCs appear in the first-order corrections. On the other hand, for µ |α µ | 2 µ |β µ | 2 , the Liouvillian gap is not closed even when S → ∞, and thus the BTC does not appear.
ρ n,q ,L 1 ρ n,q = − µ |α µ | 2 S {(S − n)(S + n + 1) + (S − n + q)(S + n − q + 1)} − |β µ | 2 S {(S + n)(S − n + 1) + (S + n − q)(S − n + q + 1)} − |γ µ | 2 q 2 S , (D.25) ρ n+1,q ,L 1 ρ n,q = µ 2|α µ | 2 S (S − n)(S + n + 1)(S − n + q)(S + n − q + 1), (D.26)
Lastly, we show that the condition µ |α µ | 2 = µ |β µ | 2 holds if the model (16) satisfies the Liouvillian symmetry (4) when we choose the parity operator to be the reflection of the basis S z . From the Liouvillian symmetry (4), there exists the Lindblad operator L µ corresponding L µ ,
L µ = α µ S + x + β µ S − x + γ µ S x = e iθ µ (−β * µ S + x − α * µ S − x + γ * µ S x ), (D.28)
where the asterisk means the complex conjugate and θ µ ∈ R. Here, the arbitrariness of the phase θ µ causes from the relation,L
D. Perturbation theory for the one-spin BTC model
Lastly, we apply the degenerate perturbation theory to the one-spin BTC model. We choose that the non-perturbative part is a coherent part in Eq. (7), namelyL 0 [·] = −i[H, ·], and the perturbative part is the dissipation part in Eq. (7). Since this model is equivalent to the model Eq.(16) for L 1 = (S x − i(S + x + S − x )/2)/ √ 2 except for a constant multiplication and it satisfies Liouvillian PT symmetry (4), the BTC appears in the first-order corrections. Now, we consider the second-order correction. The second-order eigenvalue corrections are always pure imaginary numbers since the numerator, and denominator of the fraction in Eq.(D.12) are real and pure imaginary, respectively. Following the prescription of the degenerate perturbation theory, the second-order eigenvalue corrections can be numerically calculated as in Fig.D.1.
Furthermore, we numerically find that the second-order eigenvalues correctionλ (2) m,n for each sector q is the 2-Arithmetic progressions. The 2-difference of the second-order eigenvalue correction for 1/S is plotted for each sector q in Fig.D.2 (a). Fitting with a quadratic function, it can be seen that the 2-differences approach zero when S → ∞. The 1/S dependence of the difference of the first and second maximum second-order eigenvalue correction is plotted for each sector q in Fig.D.2 (b). Fitting with a quadratic function, it can be seen that they approach zero when S → ∞. The 1/S dependence of the maximum second-order eigenvalue correction is plotted for each sector q in Fig.D.2 (c). Fitting with a quadratic function, it can be seen that they approach each sector q when S → ∞. These results show that the absolute value of the imaginary part of eigenvalues except for q = 0 decreases to 0 while retaining the commensurability of the imaginary part of eigenvalues for S 1. In other words, we can catch the behavior of the BTC phase transition up to the second-order correction.
Figure 1 .
1Illustration of dissipative spin-S models, (a) two-spin PT model (b) one-spin BTC model (c)
Figure 2 .
2Numerical analysis of the model
Figure 3 .
3Numerical analysis of the time evolution of the normalized magnetization and Liouvillian spectrum in the model with H = S x , L = S z . (a) Time evolution of the normalized magnetization S z /S for S = 20 (dashed light blue), 40 (dashed-dotted orange), and 80 (solid green) and the initial state is set as ρ(0) = |S /2 z S /2| z . (b) Liouvillian spectrum, (c) 15-minimum absolute values of real parts of the eigenvalues, (d) imaginary parts of the eigenvalues for κ/g = 1 and S = 20. Here, we have used QuTip [62] to obtain the Lindblad dynamics.
Supplemental:
Dissipative time crystals originating from parity-time symmetry Yuma Nakanishi and Tomohiro Sasamoto I. LIOUVILLIAN PT SYMMETRIC MODELS A. Two-spin model with gain and loss The Lindblad equation of the open two-spin-S model with gain and loss [47-49] is given by ∂ ∂t ρ = −ig[H, = (S +,A S −,B + H.c.)/2S . This model satisfies the criterion of Huber et al. of Liouvillian PT symmetry (4) when Γ g = Γ l and the parity transformation is the exchange of two spins (i.e. PT(S +,A S −,B + H.c.) = (S +,A S −,B + H.c.), PT (S +,A ) = S −,B , and PT (S −,B ) = S +,A )
Importantly, each term's order of spin operators decreases due to the commutation relations in Eq.(A.7). These cause the real parts of the eigenvalues to have a dependence on S −1 , and then the pure imaginary eigenvalues emerge when S → ∞. The same argument also holds when substituting (S + x ) |q| for ρ in the model (A.5). Also, such cancellations of terms are not expected in general. For example, the cancellations in Eqs.(A.6), (A.7) do not occur for the model (13) when p 0, and then (S − x ) q does not become the eigenmode. Indeed Liouvillian gap is finite even for S → ∞[31,64]. It can be calculated by converting to a quadratic bosonic diagonalized Liouvillian using the Holstein-Primakoff approximation and the non-unitary Bogoliubov transform.
Figure A. 1 .
1(a) Phase diagram of the open two-spin-S model with gain and loss when S → ∞ [47]. Red, blue, and green region indicates the ferromagnetic phase with S z,A /S = S z,B /S = 1, the ferromagnetic phase with S z,A /S = S z,B /S = −1, the anti-ferromagnetic phase with S z,A /S = 1, S z,B /S = −1, respectively. A real purple line indicates the PT phase, and a purple dashed line indicates the PT broken phase. (b) Numerical analysis for the PT -symmetry parameter Q PT (ρ ss ) for S = 7. (c) the S -dependence of Q PT in the stationary state. These show that the PT symmetry breaking in the stationary state occurs in the thermodynamic limit.
√ 1
1+ 1/S and (1 + 1/S ) S ≤ e for S > 0. Here, e is Napier's constant. Similarly, the second term on the right-hand side in Eq.(B.3) approaches 0 when S → ∞. Therefore, we can show that lim S →∞ −0 and then the stationary state (8) is PT symmetric when S → ∞.
Figure C. 1 .
1Numerical analysis of the one-spin BTC model for S = 10 at κ/g = 0.5, 0.8, 1.2, 1.5. Top: Eigenvalue structures Medium: |ρ ss | Bottom: |ρ ss − PT ρ ss PT |. Here, |ρ| means the matrix taking the absolute value for each element of the matrix ρ, and the elements are computed on the basis of the z-magnetization.
Figure C. 2 .
2Numerical analysis of the model(11) for p z = 2, p x = 1, κ + /g z = 0. (a) The normalized magnetization of the stationary state S z /S for S =23. (b) The time evolution with Lindblad dynamics (LD) for S =20,40,80,160, and (c) the quantum trajectory (QT) for S =23 of the normalized magnetization for g x /g z = 3, and (b-1), (c-1) κ − /g z = 0.5 (orange circle), (b-2), (c-2) κ − /g z = 1 (red star), (b-3), (c-3) κ − /g z = 1.5 (purple square),(b-4)
D.18) with n = {S , S − 1, · · · , −S + |q|}. For p = 0, all the matrices are symmetric, since ρ n,q ,L 1 ρ n−1,q = ρ n−1,q ,L 1 ρ n,q , (D.19) while for p 0, they are not symmetric. For example, p = q = 0, using Eqs.(D.16)-(D.18), the matrix in Eq.(D.10) can be written as the (2S + 1) symmetric tridiagonal matrix in the basis of the x-magnetization
ρ n, 0
0,L 1 ρ n,0 = − 4 S (S 2 + S − n 2 ), (D.21) ρ n+1,0 ,L 1 ρ n,0 = 2 S (S − n)(S + n + 1), (D.22) ρ n−1,0 ,L 1 ρ n,0 = 2 S (S + n)(S − n + 1), (D.23) with n = {S , S − 1, · · · , −S }. Diagonalizing the matrix (D.20)
+ n)(S − n + 1)(S + n − q)(S − n + q + 1), (D.27) with n = {S , S − 1, · · · , −S + |q|}. Comparing Eqs.(D.16)-(D.18) and Eqs.(D.25)-(D.27), these matrices are equivalent to those in the model
[
H; L µ , µ = 1, 2, · · · ] =L[H; e iθ µ L µ , µ = 1, 2, · · · ].(D.29) From Eqs.(17), (D.28), it holds thatµ |α µ | 2 + µ |α µ | 2 = µ |α µ | 2 + |β µ | 2 = µ |β µ | 2 + µ |β µ | 2 ,(D.30) and thus, the total gain and loss are exactly balanced.
Figure D. 1 .
1Comparison between the numerical Liouvillian diagonalization (orange star) and the perturbation analysis result up to the second-order corrections (blue circle) for the one-spin BTC model for κ = 0.2, 0.5, 0.8 when S = 10.
Figure D. 2 .
2(a) The 2-difference of the second-order eigenvalue correction for 1/S . (b) The difference between the first and second maximum second-order eigenvalue correction for 1/S . (c) the maximum second-order eigenvalue correction for 1/S . These results show that the absolute value of the imaginary part of eigenvalues except for q = 0 decreases to 0 while retaining the commensurability of the imaginary part of eigenvalues for S 1.
ship between the Liouvillian PT symmetry and other time crystals, such as discrete time crystals, dissipative time crystals originating from dynamical symmetry, and boundary time crystals in bosonic systems are expected to yield interesting results.Acknowledgment -We thank Kohei Yamanaka and Yukiya Yanagihara for fruitful discussions. YN also acknowledges the financial support from JST SPRING, Grant Number JPMJSP2106, and Tokyo Tech Academy for Convergence of Materials and Informatics. The work done by TS was supported by JSPS KAKENHI, Grants No. JP18H01141, No. JP18H03672, No. JP19L03665, No. JP21H04432, JP22H01143. Further, we would like to thank Editage (www.editage.com) for English language editing.
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| {'fraction_non_alphanumeric': 0.0908109404990403, 'fraction_numerical': 0.037023152591170824, 'mean_word_length': 3.583751460581483, 'pattern_counts': {'":': 0, '<': 13, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 87, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This study aims to provide evidence regarding the emergence of a class of dissipative time crystals when PT symmetry of the systems is restored in collective spin systems with Lindblad dynamics. First, we show that a standard model of boundary time crystals (BTCs) satisfies the Liouvillian PT symmetry, and prove that BTC exists only when the stationary state is PT symmetric in the large-spin limit. Also, a similar statement is confirmed numerically for another BTC model. In addition, the mechanism of the appearance of BTCs is discussed through the development of a perturbation theory for a class of the one-spin models under weak dissipations. Consequently, we show that BTCs appear in the first-order correction when the total gain and loss are balanced. These results strongly suggest that BTCs are time crystals originating from PT symmetry. arXiv:2203.06672v2 [quant-ph]', 'arxivid': '2203.06672', 'author': ['Yuma Nakanishi \nDepartment of Physics\nTokyo Institute of Technology\n2-12-1 Ookayama Meguro-ku152-8551TokyoJAPAN\n', 'Tomohiro Sasamoto \nDepartment of Physics\nTokyo Institute of Technology\n2-12-1 Ookayama Meguro-ku152-8551TokyoJAPAN\n'], 'authoraffiliation': ['Department of Physics\nTokyo Institute of Technology\n2-12-1 Ookayama Meguro-ku152-8551TokyoJAPAN', 'Department of Physics\nTokyo Institute of Technology\n2-12-1 Ookayama Meguro-ku152-8551TokyoJAPAN'], 'corpusid': 247447712, 'doi': '10.1103/physreva.107.l010201', 'github_urls': [], 'n_tokens_mistral': 24130, 'n_tokens_neox': 20867, 'n_words': 11924, 'pdfsha': '7b30ae3cc0b9bb42285e4292268a4ac9c106c014', 'pdfurls': ['https://export.arxiv.org/pdf/2203.06672v2.pdf'], 'title': ['Dissipative time crystals originating from parity-time symmetry', 'Dissipative time crystals originating from parity-time symmetry'], 'venue': []} |
arxiv |
THE ECONOMY'S POTENTIAL DUALITY AND EQUILIBRIUM
26 Oct 2022 October 27, 2022
Jacob K Goeree
AGORA Center for Market Design
UNSW
THE ECONOMY'S POTENTIAL DUALITY AND EQUILIBRIUM
26 Oct 2022 October 27, 2022Walrasian equilibriumPareto optimalitydualitypotentialroots
I introduce a concave function of allocations and prices -the economy's potentialwhich measures the difference between utilitarian social welfare and its dual. I show that Walrasian equilibria correspond to roots of the potential: allocations maximize weighted utility and prices minimize weighted indirect utility. Walrasian prices are "utility clearing" in the sense that the utilities consumers expect at Walrasian prices are just feasible. I discuss the implications of this simple duality for equilibrium existence, the welfare theorems, and the interpretation of Walrasian prices.
Introduction
A landmark of economic theory concerns the determination of prices for all goods in the economy. General equilibrium theory originated with Walras ". . . whose system of equations, defining equilibria in a system of interdependent quantities, is the Magna Carta of economic theory" (Schumpeter, 1954). The modern approach to general equilibrium theory is due to Arrow and Debreu (1954) and McKenzie (1954) who established existence of Walrasian equilibria. Duffie and Sonnenschein (1989) describe the history and importance of this existence proof, which allowed general equilibrium theory to gain the central role it now occupies in economics and finance.
Arrow and Debreu's proof pertains to "abstract economies" or "generalized games" and builds on Nash's (1950) existence proof for normal-form games. Their abstract approach establishes existence of a solution to a system of equations but does not reveal if, or why, it has desirable properties. This is the content of the welfare theorems, the modern versions of which are also due to Arrow (1951) and Debreu (1951). The first welfare theorem -that the price system results in a Pareto optimal allocation of resources -is perhaps the central result in price theory. The second welfare theorem states the converse, i.e. that every Pareto optimal allocation is Walrasian.
Despite their dual formulations, the proofs of the two welfare theorems are very different in nature. The first welfare theorem requires only positive marginal utility of income while the second welfare theorem hinges on the assumption of convexity. Duffie and Sonnenschein (1989) note that as a result of Arrow and Debreu's work ". . . the separateness of the two welfare theorems was brought into sharp focus." I demonstrate that the welfare theorems encapsulate a simple duality property of Walrasian equilibria. To this end, I introduce the economy's potential, which is a weighted sum of consumers' utilities of their allocations minus their indirect utilities at given prices. I show that, for any welfare weights, the potential is a non-positive and strictly concave function with a unique root corresponding to the Walrasian equilibrium. The intuition is that allocations solve the primal problem of maximizing weighted utilities and prices solve the dual problem of minimizing weighted indirect utilities. Walrasian prices are "utility clearing" in that the utilities consumers expect are just feasible, i.e. the potential vanishes at the Walrasian equilibrium.
Usually, the economy is parameterized by initial endowments rather than welfare weights. While the set of endowments is of higher dimension than the set of welfare weights, 1 no additional equilibria are introduced. The reason is that any endowments and resulting Walrasian price imply some set of welfare weights. Moreover, existence of Walrasian equilibrium for arbitrary endowments follows if there is at least one vector of "equilibrium weights" that produce the correct incomes. I show that existence of such equilibrium weights readily follows from the Poincare-Hopf theorem.
The next section presents a graphical illustration of duality for a simple exchange economy. Section 3 introduces the economy's potential, generalizes the duality result to any exchange economy, and shows that existence and the welfare theorems are direct corollaries of duality. Furthermore, duality provides a novel interpretation of the Walrasian price vector as the gradient of utilitarian social welfare. Section 4 discusses possible applications of the potential to non-convex economies. Proofs can be found in the Appendix.
An Example
Consider an exchange economy with two consumers with Cobb-Douglas preferences u 1 (x, y) = 2 log(x) + log(y) and u 2 (x, y) = log(x) + 2 log(y). Suppose there are three units of each good in the economy. Welfare maximization max 0 ≤ x 1 +x 2 ≤ 3 0 ≤ y 1 +y 2 ≤ 3 αu 1 (x 1 , y 1 ) + (1 − α)u 2 (x 2 , y 2 )
(1) yields the Pareto-optimal allocations (x 1 (α), y 1 (α)) = ( 6α 1+α , 3α 2−α ) and (x 2 (α), y 2 (α)) = ( 3−3α 1+α , 6−6α 2−α ). The red curve in Figure 1 shows the resulting utility pairs (u 1 (α), u 2 (α)) for α ∈ (0, 1). The shaded area corresponds to the utility possibility set.
The dual of (1) entails minimizing a weighted sum of indirect utilities with respect to prices. If prices are normalized to sum to one, i.e. the price vector is (p, 1 − p), then the indirect utility functions can be written as v 1 (p, m 1 ) = 2 log( 2m 1 3p ) + log( m 1 3(1−p) ) and v 2 (p, m 2 ) = log( m 2 3p ) + 2 log( 2m 2 3(1−p) ). The economy's total income is 3p + 3(1 − p) = 3 and a consumer's welfare weight determines her share: 2 m 1 = 3α and m 2 = 3(1 − α). The dual of (1) is thus
min 0 ≤ p ≤1 αv 1 (p, 3α) + (1 − α)v 2 (p, 3(1 − α))(2)
The blue curves show the indirect utility pairs (v 1 (p, 3α), v 2 (p, 3(1 − α)) for different values of α and p ∈ (0, 1). For α = 1 2 , the weighted sum of utilities is constant on the dashed lines and increasing in the North-East direction. The weighted sum of To see this, note that the solution to (2) is p(α) = 1 3 (1 + α) and it is readily verified that the price ratio
p(α) 1 − p(α) = 1 + α 2 − α
is equal to consumers' marginal rates of substitution at (x 1 (α), y 1 (α)) and (x 2 (α), y 2 (α)).
Hence, these allocations maximize consumers' utilities given prices (p(α), 1 − p(α)).
In other words, the Pareto-optimal allocations that follow from (1) form a Walrasian equilibrium together with the price that follows from the dual program in (2).
The Economy's Potential
Consider an exchange economy with N = {1, . . ., N} consumers and
K = {1, . . ., K } goods. For i ∈ N , let u i : R K ≥0 → R denote consumer i's utility function and ω i ∈ R K >0
consumer i's endowment. I assume the utility functions are strictly increasing, strictly concave, and differentiable. 3 For k ∈ K , let w k = i∈N ω ik denote the total amount of good k in the economy. The set of feasible allocations is
F(w) = {x ∈ R NK ≥0 | i ∈ N x ik ≤ w k ∀ k ∈ K } For vectors v, v ′ ∈ R K let 〈v|v ′ 〉 = k∈K v k v ′ k denote the usual inner product. The Fenchel dual of u i (x i ) is defined as v i (p) = max x i ≥0 u i (x i ) − 〈p|x i 〉 (3)
which is a strictly convex function of prices. An envelope-theorem argument establishes that the solution to (3) satisfies x i (p) = −∇ p v i (p), which is a simplified version of Roy's identity. The solution x i (p) is strictly decreasing in each price and further
satisfies lim p k ↓0 x ik (p) = ∞ and lim p k →∞ x ik (p) = 0.
The next lemma relates v i (p) and x i (p) to the traditional indirect utility v i (p, m i ) and the Marshallian demand x i (p, m i ) respectively.
Lemma 1 For i ∈ N , let λ i (p, m i ) solve 〈p|x i (λ i p)〉 = m i , then v i (p, m i ) = λ i (p, m i )m i + v i (λ i (p, m i )p) (4) x i (p, m i ) = x i (λ i (p, m i )p) (5) Moreover, λ i (p, m i ) equals the marginal utility of income, λ i (p, m i ) = ∂v i (p, m i )/∂m i ,
and for any price vector, λ i (p, m i ) is strictly positive and strictly decreasing in m i with
lim m i ↓0 λ i (p, m i ) = ∞ and lim m i →∞ λ i (p, m i ) = 0.
Example 1 (CES utility) For i ∈ N and ρ i < 1, consider the CES utilities
u i (x i ) = 1 ρ i log( k ∈ K a ik x ρ i ik ) with Fenchel duals v i (p) = 1 − ρ i ρ i log k ∈ K a ik p k /a ik ) ρ i ρ i −1 − 1
Roy's identity yields
x ik (p) = (p k /a ik ) 1 ρ i −1 ℓ ∈ K a iℓ (p ℓ /a iℓ ) ρ i ρ i −1 which satisfies x ik (p/m i ) = m i x ik (p) and 〈p|x i (p)〉 = 1 so λ i (p, m i ) = 1 m i . Marshallian demand is x i (p, m i ) = m i x i (p) and indirect utility is v i (p, m i ) = 1 + v i (p) + log(m i ). ■ Let U α (x) = i∈N α i u i (x i ) denote weighted aggregate utility for some positive weight vector α ∈ R N >0 . The welfare-maximization program W α (w) = max x ∈ F(w) U α (x)(6)
is equivalent to the saddle-point problem
W α (w) = min p ≥0 max x ≥0 i ∈ N α i u i (x i ) − 〈p|x i − ω i 〉
where the multiplier for the feasibility constraint i∈N (x i − ω i ) ≤ 0 is denoted p on purpose. The maximization over allocations is solved by x i (p/α i ) and using the Fenchel duals v i we can write the result as
W α (w) = min p ≥0 V α (p, w) (7) where V α (p, w) = 〈p|w〉 + i ∈ N α i v i (p/α i )(8)
is strictly convex in prices.
The economy's potential is defined as
Y α (x, p, w) = U α (x) − V α (p, w)(9)
From (6) and (7) it follows that the potential is non-positive for all feasible allocations and non-negative prices. Moreover, it is a strictly concave function of allocations and prices since U α (x) is strictly concave and V α (p, w) is strictly convex.
Let λ (−1) i (p, · ) denote the inverse of λ i (p, m i ) with respect to the income m i .
Theorem 1 If (x, p) is a Walrasian equilibrium of the economy with incomes m i for i ∈ N then Y α (x, p, w) = 0 for α i = 1/λ i (p, m i ) and i ∈ N . Conversely, for any α ∈ R N >0 , Y α (x, p, w) has a unique root (x(α), p(α)), which is the Walrasian equilibrium of the economy with incomes m i (α) = λ (−1) i (p(α), 1 α i ) for i ∈ N and i∈N m i (α) = 〈p(α)|w〉.
Evaluated at the equilibrium price, (8) can also be expressed in terms of the standard indirect utilities
V α (p(α), w) = i ∈ N α i v i (p(α), m i (α))
which follows from (4) together with λ i (p(α), m i (α)) = 1 α i and i∈N m i (α) = 〈p(α)|w〉.
Welfare Theorems
A necessary condition for the pair (x, p) to be a root of the potential is that x maximizes welfare, i.e. it is Pareto optimal. The first part of Theorem 1 thus implies the first welfare theorem and the converse part implies the second welfare theorem.
Corollary 1 Theorem 1 implies that any Walrasian allocation is Pareto-optimal and vice versa. identity. This is a consequence of the strict concavity assumption. If we relax this assumption to concavity, the tangency in Figure 1 does not necessarily occur. Nonetheless, Walrasian prices are "utility clearing," i.e. they force a zero potential.
Example 2 (Linear utility) Consider an exchange economy with two consumers
with linear preferences u 1 (x, y) = log(2x+ y) and u 2 (x, y) = log(x+2 y). Suppose there is one unit of each good in the economy. Welfare maximization yields the Pareto-optimal allocations (x 1 (α), y 1 (α)) =
(3α, 0) if α ≤ 1 3 (1, 0) if 1 3 ≤ α ≤ 2 3 (1, 3α − 2) if α ≥ 2 3
and (x 2 (α), y 2 (α)) = (1 − x 1 (α), 1 − y 1 (α)). The frontier of the utility-possibility set, depicted by the red curve in Figure 2, corresponds to the resulting utility pairs (u 1 , u 2 ).
The Pareto optimal allocations lie on the boundary of the Edgeworth box for any welfare weight and the Walrasian price does not follow from consumers' marginal rates of substitution. Instead it follows from mininizing V α (p, w), which yields
p(α) 1 − p(α) = 1 2 if α ≤ 1 3 α 1−α if 1 3 ≤ α ≤ 2 3 2 if α ≥ 2 3
This outcome is illustrated by the blue curves in Figure 2. ■
The Greedy Invisible Hand
An envelope-theorem argument applied to (7) establishes the Walrasian price vector as the gradient of utilitarian social welfare.
Corollary 2 If (x, p) is a Walrasian equilibrium then
p = ∇ w W α (w)(10)for α i = 1/λ i (p, m i ) and x i = x i (λ i (p, m i )p) for i ∈ N .
The characterization of the Walrasian equilibrium price as the gradient of utilitarian social welfare is surprising -Adam Smith's "invisible hand" steers market participants to a state of greatest happiness in a simple greedy manner. Yet, the characterization is intuitive as it implies a balance of individual and social incentives:
∂ k W α (w) ∂ ℓ W α (w) = ∂ k u i (x i ) ∂ ℓ u i (x i )(11)
for i ∈ N , k, ℓ ∈ K and α i = 1/λ i (p, m i ). In other words, individuals' marginal rates of substitution match the social marginal rate of substitution. Corollary 2 provides a simple-yet-powerful way to analytically characterize Walrasian equilibria.
Example 3 (Homogeneous CES) Suppose there are K goods and N consumers with CES utilities
u i (x) = 1 ρ log k ∈ K a k x ρ k
for ρ < 1. Consumers' endowments are ω i and the total endowments are w = i∈N ω i .
Aggregate utility U α (x) = i∈N α i u i (x i ) is maximized at x i = (α i / j∈N α j )w. The social weights are α i = 1/(∂v i /∂m i ) = m i , so the gradient of utilitarian social welfare is ∇ w W α (w) = i ∈ N m i k ∈ K w ρ k w ρ−1 = p
resulting in price ratios p k /p ℓ = (w k /w ℓ ) ρ . Using m i = 〈p|ω i 〉 and x i = (α i / j∈N α j )w yields the Walrasian equilibrium allocations
x i = k ∈ K ω ik w ρ−1 k k ∈ K w ρ k w for i ∈ N .
Note that I did not need to solve any individual consumer's maximization problem to derive the Walrasian equilibrium price and allocations. ■
Existence of Walrasian Equilibria
Theorem 1 shows there is a unique Walrasian equilibrium for any welfare weights.
No fixed-point arguments are needed. A simple duality result establishes the Walrasian equilibrium as the unique maximum, and root, of a strictly concave potential.
Usually, the economy is parameterized by endowments ω i ∈ R >0 for i ∈ N rather than welfare weights. Theorem 1 shows that if there is a Walrasian equilibrium (x, p)
then it is a root of the potential for weights that equal the inverse of the marginal utility of income: α i = 1/λ i (p, 〈p|ω i 〉). Hence, despite the set of endowments being of higher dimension (NK ) than the set of welfare weights (N), no additional Walrasian equilibria are added when parameterizing the economy by endowments.
Do Walrasian equilibria exist for any endowments? Arrow and Debreu (1954) have answered this question affirmatively by extending Nash's (1950) existence proof.
Here I present a simpler argument by showing that for any initial endowments there are "equilibrium weights" that produces the correct incomes. Since Y κα (x, κp, w) = κY α (x, p, w) for any κ > 0 we can, without loss of generality, scale the weight vector α so its entries sum to one, i.e. α belongs to the simplex Σ N . For i ∈ N , consider the fixed-point equations
〈p(α)|ω i 〉 − m i (α) = 0(12)
By Theorem 1, i∈N m i (α) = 〈p|w〉, so the left side of (12) defines a vector field on Σ N . Moreover, lim α i ↓0 m i (α) = 0 by Lemma 1, so the vector field points inward on the boundary of Σ N . By the Poincare-Hopf theorem such a vector field has at least one zero in the interior of Σ N .
Corollary 3 For any economy parametrized by endowments ω i ∈ R >0 , i ∈ N there exists a weight vector α ∈ Σ N such that the root of Y α (x, p, w) is a Walrasian equilibrium.
The solution to (12) is not necessarily unique as the next example shows.
Example 4 Consider an exchange economy with two goods and two consumers with utilities u 1 (x, y) = 3 2 x 2 3 − 1 2 y −2 and u 2 (x, y) = 3 2 y 2 3 − 1 2 x −2 and endowments ω 1 = ( 11 6 , 1 6 ) and ω 2 = ( 1 6 , 11 6 ). The Fenchel duals are and v 2 (p 1 , p 2 ) = v 1 (p 2 , p 1 ). The prices (p 1 (α), p 2 (α)) that minimize V α (p, w) solve α p 1 (α) 3 + 1 − α p 1 (α) 1 3 = 2 and the equation for p 2 (α) follows by interchanging α and 1 −α, i.e. p 2 (α) = p 1 (1 −α).
The incomes are m 1 (α) = α 3 /p 1 (α) 2 +α 1 3 p 2 (α) 2 3 and m 2 (α) = m 1 (1−α). The fixed-point condition (12) for the equilibrium weight can be written as
p 1 (α) 1 − α p 1 (α) 1 3 − 1 6 = p 1 (1 − α) α p 1 (1 − α) 1 3 − 1 6
An obvious solution is α = 1 2 and prices (p 1 , p 2 ) = ( 1 2 , 1 2 ). Two other solutions are, approximately, α ≈ 0.09, (p 1 , p 2 ) ≈ (0.16, 0.79) and α ≈ 0.91, (p 1 , p 2 ) ≈ (0.79, 0.16). ■ To summarize, parameterizing the economy using welfare weights is economical in two ways. First, there is exactly one Walrasian equilibrium for every welfare weight.
Second, this unique Walrasian equilibrium follows from duality rather than from a system of fixed-point conditions. In contrast, parameterizing the economy using endowments is uneconomical for three reasons. First, if (x, p) is a Walrasian equilibrium for the endowments ω i then it is also a Walrasian equilibrium for any endowments ω ′ i that satisfy 〈p|ω i 〉 = 〈p|ω ′ i 〉 for i ∈ N . Second, there may exist multiple Walrasian equilibria for some endowments as Example 4 shows. Third, computing Walrasian equilibria necessarily involves solving a system of fixed-point conditions, see (12). It should be noted that, since the number of goods (K ) is typically assumed to be far larger than the number of consumers (N), (12) provides a computationally efficient alternative to solving fixed-point conditions for the equilibrium prices.
Importantly, even if the economy is parameterized by endowments there are no additional Walrasian equilibria with possibly different features than the potential's roots. The Paretian-Walrasian duality derived above thus applies to all Walrasian equilibria (even when computed using standard fixed-point conditions).
Outlook
The potential provides a litmus test for equilibrium existence. Given preferences and endowments it is a mechanical exercise to compute the potential's maximum value.
A Walrasian equilibrium exists if and only if this exercise returns nil.
If not, general equilibrium theory is quiet about the allocations and prices that ensue even when there are obvious gains from trade. For instance, suppose two consumers have max(x, y) preferences and endowments ω 1 = (2, 2) and ω 2 = (1, 1). At any price vector (p, 1 − p) consumer 1 demands at least four units of one of the goods, ruling out Walrasian equilibrium. Yet, consumers can exchange one unit of either good for one unit of the other and both be better off.
Non-existence of equilibrium usually stops any further analysis, but that does not mean that gains from trade will not be seized -the economy continues to operate after all. Goeree and Roger (2022) demonstrate that the outcome in which one unit is exchanged corresponds to a maximum of the potential, albeit not a root. They term these maxima "Y equilibria" and show they exist in any economy, including nonconvex ones. As such, the potential complements general equilibrium's incomplete toolkit and provides a compass to navigate economics' terra incognita of non-convex economies.
A. Proofs
Proof of Lemma 1. The indirect utility v i (p, m i ) follows from the saddle-point prob-
lem v i (p, m i ) = min λ i ≥0 max x i ≥0 u i (x i ) − λ i (〈p|x i 〉 − m i ) (A.1)
and is related to the Fenchel dual as follows
v i (p, m i ) = min λ i ≥0 λ i m i + v i (λ i p)
Let λ i (p, m i ) be the solution to the minimization problem then
v i (p, m i ) = λ i (p, m i )m i + v i (λ i (p, m i )p) (A.2)
where the λ i (p, m i ) for i ∈ N are such that budget constraints are binding:
〈p|x i (λ i (p, m i )p)〉 = m i (A.3) From (A.1), ∇ p v i (p, m i ) = λ i (p, m i )(ω i − x i ) since m i = 〈p|ω i 〉. In addition, from (A.2), ∇ p v i (p, m i ) = λ i (p, m i )(ω i + ∇ p v i (λ i (p, m i )p)
). Combining these results yields a simplified version of Roy's identity: If (x, p) is a Walrasian equilibrium the Marshallian demands satisfy x i = −∇v i (λ i (p, m i )p), see Lemma 1. This can be inverted to λ i (p, m i )p = ∇u i (x i ), see e.g. Rockafellar (1970, Th. 26.5). 4 When α i = 1/λ i (p, m i ) we thus have p = α i ∇u i (x i ), so the x i satisfy the first-order conditions for maximizing U α . Hence, x = argmax x ′U α (x ′ ) as U α is strictly concave. The Walrasian price p is market clearing so 0 = i∈N (ω i −x i (p, m i )) = ∇ p V α (p, w) when α i = 1/λ i (p, m i ) for i ∈ N . So p satisfies the first-order condition for minimizing V α and p = argmin p ′ V α (p ′ , w) as V α is strictly convex. Since max x U α (x) = min p V α (p, w) the potential vanishes at (x, p).
x i (p, m i ) = x i (λ i (p, m i )p) = −∇ p v i (λ i (p,
For the converse part, since p(α) minimizes V α (p, w) it follows that ∇ p V α (p, w) = i∈N (ω i − x i (p(α)/α i )) = 0, i.e. the x i (p(α)/α i ) satisfy feasibility. The unique solution x(α) to max x∈F(w) U α (x) satisfies α i ∇u i (x i (α)) = q(α) for i ∈ N and some price vector q(α). This can be inverted as x i (α) = −∇v i (q(α)/α i ) = x i (q(α)/α i ). Since x(α) is feasible by construction, q(α) also minimizes V α (p, w). Strict convexity of V α (p, w) implies q(α) = p(α). By Lemma 1 the x i (p(α)/α i ) are optimal Marshallian demands at p(α) and incomes m i if α i = 1/λ i (p(α), m i ), which can be inverted as m i = λ (−1) i (p(α), 1 α i ). Taking the inner product of the feasibility condition with the price vector yields 〈p(α)|w〉 = i∈N 〈p(α)|x i (p(α)/α i )〉 = i∈N m i (α). To summarize, (x(α), p(α)) is a Walrasian equilibrium of the economy with incomes m i (α). ■
Figure 1 :
1Illustration of duality. The shaded area is the utility possibility set and the red curve its frontier. The blue curves depict indirect utilities for given welfare weights as functions of prices. For α = 1 2 , weighted utility is constant on the dashed lines and increases to the North-East while weighted indirect utility is constant on the dotted lines and decreases to the South-West. The unique point where weighted utility and weighted indirect utility match corresponds to the Walrasian equilibrium.
indirect utilities is constant on the dotted lines and decreasing in the South-West direction. There is a unique point where weighted utility is maximized and weighted indirect utility minimized and their values are equal. This point corresponds to the Walrasian equilibrium.
Figure 2 :
2Illustration of duality for the economy in Example 2.For the economy in Section 2, the Walrasian equilibrium price and allocation can be obtained from either the primal welfare-maximization program or the dual program of minimizing weighted indirect utility. For instance, the primal program yields the allocations and the price follows from consumers' marginal rates of substitution. Alternatively, the dual program yields the price and the allocations follow from Roy's
m i )p). From (A.2) it follows that ∂v i /∂m i = λ i (p, m i ). The budget constraints (A.3) imply that λ i (p, m i ) > 0 and that x i (λ i (p, m i )p) is strictly decreasing in p and strictly increasing in m i . The latter implies that λ i (p, m i ) is strictly decreasing in m i . From (A.3) it further follows that x i (λ i (p, m i )p) vanishes when m i = 0 and x i (λ i (p, m i )p) diverges when m i = ∞. Hence, lim m i ↓0 λ i (p, m i ) = ∞ and lim m i →∞ λ i (p, m i ) = 0. ■ Proof of Theorem 1.
With N > 1 consumers and K > 1 goods the set of endowments is NK dimensional while the set of welfare weights is N dimensional.
More generally, a consumer's welfare weight is equal to the inverse of her marginal utility of income, see Theorem 1 below. For the economy studied here this yields α = m 1 /3 and 1 − α = m 2 /3.
The usual assumption is that the utility functions are quasi-concave, which can be made concave by a monotone transformation. Intuitively, this construction implies that among utility functions with the same indifference curves there is one with non-increasing marginal utility. Concave functions are differentiable almost everywhere. For ease of presentation, I assume differentiability everywhere.
Theorem 26.5 inRockafellar (1970) relates the gradient of a strictly convex function to the gradient of its dual, which is also a strictly convex function. To apply the theorem use −u and v.
Existence of an Equilibrium for a Competitive Economy. K Arrow, G Debreu, Econometrica. 22Arrow, K. and G. Debreu (1954). Existence of an Equilibrium for a Competitive Econ- omy. Econometrica 22, 265-290.
An extension of the basic theorems of classical welfare economics. K J Arrow, Proceedings of the second Berkeley symposium on mathematical statistics and probability. Neyman, J.the second Berkeley symposium on mathematical statistics and probabilityBerkeley: U. of California PressArrow, K. J. (1951). An extension of the basic theorems of classical welfare economics. In Proceedings of the second Berkeley symposium on mathematical statistics and probability, Ed. Neyman, J., pp. 507-532. Berkeley: U. of California Press.
The coefficient of resource utilization. G Debreu, Econometrica. 19Debreu, G. (1951). The coefficient of resource utilization. Econometrica 19, 273-292.
Arrow and General Equilibrium Theory. D Duffie, H Sonnenschein, Journal of Economic Literature. 27Duffie, D. and H. Sonnenschein (1989). Arrow and General Equilibrium Theory. Jour- nal of Economic Literature 27, 565-598.
Y Equilibrium: A Theory For Non-Convex Markets. J K Goeree, G Roger, AGORA working paper, in preparationGoeree, J. K. and G. Roger (2022). Y Equilibrium: A Theory For Non-Convex Markets. AGORA working paper, in preparation.
On Equilibrium in Graham's Model of World Trade and Other Competitive Systems. L Mckenzie, Econometrica. 22McKenzie, L. (1954). On Equilibrium in Graham's Model of World Trade and Other Competitive Systems. Econometrica 22, 147-161.
Equilibrium points in n-person games. J F Nash, Proceedings of the National Academy of Sciences. 361Nash, J. F. (1950). Equilibrium points in n-person games. Proceedings of the National Academy of Sciences 36(1), 48-49.
R T Rockafellar, Convex Analysis. Princeton University PressRockafellar, R. T. (1970). Convex Analysis. Princeton University Press.
History of Economic Analysis. J Schumpeter, Oxford University PressNew York, USASchumpeter, J. (1954). History of Economic Analysis. New York, USA: Oxford Uni- versity Press.
| {'fraction_non_alphanumeric': 0.07559610519052365, 'fraction_numerical': 0.022004140151805567, 'mean_word_length': 3.578273078273078, 'pattern_counts': {'":': 0, '<': 2, '<?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': 22, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'I introduce a concave function of allocations and prices -the economy\'s potentialwhich measures the difference between utilitarian social welfare and its dual. I show that Walrasian equilibria correspond to roots of the potential: allocations maximize weighted utility and prices minimize weighted indirect utility. Walrasian prices are "utility clearing" in the sense that the utilities consumers expect at Walrasian prices are just feasible. I discuss the implications of this simple duality for equilibrium existence, the welfare theorems, and the interpretation of Walrasian prices.', 'arxivid': '2210.14437', 'author': ['Jacob K Goeree \nAGORA Center for Market Design\nUNSW\n'], 'authoraffiliation': ['AGORA Center for Market Design\nUNSW'], 'corpusid': 253116913, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8764, 'n_tokens_neox': 7778, 'n_words': 4940, 'pdfsha': '809332854e67292cc37eabc45ee5846922280f90', 'pdfurls': ['https://export.arxiv.org/pdf/2210.14437v1.pdf'], 'title': ["THE ECONOMY'S POTENTIAL DUALITY AND EQUILIBRIUM", "THE ECONOMY'S POTENTIAL DUALITY AND EQUILIBRIUM"], 'venue': []} |
arxiv |
Projecting to Manifolds via Unsupervised Learning
August 6, 2020
Howard Heaton
University of California
Los Angeles †
Samy Wu Fung
University of California
Los Angeles †
Alex Tong Lin
University of California
Los Angeles †
Stanley Osher
University of California
Los Angeles †
Wotao Yin
University of California
Los Angeles †
Projecting to Manifolds via Unsupervised Learning
August 6, 2020adversarial projectioninverse problemsWasserstein GANsgenerative networksop- timal transportdeep neural networksregularizationprojectionlearning to optimizecomputed tomography
We present a new framework, called adversarial projections, for solving inverse problems by learning to project onto manifolds. Our goal is to recover a signal from a collection of noisy measurements. Traditional methods for this task often minimize the addition of a regularization term and an expression that measures compliance with measurements (e.g., least squares). However, it has been shown that convex regularization can introduce bias, preventing recovery of the true signal. Our approach avoids this issue by iteratively projecting signals toward the (possibly nonlinear) manifold of true signals. This is accomplished by first solving a sequence of unsupervised learning problems. The solution to each learning problem provides a collection of parameters that enables access to an iteration-dependent step size and access to the direction to project each signal toward the closest true signal. Given a signal estimate (e.g., recovered from a pseudo-inverse), we prove our method generates a sequence that converges in mean square to the projection onto this manifold. Several numerical illustrations are provided.1. φpuq ě 0 with equality if and only if u P D true .
Introduction
Inverse problems arise in numerous applications such as medical imaging [6,7,39,53], phase retrieval [10,15,26,64], geophysics [13,27,28,34,35,42], and machine learning [21,25,36,67,69]. The underlying goal of inverse problems is to recover a signal from a collection of indirect noisy measurements. Formally stated, consider a finite dimensional Hilbert space X (e.g., R n ) with scalar product x¨,¨y and norm }¨} for the domain space, and similarly for Y (e.g., R m ) for the measurement space. Let A : X Ñ Y be a mapping between X and Y, and let b P Y be the available measurement data given by b " Apu ‹ q`ε, (1.1) where u ‹ P X denotes the true signal and ε P Y denotes the noise in the measurement. The aim in inverse problems is to Estimate u ‹ from the noisy measurements b.
(
1.2)
A difficulty in completing the task in (1.2) is that inverse problems are often ill-posed, making their solutions unstable for noise-affected data. To overcome ill-posedness, traditional approaches for solving inverse problems involve a regularized variational approach that estimates the signal u ‹ bỹ u P argmin uPX pApuq, bq`Jpuq, (1.3) where 1 : YˆY Ñ R measures the discrepancy between the measurements and the application of the forward operator A to our signal estimate (e.g., least squares). The function J : X Ñ R serves as a regularizer, which ensures that the solution to (1.3) is unique and that its computation is stable.
In addition to ensuring well-posedness, regularizers are constructed in an effort to incorporate prior knowledge of the true signal. Common model-based regularizers include, e.g., sparsity Jpuq " }u} 1 [11,16,17,24], Tikhonov Jpuq " }u} 2 [14,31], Total Variation Jpuq " }∇u} 1 [19,57], and, more recently, data-driven regularizers [3,44,50]. A related approach includes Bregman iteration methods [73]. An underlying theme in regularization is that it is commonly assumed that signals exhibit redundant representations and admit a compact low-dimensional manifold representation. However, directly approximating the manifold is highly nontrivial. Thus, a key question remains:
How do we guarantee that the reconstructed signal lies on the manifold of true signals?
Below we demonstrate that this guarantee can be ensured by using a projection algorithm. In addition, we emphasize that our approach is i) unsupervised and ii) does not require directly representing the manifold. This means that a direct correspondence between noisy signal estimate data and true signal data is not needed (e.g., we may even have different amounts of samples from each data set).
Remark 1.1 Throughout this work we refer to reconstructing signals. This phrase is meant in a general sense to describe an object of interest that can be represented mathematically. This includes, e.g., images, parameters of a differential equation, and points in a Hilbert space.
Contribution
We present adversarial projections, a new framework for solving inverse problems. Our core result is to demonstrate how unsupervised learning can be used to project signal estimates onto the underlying low-dimensional manifold of true signals. This is accomplished without making a direct representation of the manifold. The training process consists of solving a sequence of minimization problems (related to the inner maximization in general adversarial networks, discussed below). During implementation, our proposed algorithm forms a Halpern-type method with relaxed projections, which we prove converges in mean square to the projection of the initial estimate onto the manifold. At the level of individual signals, this work may also be interpreted as learned gradient descent with a sequence of expert-like regularizers [30]. And at the aggregate level of distributions, it may be viewed as a subgradient method for minimizing the Wasserstein-1 distance between the distribution of initial estimates and the true distribution. The remainder of this paper is organized as follows. In Section 2, we provide an overview of generative adversarial networks (GANs) and their connections to optimal transport (OT), adversarial regularizers, and expert regularizers. In Section 3, we describe our adversarial projections approach. The convergence analysis is covered in Section 4. In Section 5, we review the related works. In Section 6, we show the potential of adversarial projections on a two-dimensional distribution as well as two low-dose parallel beam computed tomography experiments. We conclude with a brief discussion in Section 7.
Background
In this section, we briefly review adversarial regularizers [50], Wasserstein GANs [5,32] and their connections to optimal transport [46,63], and expert regularizers [30]. These topics will be useful for interpreting adversarial projections.
Wasserstein GANs and Optimal Transport
In GANs [5,32], access is given to a discriminator and generator, and the goal is to train the generator to produce samples from a desired distribution. The generator does this by taking samples from a known distribution N and transforming them into samples from the desired distribution D true . Meanwhile, the purpose of the discriminator is to guide the optimization of the generator. Given a generator network G θ and a discriminator network D ω , the goal in Wasserstein GANs is to find a saddle point solution to the minimax problem
inf G θ sup Dω E u"Dtrue rD ω puqs´E z"N rD ω pG θ pzqqs , s.t. }∇D ω } ď 1, (2.1)
Here, the discriminator attempts to distinguish real images from fake/generated images, and the generator aims to produce samples that "fool" the discriminator by appearing real. The supremum expression in (2.1) is the Kantorovich-Rubenstein dual formulation [66] of the Wasserstein-1 distance, and the discriminator is required to be 1-Lipschitz. Thus, the discriminator computes the Wasserstein-1 distance between the true distribution D true and the distribution of fake images generated by G θ pzq. Originally, weight-clipping was to enforce the Lipschitz condition of the discriminator network [5], but an improved method using a penalty on the gradient was used in [33].
Adversarial Regularizers
A good regularizer J : X Ñ R is able to distinguish between signals drawn from the true distribution D true and drawn from an approximate distributionD -taking low values on signals from D true and high values otherwise [12]. Such regularizer plays a similar role as the discriminator described in Section 2.1; however, this setting is different in that D ω assigns high values to true signals instead. Mathematically, J "´D ω . These regularizers are called adversarial regularizers [50]. They are trained a priori in a GAN-like fashion, and are then used to solve a classical inverse problem (1.2). These regularizers were shown to have desired distributional properties in that their gradients provide a descent direction for the Wasserstein-1 distance [50, Section 3.2]. Our work takes advantage of this fact to provide convergence guarantees (see Section 4).
Expert Regularizers
For many inverse problems, well-posed reconstructions can be obtained by incorporating additional knowledge about the signals to be recovered. Expert regularizers [30] are functions used for accomplishing this task that attain small values at signals similar to the distribution of true signals and larger values at signals drawn elsewhere. Inclusion of experts regularizers, thus, should encourage recovery of signals from the true distribution D true while not introducing additional artifacts. Formally stated, given constants β, µ P p0, 8q, desirable properties include 2. For all ε ą 0, there exists δ ą 0 such that if v P D true , then }v´u} ď δ implies φpuq ď ε.
3. For all v P D true , }v´u} ě µ implies φpuq ě βµ.
Note that, if D true is closed and convex, all of the above properties are satisfied by the function that measures the distance between u and the set D true . In addition, the second item is automatically satisfied if the first item holds and φ is Lipschitz. The primary task in the training process (Algorithm 1) for our proposed method may be viewed as finding a sequence of regularizers tφ k u that approximate the properties of expert regularizers. Our adversarial projection method (Algorithm 2) then performs a sequence of gradient descent steps successively using each φ k .
Adversarial Projections
Herein a projection method is proposed to solve the inverse problem (1.2). Suppose access is provided to a reasonable estimateũ of the true signal u ‹ and that u ‹ is contained in a compact manifold M (formally stated in Section 4). The key idea is that the point in M closest to the estimateũ forms an improved approximation of u ‹ . That is, u ‹ « P M pũq where P M is the projection operator onto M defined by 2 P M puq :" argmin vPM }v´u}.
(3.1)
In most practical settings we do not have direct access to the manifold M to determine this projection. However, below we indirectly form projections using the pointwise distance function
d M puq :" inf vPM }v´u}. (3.2)
Indeed, for α P R and λ " α¨d M puq, we obtain the inclusion relation 3 u`α pP M puq´uq P u´λBd M puq, (3.3) and the left hand side is called the α-relaxed projection of u onto M. In particular, we can directly obtain the left hand side from the subgradient expression on the right (described below). With additional assumptions (see Section 4), we find that when the estimateũ is drawn from a distribution of estimatesD and the true signal u ‹ is drawn from the distribution of true signals D true ,
d M P argmax }f } L ď1 E u"D rf puqs´E u"Dtrue rf puqs ,(3.4)
i.e., the pointwise distance function d M is a maximizer of the expression on the right hand side.
(Here }f } L ď 1 denotes the set of all 1-Lipschitz functions f .) Thus, our task is to solve (3.4), which is a form of unsupervised learning, and then use our estimate of d M to form a relaxed projection. This also illustrates that the name adversarial projection derives from the fact that the relaxed projection operation we implement using (3.3) and (3.4) comes from the inner expression in (2.1) used for WGANs.
In practice, the estimate obtained for d M may be a rough approximation. In light of this, our method uses a small fixed step size common for allũ PD when performing each update (rather 2 The projection is well-defined precisely when the minimization problem admits a unique solution. 3 For completeness, this statement is proven in Lemma 8.1 of the Appendix. than individualized step sizes), mimicking a gradual and (hopefully) stable flow of the distributioñ D toward D true . In some cases, this causes the updates to overshoot the manifold in such a way that the projection of the new update onto the manifold is not the point P M pũq. However, we can still ensure the sequence generated by our method converges to the P M pũq by incorporating a form of anchoring (i.e., pulling updates closer toũ). Given a sequence of real numbers tγ k u Ă p0, 1q and a 1-Lipschitz operator T : X Ñ X (i.e., nonexpansive), Halpern [37] proposed finding the projection of u 1 "ũ onto the fixed point set of the operator T pi.e., the points such that u " T puqq by generating a sequence tu k u of the form u k`1 " γ k u 1`p 1´γ k qT pu k q, for all k P N, (3.5) where each update is a convex combination of u 1 and T pu k q. Our method takes a related form u k`1 " γ k u 1`p 1´γ k q´u k`α k pP M pu k q´u k q¯, for all k P N, (3.6) where we note the expression replacing T on the right is not necessarily 1-Lipschitz for our choice of sequence tα k u; however, this expression has the desired fixed point set M since the terms multiplied by α k cancel when u k P M. We choose step sizes so that typical updates in our method will have α k P p0, 1q, resulting in an under-relaxed projection and convergence to P M pũq in probability (see Theorem 4.9). Algorithm 1 articulates the training procedure for identifying the parameters needed for our adversarial projection scheme in Algorithm 2.
Remark 3.1 The pointwise distance function d M puq is distinct from the Wasserstein-1 distance WasspD, D true q between the distributionD of estimates and the true signal distribution D true . The former measures the distance from an individual point to a set while the latter is a metric for distributions of points. The connection between these, in our setting, is that the expected value of the distance to the manifold among allũ "D is equivalent to the Wasserstein-1 distance, i.e.,
Eũ "D rd M pũqs " WasspD, D true q. Algorithm 1 for flowing the training data distribution toward the true distribution may be described as follows. First note that pΩ, F, Pq is a probability space, where Ω, F, and P are the sample space, σ-algebra, and probability measure, respectively. Rather than write a composition of operations applied to ω P Ω, we adopt the notation convention that u 1 pωq gives the initial estimate recovered from the measurement data 4 b and, for all k P N, u k pωq is the k-th iterate of the method. The output of Algorithm 1 is a collection of parameterized functions and step sizes. The relaxation constant α in Step 2 follows its use in (3.3) to determine the step size for relaxed projections. The anchoring sequence tγ k u is chosen to pull successive updates closer to the initial iterate (e.g., γ k " 1{k). The function parameterization I defines the collection tJ θ u θPI of functions over which the optimization occurs in Step 3, which in practice forms an approximation of the set of all 1-Lipschitz functions. The initial collection of signal estimates is denoted by D 1 and for k-th iterate we write D k :" tu k pωq : ω P Ωu so that
E ω"Ω " J θ pu k pωqq ‰ " E u"D k rJ θ puqs . (3.8)
A for loop occurs from Lines 5-9 with each index k giving rise to a distribution D k of signal estimates. Flipping the sign of the problem in (3.4) yields the minimization problem in Line 6, which can be solved using ADAM [43]. Line 7 then defines the step size λ k , which is proportional to the average distance between points in D k and D true . Line 8 defines the updates for each iterate u k pωq using the Halpern-type update described in (3.6). There we use the definition
g k puq :" # u`λ k¨∇ J θ k puq if J θ k is differentiable at u, u otherwise. (3.9)
so that, upon flipping signs to assume d M "´J θ k , we obtain the relaxed projection
g k puq " u`λ k d M puq loomoon ":α k puq pP M puq´uq " u`α k puq pP M puq´uq P u´λ k Bd M puq, (3.10)
where α k puq is defined to be the underbraced term and we adopt the convention of taking α k puq " 0 when d M puq " 0. (This is justified since d M puq " 0 implies P M puq " u.) This Halperntype update forms a convex combination of the initial iterate u 1 and the relaxed projection g k pu k q. Upon completion of this training process, inferences can be performed using the learned quantities tθ k , λ k u by applying Algorithm 2.
Remark 3.2 In practice, because we perform numerical differentiation, we abusively write
g k puq " u`λ k¨∇ J θ k puq, (3.11)
which is the expression used in our experiments, with J θ k analogous to discriminators in GANs.
Explanation of Algorithm 2 is as follows. First the parameters tθ k u, step sizes tλ k u, and anchoring sequence tγ k u are chosen according to Algorithm 1. Then in Line 3 the point u 1 is initialized to an initial estimate of u ‹ . This estimate can be generated using, for example, a pseudo inverse or a solution to an associated regularized problem. Then, for each k, the Halpern-type update is computed using a relaxed projection with g k (Line 6). Note g k is defined using λ k and θ k . Upon repeating this process the same number of times as the training iterations, we obtain our estimate u k in Line 7. The following section proves convergence of adversarial projections.
Algorithm 2: Adversarial Projection (How to Reconstruct Individual Signal)
Result: True Signal Estimate u k from Data b 1 Choose weights tθ k u, step sizes tλ k u, and sequence tγ k u from Algorithm
1 2 AdvProj(b) 3 u 1 Ðũpbq Ÿ Initialize to pseudo inverse or solution to (1.3) 4 for k " 1, 2, . . . do 5 u k`1 Ð γ k u 1`p 1´γ k qg k pu k q Ÿ Halpern-type
Convergence Analysis
This section formalizes the assumptions and states the main convergence result for the adversarial projections method. We first articulate a form of the intuitive idea that the true data is contained in a low dimensional manifold M. Then we assume the initial distribution estimate is bounded and each successive distribution D k is not "too noisy" (i.e., the observed signal is not missing significant features from the true signal, commonly known as collapsed modes).
Remark 4.4
This assumption effectively states the noise is not "too large" and that the method used to obtain the initial distribution is sufficiently representative and does not collapse modes. This is a weaker statement than assuming each individual signal can be recovered from its measurements. And, if our method is actually making progress, the assumption holding for k " 1 should naturally imply it also holding for all subsequent values of k.
Using the above assumptions, [50] provides an equivalent variation of the following theorem, which relates the set of 1-Lipschitz functions to the distance function d M from points to the manifold M. Figure 1: Each point in the distribution D k (blue) is updated by the relaxed projection g k (purple) for the projection P M onto the manifold M (red). We label the over-relaxation g k pvq and underrelaxation g k pwq. The point p (brown) illustrates an approximate average of the points in D k with distance E u"D k rd M puqs to the manifold M (i.e., the distance between p and M equals the average distance from each point u P D k to M).
sup }f } L ď1 E u"D k rf puqs´E u"Dtrue rf puqs . (4.2) That is, E u"D k rd M puqs " sup }f } L ď1 E u"D k rf puqs´E u"Dtrue rf puqs . (4.3) • p • • • • • • • • • • • • • • P M (p) v g k (v) P M (v) w g k (w) P M (w) λ k λ k d k M
Theorem 4.5 is incredibly useful for our task as it provides a way to possibly approximate the pointwise distance function d M . Note the set of maximizers of (4.2) is not unique. For example, if f puq is a maximizer, then for any c P R so also is gpuq :" f puq`c. However, in the practical settings we have considered, we find that the gradient of our estimate of a maximizer of (4.2) adequately approximates the gradient of d M on points in D. In order to apply Theorem 4.5, we utilize the following assumption.
Assumption 4.6 The parameter set I is such that tJ θ u θPI forms the set of 1-Lipschitz functions.
Remark 4.7 In practice, Assumption 4.6 can be approximately implemented by, e.g., adding a gradient penalty to the loss function [33] or using sorting [4].
Finally, together the above assumptions with the following conditions on the anchoring sequence tγ k u, we state our main convergence result.
Assumption 4.8 The sequence tγ k u satisfies the following properties: i) γ k P p0, 1s for all k P N, ii) lim kÑ8 γ k " 0, and iii) ř kPN γ k " 8. If the sequence tu k u is generated by Algorithm 2 and the minimizer θ k in Line 6 is chosen so that J θ k "´d M for all k P N (as permitted by Theorem 4.5), then the sequence tu k u converges to P M pu 1 q in mean square.
Recall that convergence in mean-square implies convergence in probability. And, by the definition of convergence in probability, this theorem implies that, given ε ą 0, the probability that }u k8 P M pu 1 q} ą ε goes to zero as k Ñ 8. That is,
lim kÑ8 P " tω P Ω : }u k pωq´P M pu 1 q} ą εu ‰ " 0. (4.4)
In common language, this may be interpreted as saying the probability of the sequence tu k u "not converging to P M pu 1 q" becomes smaller and smaller as the sequence progresses.
Below we present a lemma about the adversarial projections that can find use during training (i.e., in solving the minimization problem in Line 6 of Algorithm 1).
Lemma 4.10
In the same setting as Theorem 4.9, for λ k ą 0, choosing
η k :" 1 λ 2 k¨E u"D k " }g k pu k q´u k } 2 ‰ (4.5)
yields an upper bound η k P r0, 1s on the proportion of the distribution D k that is not contained in the manifold M, i.e.,
|D k´M | ď η k ď 1, (4.6)
where |¨| denotes the measure of the set.
Remark 4.11 Note (4.5) can be abusively rewritten as
η k " E u"D k " }∇J θ k puq} 2 ‰ , (4.7)
where the abuse comes from the fact in practice each differentiation is performed numerically.
Remark 4.12 During training, Lemma 4.10 can be used to determine when a good estimate of θ k has been found. Initially, by evaluating (4.7) our estimate of η k will be small (less than unity). As we get closer to the optimal weights θ k , this lemma indicates that our estimate of η k should increase as our estimate of J θ k is better able to distinguish between points in D k and points in the manifold M. We can then use the fact η k is bounded to the interval r0, 1s to identify a pair of stopping criteria. Namely, if tη k u is a sequence indexed by corresponding to the sequence of weight estimates tθ k u of optimal weights θ k , and if ε 1 , ε 2 P p0, 1q, then training can terminate if either
1. η k ě 1´ε 1 , or 2. η `1 k ď η k`ε 2 .
The first condition holds if there is negligible overlap of D k and M and our estimate of η k is close to the ideal value (unity). The second condition halts training when progress becomes small and this condition puts an upper bound on the number of epochs used to perform training (i.e., ă 1{ε 2 ).
Related Works
Our work bears connections with GANs [5,32], and its applications to inverse problems [61]. Our approach can be viewed as training a GAN, except that rather than solving a minimax problem, we solve a sequence of minimization problems. In this case, J is the discriminator network that distinguishes between signals coming from the "fake" distribution (i.e., our approximate distribution) and the true distribution, and g η is the generator which tries to generate signals that resemble those from the true distribution. Our work also bears connections with optimal transport [60,66]. In particular, under certain assumptions (see Section 4), the adversarial projections can be interpreted as a subgradient flow that minimizes the Wasserstein-1 distance, where the function J corresponds to the Kantorovich potential [5,46,47,52,63], or in the context of mean field games and optimal control, the value function [46,59]. Analogous to classical physics, the signals flow in a manner that minimize their potential energy. Our approach learns a sequence of these potential functions that project (or "flow") the distribution of signals towards the true distribution of signals.
From an inverse problems perspective, our approach falls under the category of using deep learning to solve inverse problems [68]. One approach, known as post-processing, first applies a pseudoinverse operator to the measurement data (e.g., FBP) and then learns a transformation in the image space. This approach has been investigated and found effective by several authors [20,33,41,56]. Another approach is to learn a regularizer, and then use it in a classical variational reconstruction scheme according to (1.2). Other works investigate using dictionary learning [71], variational autoencoders [51], and wavelet transforms [23] for these learned regularizers. Perhaps the most popular schemes are learned iterative algorithms such as gradient descent [2,38,44], proximal gradient descent or primal-dual algorithms [3,62]. These iterative schemes are typically unrolled, and an "adaptive" iteration-dependent regularizer is learned. One key difference between adversarial projections and the aforementioned data-driven approaches is that our approach is unsupervised. That is, we do not need a correspondence between the measurement b and the true underlying signal u ‹ . The adversarial projections simply requires a batch of true signals and a batch of measurements, regardless of whether these directly correspond to each other (i.e., an injective map between the two might not be available); this is especially useful in some applications (e.g., medical imaging) where the true image corresponding to the measurement is often not available. Another set of work uses deep image priors (DIP) [8,65], which attempt to parameterize the signal by a neural network. The weights are optimized by a gradient descent method that minimizes the data discrepancy of the output of the network. The authors in [8] show that combining DIPs with classical regularization techniques are effective in limited-data regimes. Our work is perhaps most similar to adversarial regularizers [50], where a regularizer J is trained in a GAN-like process (see Section 2.2). The regularizer learns to discriminate between FBP reconstructions and the training data, and it is used to reconstruct an approximate signal by solving the variational problem (1.3) using gradient descent (see Algorithms 1 and 2 in [50]). On the other hand, our approach learns a sequence of regularizers tJ θ k u. Each of these regularizers are used project the k th distribution D k toward the manifold M of true signals. Under certain assumptions (see Section 4), these regularizers can be viewed as potential functions that "flow" our current estimate signals to the distribution of true signals by performing a subgradient descent on the Wasserstein-1 distance at each iteration k. In our setting, each J θ k can be viewed as approximations to expert regularizer [30] (see Section 2.
3) for the current distribution D k .
Numerical Experiments
In this section, we outline the potential of adversarial projections. We begin with a distributional illustration showing how adversarial projections project (or "flow") a distribution onto another. We then test our approach on computed tomography (CT) examples using two standard datasets: a synthetic dataset comprised of randomly generated ellipses as well as the Low-Dose Parallel Beam Progressing from left to right and top to bottom, snapshots are provided at k " 1, 5, 25, 300. These plots agree with Theorem 4.9 in verifying that, as k Ñ 8, the probability that each u k P D k is also in M goes to unity.
(LoDoPaB) dataset [45]. We focus on the unsupervised learning setting, where we do not have a correspondence between the distribution of true signals and approximate signals. Therefore, we set adversarial regularizers (an unsupervised learning approach) as our benchmark. The quality of the image reconstructions are determined using the Peak Signal-To-Noise Ratio (PSNR) and structural similarity index measure (SSIM). For all experiments, we use the PyTorch deep learning framework [54] and the ADAM [43] optimizer. We also use the Operator Discretization Library (ODL) python library [1] to compute the TV and filtered backprojection (FBP) solutions. The experiments are run on a single NVIDIA TITAN X GPU with 12GB RAM.
Remark 6.1 Although for practical reasons we consider linear inverse problems in our experiments, we emphasize that our presented methodology applies even when u ‹ is recovered from nonlinear measurements (i.e., when A is a nonlinear operator).
Distributional Illustration
In this section, we show a toy example to provide intuition for the flow from an initial distribution estimate D 1 to a true distribution D true contained in a manifold M. For simplicity, here we take D true " M. To coincide with the assumption that the manifold M admits a lower dimensional representation, we let it take the form of a curve in 2D. The initial distribution D 1 takes the form Figure 2. The expected distance between points and the manifold is given by (4.3), which, in this setting, is effectively equivalent to the Wasserstein-1 distance WasspD k , Mq. The nonoverlap proportion tη k u provides an upper bound on the measure of the difference between the distributions (i.e., |D k´M |), and so its going to zero also provides an insight into convergence.
• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Expected• • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • Nonoverlap Proportion η k
of a collection of finite samples from a Gaussian distribution in 2D. This is illustrated in Figure 2a.
Here 600 samples are drawn from each distribution. Algorithm 1 is then used to successively update each point in D k to flow all of the points toward the manifold. Snapshots of this flow are illustrated through the sequence of photos in Figure 2, which demonstrates that the expected distance converges to zero. Figure 3 provides a plot of the distance converging to zero and reveals that the nonoverlap proportion bound tη k u also decreases, as expected.
In this example, it is safe to suppose Assumptions 4.1 holds by our choice of M and roughly uniform sampling there. Although D 1 was sampled from a Gaussian, 4.2 holds because we use finitely many samples. However, Assumption 4.3 does not actually hold, although it is "close" to being true. This is because most of the points in D 1 would project directly onto the vertical portion of M rather than the curved ends. Despite this, the flow behaves as one would hope in the sense that it spreads D 1 out to the points covering M while, roughly speaking, being close to their original projections onto the manifold. Assumption 4.6 approximately holds for this example since the network used follows the neural network structure proposed by [4], which possesses the property of being universal 1-Lipschitz function approximators as the number of parameters/layers increase. Lastly, we ensured Assumption 4.8 by choosing γ k " 1{k.
Low-Dose Computed Tomography
We now demonstrate adversarial projections on two low-dose CT examples.
Ellipse Phantoms We use a synthetic dataset consisting of random phantoms of combined ellipses as in [2]. The images have a resolution of 128ˆ128 pixels. Measurements are simulated with a parallel beam geometry with a sparse-angle setup of 30 angles and 183 projection beams. Moreover, we add Gaussian noise with a standard deviation of 2.5% of the mean absolute value of the projection data to the projection data. In total, the training set contains 10,000 pairs, while the validation and test set consist of 1,000 pairs each. Human Phantoms As a more realistic dataset, we use human phantoms consisting of chest CT scans from the Low-Dose Parallel Beam dataset (LoDoPaB) [45]. In our setup, we use 20,000 training images and 2,000 validation images of size 128ˆ128. Similar to the ellipse phantoms, we simulate the data using 30 angles and 183 projection beams. We also add Gaussian noise with a standard deviation of 2.5% of the mean absolute value of the projection data to the projection data.
Network Structure We use a simple 5 layer neural network containing 38,534 trainable parameters. The first three being convolution layers with kernel size 4 and stride 2, with output channels 32, 64, and 1 for layers one, two, and three, respectively. For the last two layers, we use fully connected layers to bring the dimensions back to a scalar. As nonlinear activation function,we choose the Parametric Rectified Linear Units (PReLU) functions which was shown to be effective in other applications such as classification [40].
σ c pxq " # x if x ě 0 cx else ,
For adversarial regularizers, we use the network structure described in [50], which consists of an 8-layer CNN with Leaky-Relu activation. The network contains 2,495,201 parameters. More details can be found in [50,Appendix B].
Training Setup To train the adversarial projections, we begin with an initial distribution obtained from the TV reconstructions. We update the distribution whenever 200 epochs have passed since the last update for both ellipses and human phantoms datasets. We also update the distributions if the conditions in Remark 4.12 is satisfied for η k " 10´4. As stopping criterion, we set a maximum of 50 iterations, i.e., generator updates, in Algorithm 1. We note that in practice, the number of epochs and η k are hyperparameters that need to be tuned. These are of particular importance since it determines how well we approximate Line 6 in Algorithm 1.
In the ADAM optimizer, we use a learning rate of 10´5, and use a batch size of 16
• • • • • • • • • • • • •• ••••• •• ••••••••••• ••• ••• ••••• •• •••• •• Expected Distance to M 1 25 50 0.75 1.00 • • • • • • • • • ••••• • •• ••••• • •• • •••• •• •• • ••• •• •• •••• •• •• •
Nonoverlap Proportion η k Figure 6: Convergence plots for the ellipse dataset. The expected distance between points and the manifold is given by (4.3), which, in this setting, is effectively equivalent to the Wasserstein-1 distance WasspD k , Mq. The nonoverlap proportion tη k u provides an upper bound on the measure of the difference between the distributions (i.e., |D k´M |)
update the samples/distribution in Algorithm 1, we use step-size constant α " 0.5. To ensure Assumption 4.8 is satisfied, we choose γ k " 10´8{k. Finally, to approximately satisfy Assumption 4.6, we enforce J to be 1-Lipschitz by adding a gradient penalty [33].
To train the adversarial regularizers, we use the code provided in [49]. Here, for the ellipses we use a learning rate of 10´4, a batchsize of 16, and a gradient-norm-weight of 20, and for the LoDoPaB dataset we use a learning rate of 10´3, a batchsize of 32, and a gradient-norm-weight of 1. We note that the setup for adversarial regularizers in [50] adds white Gaussian noise independent of the data, and is therefore different from our setup. As a result, we re-train the adversarial regularizers and tune the reconstruction parameters to the best of our ability. In particular, after the regularizer is trained, we tune the regularization parameter, stepsize, and number of gradient steps (see Algorithm 2 in [50]) for the highest PSNR. For a fair comparison, the adversarial regularizer is also trained on TV reconstructions as the initial distribution.
Experimental Results
In Tables 1 and 2, we compare the average PSNR and SSIM on the validation datasets (1,000 images) for the ellipse dataset and LoDoPaB dataset (2,000 images), respectively. These results compare adversarial projections with FBP, TV, and adversarial regularizers. We also show an ellipses image in Figure 4 and a LoDoPab image in Figure 5. For the adversarial regularizers, we find that using 25 steps with a stepsize of 0.05 and a regularization parameter of 2 leads to the highest PSNR on the ellipse dataset. Similarly, we find that using 25 steps with a stepsize of 0.01 and a regularization parameter of 2 leads to the highest PSNR on the LoDoPaB dataset.
In Figures 6 and 7, we observe that the approximate expected distance to the manifold (i.e., our approximation of the Wasserstein distance) decreases as we update the distribution. We also show the values of η k , which provide a bound on the nonoverlap proportion. While adversarial projections performs the best, we note that, e.g., some ellipses are not reconstructed in adversarial projections (this is also seen in adversarial regularizers). This is due to the fact that the initial TV reconstruction completely erases some ellipses due to the sparse angle setup.
In this case, we have that some modes collapse, and Assumption 4.3 is not entirely satisfied. In particular, we obtain that the pushforward is simply a subset of the true manifold M. More image reconstructions can be found in Appendix 8.2.
Convergence Plots for LoDoPab dataset 1 25 50 0.10 1.00
10.0 • • • • • • •• •• • • ••• • • •• •• •• ••• ••• • •••• •• • • •• •• •• •• • • • • • Expected Distance to M 1 25 50 0.75 1.00 • • • • • • • •• • • •• •• • ••• •••• •• • • • • • • • • •• • • • ••• • • • • • • • • • •
Nonoverlap Proportion η k Figure 7: Convergence plots for the LoDoPab dataset. The expected distance between points and the manifold is given by (4.3), which, in this setting, is effectively equivalent to the Wasserstein-1 distance WasspD k , Mq. The nonoverlap proportion tη k u provides an upper bound on the measure of the difference between the distributions (i.e., |D k´M |)
Conclusion
We present adversarial projections, a new framework for solving inverse problems. The main idea is that, by solving unsupervised learning problems, we can project signal estimates onto the underlying low-dimensional manifold of true signals. The training process consists of solving a sequence of minimization problems, which can be interpreted as training a sequence of discriminator networks that attempt to distinguish between signals in the approximate and true distributions. During implementation, our proposed algorithm forms a Halpern-type method with relaxed projections, which we prove converges in mean square to the projection of the initial estimate onto the manifold. At the level of individual signals, this work may also be interpreted as learned gradient descent with a sequence of expert-like regularizers. At the aggregate level of distributions, adversarial projections may be viewed as a subgradient method for minimizing the Wasserstein-1 distance between the distribution of initial estimates and the true distribution. Our numerical experiments show that adversarial projections outperform adversarial regularizers, a state-of-the-art unsupervised learning method for inverse problems. An extension to our work we intend to investigate the semi-supervised regime, where we have labels for some of the data, and to investigate inclusion of the measurement data into the projection scheme. We also intend to investigate guidelines on the design of more effective network architectures such as PDE-based neural networks [36,58].
Appendix
Proofs
We begin with the elementary result stated in Section 3.
∇d M puq " u´P M puq }u´P M puq} " u´P M puq d M puq . (8.3)
Thus, direct substitution reveals 4) and the proof is complete.
u´λ∇d M puq " u`α pP M puq´uq ,(8.
Below we restate and prove the lemma about the sequence tη k u.
Lemma 4.10 In the same setting as Theorem 4.9, for λ k ą 0, choosing
η k :" 1 λ 2 k¨E u"D k " }g k pu k q´u k } 2 ‰ (8.5)
yields an upper bound η k P r0, 1s. This η k represents the proportion of the distribution D k that is not contained in the manifold M, i.e.,
|D k´M | ď η k ď 1, (8.6)
where |¨| denotes the measure of the set. Proof: For notational brevity, below we write u k " u k pωq and use α k " α k puq, as defined in (3.10). For each u k R M, observe that
}g k pu k q´u k } 2 " }α k pu k qpP M pu k q´u k q} 2 (8.7) " α 2 k pu k qd M pu k q 2 (8.8) " λ 2 k d M pu k q 2¨d M pu k q 2 (8.9) " λ 2 k . (8.10)
And, if u k P M, then g k pu k q " u k . Thus, for λ k ą 0,
1 λ 2 k¨} g k pu k q´u k } 2 " # 0 if u k P M, 1 if u k R M,(8.11)
Hence taking the expectation yields
Eω"Ω " 1 λ 2 k¨} g k pu k q´u k } 2 " P "
tω P Ω : u k pωq P Mu ı¨0`P " tω P Ω : u k pωq R Mu ı¨1 (8.12)
" P " tω P Ω : u k pωq R Mu ı (8.13)
" |D k´M |, (8.14) which gives the first equality for η k . Since |D k | " 1, the final inequality also holds, and the proof is complete.
The following lemma can be found in various forms in the literature (e.g., see [48,55,70]).
Lemma 8.2
If tδ n u is a sequence of nonnegative real numbers such that δ k`1 ď p1´γ k qδ k`γk σ k , for all k P N, (8.15) where tγ k u is a sequence in p0, 1s and tσ k u is a sequence in R such that Below is a proof of the main result, Theorem 4.9. The analysis for the Halpern iteration closely follows the approach in [72]. For completeness, we first restate the theorem. If the sequence tu k u is generated by Algorithm 2 and the minimizer θ k in Line 6 is chosen so that J θ k "´d M for all k P N (as permitted by Theorem 4.5), then the sequence tu k u converges to P M pu 1 q in mean square.
Proof: Before beginning the proof, we define the following quantities that are used throughout. Let pΩ, F, Pq be a probability space, where Ω, F, and P are the sample space, σ-algebra, and probability measure, respectively. We take D 1 " tu 1 pωq : ω P Ωu, zpωq :" P M pu 1 q, and, for all k P N,
δ k :" E ω"Ω " }u k´z } 2 ‰ , (8.19) d k :" E ω"Ω " }u k´z } ‰ , (8.20) σ k :" E ω"Ω " 2 xu k`1´z , u 1´z y ‰´1´γ k γ k¨α p2´αqd 2 k . (8.21)
For notational brevity, below we write u k " u k pωq, z " zpωq, and similarly for other quantities defined in terms of these expressions. We also use α k " α k puq, as defined in (3.10). Note α is a fixed constant whereas α k puq varies by iteration k and iterate u.
We proceed in the following manner. First an inequality is derived bounding the expectation of }g k pu k q´z} 2 (Step 1). This is used to show the sequence tδ k u is bounded (Step 2) and then obtain an inequality relating δ k`1 , δ k , and σ k as in (8.15) (Step 3). We next verify the limit supremum of the sequence tσ k u is finite (Step 4), which enables us to deduce that a subsequence of td k u converges to zero (Step 5). This implies lim sup kÑ8 σ k ď 0, and so δ k Ñ 0 (Step 6), completing the proof.
Step 1. We first derive a descent inequality for relaxed projections. Define the residual operator
Spuq :" u´P M puq. (8.22)
Fix any k P N. Using (3.10) with α k " α k puq, observe
}g k pu k q´z} 2 " }u k`α k pP M pu k q´u k q´z} 2 (8.23) " }u k´z } 2´2 α k xu k´z , u k´P M pu k qy`α 2 k }u k´P M pu k q} 2 (8.24) " }u k´z } 2´2 α k xu k´z , Spu k q´Spzqy`α 2 k }Spu k q} 2 ,(8.25)
where we note Spzq " 0 since z " P M pu 1 q P M. Furthermore, S is firmly nonexpansive (e.g., see Prop. 4.16 in [9]), which implies xu k´z , Spu k q´Spzqy ě }Spu k q´Spzq} 2 " }Spu k q} 2 . " }u k´z } 2´α k p2´α k qd M pu k q 2 (8.28)
" }u k´z } 2´λ k p2d M pu k q´λ k q. (8.29) Thus, taking the expectation of (8.29),
E ω"Ω " }g k pu k q´z} 2 ‰ ď E ω"Ω " }u k´z } 2 ‰´E ω"Ω " λ k p2d M pu k q´λ k q ‰ (8.30) ď E ω"Ω " }u k´z } 2 ‰´λ k p2d k´λk q (8.31) " E ω"Ω " }u k´z } 2 ‰´α p2´αqd 2 k (8.32) " δ k´α p2´αqd 2 k . (8.33)
The first equality above holds by the definition of λ k in Line 7 of Algorithm 1 and d k in (8.20), noting that J θ k "´d M and recalling (4.3). Also, the rightmost term in the final line (8.33) is nonnpositive since α P p0, 2q.
Step 2. Expanding the expression for δ k`1 , we deduce Step 3. To establish a useful inequality bounding δ k`1 , we expand this expression once again to obtain, for all k P N,
δ k`1 " E ω"Ω " }u k`1´z } 2 ‰ (8.34) " E ω"Ω " }γ k u 1`p 1´γ k qg k pu k q´z} 2 ‰ (8.35) ď E ω"Ω " γ k }u 1´z } 2`p 1´γ k q}g k pu k q´z} 2 ‰ (8.36) " γ k¨Eω"Ω " }u 1´z } 2 ‰`p 1´γ k qE ω"Ω " }g k pu k q´z} 2 ‰ (8.37) ď γ k δ 1`p 1´γ k qδ k (8.38) ď max pδ 1 , δ k q ,(8.δ k`1 " Eω"Ω " }u k`1´z } 2 ı (8.41) " Eω"Ω " }γ k u 1`p 1´γ k qg k pu k q´z} 2 ı (8.42) " Eω"Ω " γ 2 k }u 1´z } 2`p 1´γ k q 2 }g k pu k q´z} 2`2 γ k p1´γ k q xu 1´z , g k pu k q´zy ı (8.43) ď p1´γ k q¨Eω"Ω " }g k pu k q´z} 2 ı`2 γ k¨Eω"Ω " xu k`1´z , u 1´z y ı (8.44) ď p1´γ k q`δ k´α p2´αqd 2 k˘`2 γ k¨Eω"Ω " xu k`1´z , u 1´z y ı (8.45) " p1´γ k qδ k`γk " Eω"Ω " 2 xu k`1´z , u 1´z y ı´1´γ k γ k¨α p2´αqd 2 k ,(8.46)
where we leverage the definition of u k`1 and the inclusions γ k , p1´γ k q P r0, 1s. Substituting the definition of σ k from (8.21) into (8.46) yields the inequality δ k`1 ď p1´γ k qδ k`γk σ k , for all k P N. (8.47)
Step 4. We now show the limit supremum of tσ k u is finite. Indeed, the fact that, for all k P N, Step 5. Because (8.59) shows the limit supremum of the sequence tσ k u is finite, there is a convergent subsequence tσ n k u Ď tσ k u satisfying lim sup kÑ8 σ k " lim kÑ8 σ n k (8.60)
σ k ď E ω"Ω " 2 xu k`1´z , u 1´z y ‰ (8.48) ď E ω"Ω " }u k`1´z } 2`} u 1´z } 2 ‰ (8.49) " δ k`1`
" lim kÑ8 " E ω"Ω " 2 xu n k`1´z , u 1´z y ‰´1´γ n k γ n k¨α p2´αqd 2 E ω"Ω "ˇˇx u n k`1´z , u 1´z yˇˇ‰ ď E ω"Ω " 1 2`} u n k`1´z } 2`} u 1´z } 2˘ (8.62) " 1 2 pδ n k`1`δ 1 q (8.63)
ď δ 1 ,(8.64)
and so tE ω"Ω rxu n k`1´z , u 1´z ysu is a bounded sequence of real numbers. Thus, it contains a convergent subsequence txu m k`1´z , u 1´z yu (i.e., tm k u Ď tn k u). This implies, when combined with the convergence of tσ m k u and (8.61), existence of the limit lim kÑ8 1´γ m k γ m k¨α p2´αqd 2 m k . i.e., a subsequence td m k u of td k u converges to zero.
Step 6. Observe, for all k P N, E ω"Ω " }u k`1´uk } ‰ ď γ k E ω"Ω " }u k´u1 } ‰`p 1´γ k qE ω"Ω " }g k pu k q´u k } ‰ (8.67) ď γ k d 1`p 1´γ k q E ω"Ω rλ k s (8.68)
" γ k d 1`p 1´γ k qλ k (8.69)
" γ k d 1`p 1´γ k qαd k , (8.70) where (8.68) holds since, by the choice of g k in (3.10), g k puq´u P λ k Bd M puq ùñ }g k puq´u} ď λ k , (8.71) with the implication following from the fact that Bd M puq is a subset of the unit ball centered at the origin since d M is 1-Lipschitz. Utilizing (8.66) and the fact γ m k Ñ 0, we deduce where the final inequality holds by application of the Cauchy Schwarz inequality, and utilizing the fact that z " P pu 1 q and, by the projection identity (e.g., see Thm. 3.16 in [9], Thm 4.1 [22], and Thm 7.45 in [29]), xv´P M pu 1 q, u 1´P M pu 1 qy ď 0, for all v P M.
lim kÑ8 E ω"Ω " }u m k`1´u m k } ‰ ď lim kÑ8 γ m k d 1`p 1´γ m k qαd m k " 0.
Assumption 4. 1
1The true distribution D true is supported on a convex, compact set M Ă X .
Assumption 4. 2
2The initial distribution D 1 is uniformly bounded.
Assumption 4. 3
3For all k P N, the distribution D k is such that the push forward of the projection operation onto the manifold M recovers the true signal distribution D true up to a set of measure zero, i.e., D true " pP M q # pD k q :" tP M puq : u P D k u " tP M pu k pωqq : ω P Ωu.(4.1)
Theorem 4. 5
5Under Assumptions 4.1 and 4.3, for all k P N, d M is a solution to
Figure 2 :
2Flow of distribution D k toward the manifold M via Algorithm 1.
Figure 3 :
3Convergence plots for illustration used to describe the flow of distribution D k toward the manifold M in
Figure 4 :
4Reconstruction on a validation sample obtained with Filtered Back Projection (FBP) method, TV regularization, Adversarial Regularizer, and Adversarial Projections (left to right). Bottom row shows expanded version of corresponding cropped region indicated by red box.
Figure 5 :
5Reconstruction on a validation sample obtained with Filtered Back Projection (FBP) method, TV regularization, Adversarial Regularizer, and Adversarial Projections (left to right). Bottom row shows expanded version of corresponding cropped region indicated by red box.
Lemma 8. 1
1Let u P X and α P R. If λ " α¨d M puq, then the inclusion relation in (3.3) holds.Proof: First consider the case where u P M. Since d M is a metric, it is nonnegative. And,d M puq " inf vPM }v´u} ď }u´u} " 0 ùñ d M puq " 0. (8.1)Thus, u is a minimizer of d M . Because d M is convex and u is a minimizer, it follows that 0 P Bd M puq. Additionally, P M puq " u. Combining these results reveals u`α pP M puq´uq " u`α pu´uq " u`0 P u´λBd M puq.(8.2) Now suppose u R M. Then d M puq ą 0 and, by Lemma 2.2.28 in [18],
k pu k q´z} 2 ď }u k´z } 2´α k p2´α k q}Spu k q} 2(8.27)
noting that the boundedness of D 1 and M implies there exists a constant C ą 0 such that2}u 1 pωq´zpωq} ď 2p}u 1 pωq}`}zpωq}q ď C, for all ω P Ω, xP M pu m k`1 q´z, u 1´z y ‰ (8.78) E ω"Ω " 2 xu m k`1´P M pu m k`1 q, u 1´z y ‰ (8.79) ď lim kÑ8 C¨E ω"Ω " }u m k`1´P M pu m k`1 q} ‰ ,(8.80)
the triangle inequality and using the fact the projection P M is 1-Lipschitz yieldslim sup kÑ8 σ k ď lim kÑ8 C¨E ω"Ω " }u m k`1´u m k }`}u m k´P M pu m k q} ‰ (8.82) E ω"Ω " }P M pu m k q´P M pu m m k`1´u m k }`}u m k´P M pu m k q} ‰
Figure 9 :
9Additional human phantom reconstructions on a validation sample obtained with Filtered Back Projection (FBP) method, TV regularization, Adversarial Regularizer, and Adversarial Projections (left to right). Bottom row shows corresponding cropped region indicated by red box.
Table 2 :
2Average PSNR and SSIM on a validation dataset consisting 2,000 images of human phan-
toms.
ground truth
FBP
TV
Adv. Reg.
Adv. Proj.
SSIM: 0.361
SSIM: 0.656
SSIM: 0.715
SSIM: 0.758
PSNR: 14.67
PSNR: 17.46
PSNR: 22.88
PSNR: 25.93
holds by Assumption 4.8i. Through induction, it follows that tδ k u is bounded since δ k`1 ď δ 1 ă 8, for all k P N.(8.40) 39)
where (8.36) follows from (8.35) by Jensen's inequality, (8.38) holds by applying (8.33), and the
final inequality
This implies there exists N 1 P N such that σ k ď´1, for all k ě N 1 ,(8.55) and so δ k`1 ď p1´γ k qδ k´γk ď δ k´γk , for all k ě N 1 .which induces a contradiction since the sequence tδ k u is nonnegative. This proves (8.54) is false, and so´1 ď lim supδ1
(8.50)
ď 2δ 1
(8.51)
ă 8
(8.52)
implies
lim sup
kÑ8
σ k ă 8.
(8.53)
Next, by way of contradiction, suppose
lim sup
kÑ8
σ k ă´1.
(8.54)
(8.56)
By induction, it follows that
δ k`1 ď δ N1´k
ÿ
"N1
γ .
(8.57)
Applying Assumption 4.8iii and letting k Ñ 8 reveals
lim sup
kÑ8
δ k ď δ N1´8
ÿ
"N1
γ "´8,
(8.58)
kÑ8
σ k ă 8.
(8.59)
86 )
86where (8.86) follows from (8.85) by(8.66) and(8.73). Now, since the limit supremum of tσ k u is nonpositive, we may apply Lemma 8.2 to (8.47) to deduce δ k Ñ 0 ùñ limFigure 8: Additional ellipse reconstructions on a validation sample obtained with Filtered Back Projection (FBP) method, TV regularization, Adversarial Regularizer, and Adversarial Projections (left to right). Bottom row shows corresponding cropped region indicated by red box.kÑ8
E ω"Ω
"
}u k´z } 2 ‰
" 0,
(8.87)
completing the proof.
8.2 More Reconstructions
ground truth
FBP
TV
Adv. Reg.
Adv. Proj.
SSIM: 0.253
SSIM: 0.769
SSIM: 0.794
SSIM: 0.836
PSNR: 19.19
PSNR: 28.85
PSNR: 29.04
PSNR: 29.34
SSIM: 0.301
SSIM: 0.812
SSIM: 0.822
SSIM: 0.872
PSNR: 18.53
PSNR: 28.26
PSNR: 28.30
PSNR: 29.36
ground truth
FBP
TV
Adv. Reg.
Adv. Proj.
SSIM: 0.336
SSIM: 0.626
SSIM: 0.659
SSIM: 0.701
PSNR: 13.86
PSNR: 15.26
PSNR: 22.20
PSNR: 22.82
SSIM: 0.331
SSIM: 0.634
SSIM: 0.642
SSIM: 0.772
PSNR: 14.82
PSNR: 17.93
PSNR: 22.53
PSNR: 28.12
Here we use R " R Y t8u.2
For all practical purposes, we can assume ω " b and that Ω is the set of all plausible measurement data.
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Optimal transport: old and new. Cédric Villani, Springer Science & Business Media338Cédric Villani. Optimal transport: old and new, volume 338. Springer Science & Business Media, 2008.
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| {'fraction_non_alphanumeric': 0.0635318243173087, 'fraction_numerical': 0.033200083065102276, 'mean_word_length': 4.115116510655248, 'pattern_counts': {'":': 1, '<': 0, '<?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': 106, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We present a new framework, called adversarial projections, for solving inverse problems by learning to project onto manifolds. Our goal is to recover a signal from a collection of noisy measurements. Traditional methods for this task often minimize the addition of a regularization term and an expression that measures compliance with measurements (e.g., least squares). However, it has been shown that convex regularization can introduce bias, preventing recovery of the true signal. Our approach avoids this issue by iteratively projecting signals toward the (possibly nonlinear) manifold of true signals. This is accomplished by first solving a sequence of unsupervised learning problems. The solution to each learning problem provides a collection of parameters that enables access to an iteration-dependent step size and access to the direction to project each signal toward the closest true signal. Given a signal estimate (e.g., recovered from a pseudo-inverse), we prove our method generates a sequence that converges in mean square to the projection onto this manifold. Several numerical illustrations are provided.1. φpuq ě 0 with equality if and only if u P D true .', 'arxivid': '2008.02200', 'author': ['Howard Heaton \nUniversity of California\nLos Angeles †\n', 'Samy Wu Fung \nUniversity of California\nLos Angeles †\n', 'Alex Tong Lin \nUniversity of California\nLos Angeles †\n', 'Stanley Osher \nUniversity of California\nLos Angeles †\n', 'Wotao Yin \nUniversity of California\nLos Angeles †\n'], 'authoraffiliation': ['University of California\nLos Angeles †', 'University of California\nLos Angeles †', 'University of California\nLos Angeles †', 'University of California\nLos Angeles †', 'University of California\nLos Angeles †'], 'corpusid': 221045076, 'doi': None, 'github_urls': ['https://github.com/lunz-s/'], 'n_tokens_mistral': 24683, 'n_tokens_neox': 21475, 'n_words': 13031, 'pdfsha': '121c4629a70f8ee50e3d0b29efb7e01291f1708d', 'pdfurls': ['https://arxiv.org/pdf/2008.02200v1.pdf'], 'title': ['Projecting to Manifolds via Unsupervised Learning', 'Projecting to Manifolds via Unsupervised Learning'], 'venue': []} |
arxiv |
Influence of Defects on the Valley Polarization Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition
10 Dec 2022
Faiha Mujeeb [email protected]
Department of Physics
Indian Institute of Technology Bombay
400076PowaiMumbaiIndia
Poulab Chakrabarti
Department of Physics
Indian Institute of Technology Bombay
400076PowaiMumbaiIndia
Vikram Mahamiya
Department of Physics
Indian Institute of Technology Bombay
400076PowaiMumbaiIndia
Alok Shukla
Department of Physics
Indian Institute of Technology Bombay
400076PowaiMumbaiIndia
Subhabrata Dhar
Department of Physics
Indian Institute of Technology Bombay
400076PowaiMumbaiIndia
Influence of Defects on the Valley Polarization Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition
10 Dec 20221 arXiv:2212.05247v1 [physics.optics]
Here, the underlying mechanisms behind valley de-polarization is investigated in chemical vapor deposited 1L-MoS 2 . Temperature dependent polarization resolved photoluminescence spectroscopy was carried out on as-grown, transferred and capped samples. It has been found that the momentum scattering of the excitons due to the sulfur-vacancies attached with air-molecule defects have strong influence in valley de-polarization process. Our study reveals that at sufficiently low densities of such defects and temperatures, long range electron-hole exchange mediated intervalley transfer due to momentum scattering via Maialle-Silva-Sham (MSS) mechanism of excitons is indeed the most dominant spin-flip process as suggested by the theory 1 . Rate of momentum scattering of the excitons due to these defects is found to be proportional to the cube root of the density of the defects. Intervalley transfer process of excitons involving Γ-valley also has significance in valley de-polarization process specially when the layer has tensile strain or high density of V S defects as these perturbations reduces K to Γ-energy separation. Band-structural calculations carried out within density functional theory framework validate this finding. Experimental results further suggest that exchange interactions with the physisorbed air molecules can also result in the intervalley spin-flip scattering of the excitons and this process give an important contribution to valley depolarization specially at the strong scattering regime.
Two-dimensional transition metal dichalcogenides (TMDs) offer valley degree of freedom, which can be exploited to design next-generation valley based electronics or 'valleytronics' 2 .
The broken inversion symmetry, together with strong spin-orbit coupling, results in the valley-dependent optical selection rules in monolayer (1L)-MoS 2 . This property enables an exciton to sustain its valley character throughout the time of its existence. In fact, as high as 100% valley polarization has been reported in exfoliated 1L-MoS 2 samples 2-6 .
Whereas, 1L-MoS 2 films grown by chemical vapour deposition (CVD) technique, which is frequently used to grow large area films on different substrates, show only moderate values of polarization (less than 50%) 7 . Since large area coverage of the monolayer film has to be ensured for any practical application of the material, it is imperative to understand the reason for moderation of valley polarization in CVD grown 1L-MoS 2 . Note that the optical and electrical properties of CVD grown layers often suffer from the presence of a high density of sulfur vacancy defects (V S ) and the residual strain [8][9][10][11][12][13][14] . Since the valley and spin properties are closely related to the crystal symmetry, both the strain 14 tensile strain 20,21 . However, the underlying mechanism through which defects govern VP in this system is yet to be systematically investigated.
In an ideal scenario, bright excitons generated in one of the K-valleys through circularly polarized (σ + or σ − polarization) photons are expected to stay in the same valley until recombination. One may think that intervalley phonon scattering along with spin flipping of both electron and hole are necessary to transfer a bright exciton between K to K -valleys.
However, such processes are rare because neither D'yakonov-Perel' (DP) nor Elliott-Yafet (EY) mechanism can result in spin relaxation of electrons/holes as the out-of-plane spin component is conserved for both the carriers 2,22-24 . But in reality, excitons do move between K to K -valleys and in certain cases, the de-polarization rate is shown to be extremely fast even in exfoliated 1L-MoS 2 samples 25-27 . A recent theory suggests that long range part of the electron-hole exchange interaction can virtually transfer excitons between K to K -valleys 1 without directly involving any phonon. In this process, excitons can experience in-plane effective magnetic field Ω(P ex ) that depends upon its in-plane centre of mass momentum In the room temperature PL spectra recorded with 532 nm laser excitation for the three samples as shown in Fig. 1(a), A-excitonic features are found at almost the same energy positions for the as-grown and the capped samples, whereas the feature appears at a higher energy position for the transferred sample. This blue shift implies the release of the tensile strain after transfer of the monolayer from the sapphire to the amorphous SiO 2 /Si substrate 13,14 . Note that the as-grown 1L-MoS 2 layer on sapphire is expected to be under a tensile biaxial strain due to the mismatch in the thermal expansion coefficient and/or lattice constant 28-30 . The broad luminescence feature (D) appearing at ∼ 1.75 eV in Fig. 1
(b),
where 85 K PL spectra are compared, can be attributed to those V S sites where air molecules, such as oxygen and water, are physisorbed 11,15,31 . Evidently, the intensity of D peak is significantly less in the transferred sample M3 as compared to that of the as-grown sample M1. D-peak is almost fully suppressed in the capped sample M2. Annealing followed by capping in the preparation process of samples M2 and M3 are found to be the reason for the reduction of D feature 15 . It is interesting to note that the trion peak is stronger than the excitonic feature in sample M3, implying a large enhancement of electron concentration, which can be attributed to polystyrene. As an aromatic hydrocarbon, PS has the potential to act as donors 15 . with I −/+ as the intensity of σ −/+ light is also plotted as functions of photon energy in respective panels. Evidently, in all cases, polarization could only be observed at the Aexciton/trion features, while the D-band does not show any polarization at all. Note that D-feature is almost completely suppressed in the sample M2 and the sample shows higher P than that is typically obtained in as-grown samples. This finding highlights the role of V S -Air defects in determining the valley polarization property of the material. Interestingly, P goes as high as 82% in case sample M3, where the intensity of the D-peak is significantly reduced as compared to that is generally found in as grown samples, but the reduction is not as much as it found in sample M2. Observation of higher P in sample M3 than M2 even when V S -Air defect density is larger in the former, can be attributed to the relaxation of biaxial strain in the MoS 2 film after the transfer 15 . Polarization resolve PL spectra are recorded at several spots on each sample. The relative intensity of the D-feature with respect to A-exciton complex I D /I A at each sampling point can serve as a measure for the density of V S -Air defects at that location. Note that in case of the as-grown samples, even though the ratio is found to vary significantly from spot to spot, the position of both D-and A-features does not change much. In the case of capped 2(Q s /γ)(γ r γ D /βγ r D ) α . Note that γ r and γ r D are independent of the defect concentration. At low defect densities and sufficiently low temperatures, recombination of excitons takes place mostly through radiative pathways and hence γ ≈ γ r and γ D ≈ γ r D . S can thus be treated as independent of N D . The rapid initial fall in P versus I D /I A plot shown in Fig. 3(a) can now be explained. In Fig. 3(b), ln (P −1 − 1) is plotted versus ln (I D /I A ) for all data points shown in Fig. 3(a). Data obtained for sample M1 and M2 clearly follow a straight line.
While those from sample M3 can be fitted with a separate but parallel straight line with a slope of α =0.33. This further establishes the validity of the P versus I D /I A relationship in explaining the experimental results in the weak scattering regime. It should be noted that MSS mechanism predicts γ s ∝ r −1 p in the strong scattering regime, meaning P should increase with r p , while r p is expected to increase with T . In 3(c) and (d), the observation of the initial decrease and increase of P with the rising temperature when I D /I A is low and high, respectively, can be assigned to the weak and strong scattering regimes, respectively.
At sufficiently high temperatures, P has been found to decrease with increasing T in all cases. This can be attributed to the increase of center of mass momentum of the excitons p ex . Since the band gap of the material decreases with the increase of T , for the same photon energy of excitation, the probability of generation of excitons with higher p ex increases, and according to the theory 1 , γ s increases with p ex . Polarization data presented in Fig. 3(a) show a plateauing tendency beyond I D /I A ≈ 1.5 before abruptly dropping down to zero at I D /I A ≈ 6. However, theory predicts P to enhance with I D /I A beyond a certain point as the defect density moves from the weak to strong scattering regime. This may indicate the presence of other competing mechanisms, which increase the excitonic spin relaxation rate with I D /I A .
One of the possible candidates might be the exchange interaction of the excitons with the air molecules attached at the V S -sites. Note that certain air-molecules, such as O 2 , H 2 O, possess magnetic moment 32 . Physisorption of such molecules at the V S -sites, can interact with the excitons through exchange coupling. Spin relaxation rate of the excitons due to these scattering processes is expected to be proportional to the density of these defects 33,34 .
We believe that the sudden drop of P to zero when plotted as a function of I D /I A (at ∼ 6) in Fig. 3(a) Calculated band structures for 1L-MoS 2 , when the layer is pristine, biaxially strained (tensile), unstrained but with bare V S defects and unstrained but with V S -O 2 defects, are compared in Fig. 4. It is noticeable that as compared to the pristine layer, the energy separation between the K/K -and the Γ-valley is reduced whenever the layer is either under a tensile biaxial strain or incorporated with the defects. Reduction of the energy gap can enhance the chance of the holes to transfer between the K and K-valleys via Γ-valley through phonon assisted inter-valley processes. However, it has to be noted that the z-component of hole spin is still a good quantum number even for the Γ-valley. DY mechanism can not thus be a dominant process for spin relaxation in this path way 1 . Rather, Elliott-Yafet (EY) mechanism should play more significant role in the hole spin relaxation process at the Γ-valley.
In conclusion, the momentum scattering of excitons due to V S -air defects has been found to play a vital role in valley de-polarization process of CVD grown 1L-MoS 2 . The study clearly demonstrates that at sufficiently low defect densities and temperatures, long range electron-hole exchange mediated transfer of excitons between K/K -valleys indeed happens due to momentum scattering via MSS mechanism as theoretically proposed 1 . Momentum scattering rate of the excitons due to these defects comes out to be proportional to the cube root of the defect density. Intervalley transfer process of excitons involving Γ-valley also play substantial role specially when the layer has tensile strain or high density of V S defects as K to Γ-energy separation decreases with these perturbations. The study further suggests that the exchange interaction between the excitons and the physisorbed air molecules can also lead to intervalley spin-flip scattering. Such processes also give substantial contribution to valley depolarization specially at strong scattering regime. Hz. The base pressure of the chamber was measured to be less than 1 x 10 −5 mbar. The deposition is performed under N 2 atmosphere, keeping a pressure of 2 x 10 −2 mbar inside the chamber under a flow of 5N pure N 2 gas at 100 sccm. The substrate to target working distance was kept at 5 cm. The Scanning electron microscope (SEM) image of the deposited film is given in Fig. 6(c). The thickness of the deposited film is found to be ∼ 22 nm using cross-sectional SEM, shown in the inset of Fig. 6(c). The XPS spectra of both B 1s and N 1s levels of the deposited layer are given in Fig.7 (a,b), respectively. After a Shirley background subtraction, the spectra are deconvoluted using mixed Gaussian (80%)-Lorentzian (20%) functions. As shown in the Fig. 7 (a)
IV. RAMAN SPECTRA
Raman spectra on these samples were recorded at room temperature in back scattering geometry with 532 nm diode laser using Horiba JobinYvon HR800 confocal Raman spectrometer. Results are shown in Fig. 8. In all cases, the characteristic in-plane E 1 2g and out-of-plane A 1g vibrational modes for the zone-center phonons could be seen at ∼385 and ∼405 cm −1 , respectively. The separation between the two featires, which serves as a good indicator of the layer thickness, comes out to be ∼ 20 cm −1 for every sample. This further demonstrates monolayer nature of these MoS 2 flakes. The population of exciton at K/K valleys (X/X ) and bound excitons (X D ) can be written as,
dX dt = G − γX − β(N D − n D )X − (X − X )γ s (1) dX dt = −γX − β(N D − n D )X − (X − X)γ s (2) dX D dt = β(N D − X D )(X + X ) − γ D X D(3)
Where G is the pumping rate of neutral exciton at K valley by left circularly polarized light. And, γ, γ s and γ D are total recombination rate of exciton, inter-valley relaxation rate and the recombination rate of bound exciton, respectively. N D , n D and β is the defect concentration in the sample, the population of neutral exciton bound to the defect state and the coefficient of transition of exciton from free to bound (defect) state.
Under steady state condition, dX(X )
dt = dX D dt = 0. β(N D − n D )(X + X ) − γ D X D = 0(4)
Under N D >> X D ,
X D (X + X ) = βN D γ D(5)
Also,
− γX − β(N D − n D )X − (X − X)γ s = 0(6)
The equation 6 can be rearranged to,
X X = 1 1 + γ+βN D γs(7)
The polarization helicity (P) can be expressed as,
P = I − − I + I − + I + = X − X X + X(8)
Assuming γ and γ s depends on the defect concentration N D as follows,
γ s = Q s N α D (10) γ = γ 0 + Q A N α 1 D(11)
Where γ 0 is the rate of recombination when the sample with no defect, Q s and Q A are constants.
P = 1 1 + 2 QsN α D γ 0 +Q A N α 1 D +βN D(12)
When the defect concentration is low, N D → 0, γ 0 >> Q A N α 1 D + βN D , γ 0 ∼ γ equation 12 becomes,
P = 1 1 + 2 QsN α D γ(13)
The PL intensity of D feature I D , normalized on total exciton intensity I A = I K A + I K A ,
I D I A = X D γ r D (X + X )γ r = βN D γ r D γ D γ r(14)
Using
,15 and the defects 16-19 are expected to have certain impacts on the valley polarization (VP) property of 1L-MoS 2 grown by CVD technique. It has indeed been experimentally demonstrated that VP decreases with increasing tensile strain in the 1L-MoS 2 14,15 . This has been explained in terms of longitudinal acoustic (LA) phonon assisted intervalley scattering of the excitons via Γ valley as the K to Γ-valley energy separation decreases with the increase of biaxial
Fig. 2
2shows the polarization resolved PL spectra with σ − excitation recorded at 85 K on the three samples. Degree of valley polarization, which is defined as P = (I − −I + )/(I − +I + )
FIG. 2 .
2Circular polarization resolved PL spectra recorded at 85 K for the sample (a)M1, (b)M2 and (c)M3. Degree of polarization (P ) is also plotted as a function of photon energy in these figures (dashed blue lines).
FIG. 3 .
3(a) Degree of circular polarization (P ) as a function of I D /I A obtained at various sampling points on different samples. (b) Plot of ln (P −1 − 1) vs ln (I D /I A ) for all the sampling points. Temperature dependence of P for (c) as-grown samples with two different I D /I A ratios and (d) the transferred sample.and transferred samples, neither the I D /I A ratio nor the peak positions show much spatial variation. Degree of polarization P obtained at 85 K from various parts of these samples is plotted versus I D /I A inFig. 3(a). In case of the as-grown and the capped samples (M1 and M2), P obtained at a fixed energy of 1.945 eV, while for the transferred sample, P measured at the peak of the A-exciton/trion complex is used for the plot. Since the higher energy side of the PL feature corresponding to the A-exciton/trion complex can not be visible with the 633 nm (1.96 eV) excitation, I D /I A ratio is obtained from the PL spectra recorded with 532 nm laser excitation at the same spot in all cases. Evidently, for the as-grown and the capped samples, all the data obey a trend of rapid initial decrease followed by a plateauing as I D /I A ratio increases. Interestingly, beyond a certain I D /I A ratio, P suddenly drops to zero. Note that the data obtained from the transferred sample stay clearly isolated from other data points in the plot. But, they also show a reduction as I D /I A increases. These observations clearly demonstrate the correlation between the V S -Air defects and P .Fig. 3(c) compares the temperature (T ) variation of P recorded for an as-grown sample at two spots with different I D /I A ratios. Interestingly, P shows a monotonous decrease with the increase in temperature when I D /I A is only 0.19. While P initially increases and then decreases with rising temperature for I D /I A =1.87. We have investigated several spots with different I D /I A ratios, and P is found to consistently exhibit an initial trend of either reduction or enhancement with increasing T depending upon whether I D /I A is sufficiently low or high, respectively.Fig. 3(d)plots P as a function of T for the transferred sample M3 at a spot with I D /I A =0.2. P , in this case, shows a reduction followed by a plateauing tendency as T increases. Beyond ∼250 K, the polarization suddenly drops to zero.Upon illumination with a circularly polarised light falling perpendicularly to the layer plane, A-excitons/trions are generated in one of the K-valleys depending upon the helicity of the incident light. Generated excitons can either be transferred to other non-equivalent K valleys through inter-valley transition processes or can be captured by the V S -Air defect centers before recombination. One can consider that the excitons are generated at a rate of G in only one of the valleys (say K-valley). At the steady state condition, the population of excitons in the K, K -valleys (X and X ) and the defect sites (X D ) can be obtained in terms of G, total recombination (radiative plus non-radiative) rate γ of the excitons, intervalley relaxation rate γ s of the excitons, total recombination rate of the bound excitons γ D , coefficient of transition of A-excitons to the defect bound state β and the defect concentration N D by solving the rate equations. Considering X D /N D << 1, polarization can be obtained as P = 1/[1 + 2γ s /(γ + βN D )]. One may further contemplate that the rate of recombination of A-excitons γ is much higher than the rate of their capture at the defect sites βN D . Polarization can then be expressed as P = 1/[1 + 2γ s /γ]. More details of these calculations could be found in the supplementary. According to the theory proposed by Yu and Wu, inter-valley spin scattering rate γ s of the A-excitons should depend on the momentum scattering rate (r p ) of the excitons through Maialle-Silva-Sham (MSS) mechanism 1 . It is quite reasonable to believe that the presence of air-molecules at the S-vacancy sites introduces certain additional local vibrational modes, which can take part in the momentum scattering of excitons. One may thus consider that r p ∝ N α D , where α is a constant. In the weak scattering regime, γ s ∝ r p and hence γ s = Q s N α D , where Q s is a constant. One can also express defect concentration N D in terms of the intensity ratio I D /I A as N D = (γ r γ D /βγ r D )(I D /I A ), where γ r and γ r D are the radiative recombination rate of A-excitons/trions and the defect bound excitons, respectively (see supplementary). P can now be given by P = 1/[1 + S(I D /I A ) α ], where S =
FIG. 4 .
4is due to the change in the band structure as a result of the inclusion of a large density of disorder in the lattice at such a high defect concentration. In fact, the reduction of the valley polarization with the increase of disorder in 1L-MoS 2 has been reported 35 . Calculated band structures of 1L-MoS 2 : when the film is (a) pristine, (b) biaxially strained (tensile), (c)unstrained but with bare V S defects and (d) unstrained but with V S -O 2 defects. Here, we have carried out band structural calculations under the framework of density functional theory (DFT) to understand the effect of different perturbations such as V Sformation, physi-adsorption of air-molecules with the V S -sites and biaxial strain on the band structure of 1L-MoS 2 . More details about the calculation can be found in the supplementary.
Supplementary:FIG. 5 .
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Physics Reports, 493(2-4):61-236, 2010. 34 WH Koschel and M Bettini. Zone-centered phonons in aibiiis2 chalcopyrites. physica status solidi (b), 72(2):729-737, 1975. 35 Qinsheng Wang, Shaofeng Ge, Xiao Li, Jun Qiu, Yanxin Ji, Ji Feng, and Dong Sun. Valley carrier dynamics in monolayer molybdenum disulfide from helicity-resolved ultrafast pump-probe spectroscopy. ACS nano, 7(12):11087-11093, 2013. 36 P. K. Mohapatra, S. Deb, B. P. Singh, P. Vasa, and S. Dhar. Strictly monolayer large continuous MoS 2 films on diverse substrates and their luminescence properties. Appl. Phys. Lett., 108:042101, 2016. 37 Alper Gurarslan, Yifei Yu, Liqin Su, Yiling Yu, Francisco Suarez, Shanshan Yao, Yong Zhu, Mehmet Ozturk, Yong Zhang, and Linyou Cao. Surface-energy-assisted perfect transfer of centimeter-scale monolayer and few-layer mos2 films onto arbitrary substrates. ACS nano, 8(11):11522-11528, 2014. Supplementary Information Influence of Defects on the Valley Polarization Properties of Monolayer MoS 2 Grown by Chemical Vapor Deposition Faiha Mujeeb, Poulab Chakrabarti, Subhabrata Dhar Department of Physics, Indian Institute of Technology Bombay, Powai, Mumbai-400076, India I. GROWTH OF 1L-MOS 2 USING CHEMICAL VAPOR DEPOSITION 1L-MoS 2 films were grown on double-sided polished c-plane sapphire using microcavity based chemical vapor deposition technique. High purity MoO 3 (99.5 %) and S (99.7 %) were used as precursors and Argon was used as a carrier gas. The MoO 3 (6 g) was filled inside a ceramic boat and placed at the center of the 2-inch furnace. Another boat was filled with S powder (350 g) and placed at one end. Three substrates were used, in which two were placed parallel to each other on top of the MoO 3 filled boat, and the third one was kept on top of both in the middle, and it is supported by two sapphire strips. The gap formed between the substrate and sapphire strips acts as a natural reactor cavity. More details about the growth method can be found elsewhere 36 . The AFM image of the film grown on sapphire substrate is shown in Fig. 6. The average height (after sampling at different locations of the image) is found to be 0.7 nm, which is typically reported for a 1L-MoS 2 film. AFM image for the as-grown sample. The inset shows the height distribution profile along the blue line drawn across 1L-MoS 2 -substrate boundary.II. TRANSFER OF 1L-MOS 2 ONTO SIO 2 /SI WAFER 1L-MoS 2 films are transferred from the sapphire substrates to SiO 2 /Si wafers using a surface energy assisted transfer procedure using Polystyrene (PS) as the carrier polymer 37 .The PS solution was spin-coated on as-grown MoS 2 /sapphire samples and then baked at 80-90 • C for 35 min followed by 120 • C for 10 min. MoS 2 /PS assembly was gently poked at the edge, while water was dropwise added onto the film, which allows the water to go underneath the MoS 2 . This helps MoS 2 /PS assembly to be lifted from the sapphire substrate.The MoS 2 /PS assembly was then transferred onto SiO 2 /Si wafer. After transferring to the target substrate, it was again baked for 80-90 • C for 35 minutes followed by 120 • C for 10 minutes. Finally, the PS layer is removed by rinsing it in toluene. The AFM image for the transferred sample is shown inFig. 6. The average height between the 1L-MoS 2 and the substrate for the transferred sample is found to be of ∼3 nm with a rms roughness of 0.8 nm. This suggests that a ∼2.3 nm thick coating of PS is still remaining on top of the film even after rinsing in toluene for several times.
FIG. 6 .
6AFM image for the transferred sample. The inset shows the height distribution profile along the blue line drawn across 1L-MoS 2 -substrate boundary. III. CAPPING OF 1L-MOS 2 USING HBN PELLETS The 1L-MoS 2 samples were vacuum annealed in a pulsed laser deposition chamber and capped using the deposit formed using the laser ablation of the h-BN pellet. A KrF excimer laser with a wavelength of 248 nm and pulse width of 25 ns was used to ablate the h-BN pellet. The energy density of the laser pulse was kept at 1.4 J cm −2 at a frequency of 5
FIG. 7 .
7XPS core level spectra for (a) B 1s and (b) N 1s levels of the capped film. (c) SEM image of the deposited layer using BN pellets. The cross-section image is shown in the inset.
spectrum is deconvoluted with five peaks attributed to B 2 O 3 (192 eV) 38 , h-BN (191 eV) 39 , B-C (189.2 eV) 40 , BCN (188 eV) 41 and B 6 O (187.4 eV) 42 . The N 1s spectrum is deconvoluted with three peaks corresponding to h-BN (398.2 eV) 43 , N-C (399.2 eV) 44 and N-O (401 eV) 45 .
FIG. 8 .
8Room temperature Raman spectra normalized at A 1g peak for as-grown (sample M1) and capped (sample M2) and transferred (sample M3) samples. V. DYNAMICS OF OPTICAL PUMPING OF VALLEY POLARIZATION IN 1L-MOS 2
carried out density functional theory (DFT) simulations 46 by employing the Vienna ab-initio simulation package (VASP) 47,48 to investigate the influence of strain and sulfur vacancies on the valley polarization properties of MoS 2 . We have taken the Perdew-Burke-Ernzerhof (PBE) 49,50 exchange-correlation functional along with the generalized gradient approximation and considered the plane wave basis expansion up to a kinetic energy cut-off of 450 eV for the calculations. The Monkhorst-pack k -grids of 5×5×1 and 7×7×1 kpoints were taken for the geometry relaxation and density of states calculations, respectively. We apply a convergence limit of 0.02 eV/Å and 10 −5 eV for the calculations of Hellman-Feyman forces and total energy, respectively. The effects of spin-orbit coupling is incorporated and the van der Waals interactions are taken into account by employing Grimme's dispersion corrections of DFT-D3 type 51 . A 4×4 supercell of MoS 2 is considered for the simulations, and a vacuum space of 20Å is introduced along the z-direction to avoid the periodic interactions in our system. One sulfur vacancy is introduced in the 4×4 supercell of MoS 2 , and an oxygen molecule is adsorbed onto the sulfur vacancy. The top and side views of the relaxed structure of 4×4 supercell of MoS 2 with one passivated sulfur vacancy are shown in
Fig.
??(a,b), respectively.
FIG. 9 .
9Optimized structure of 4×4 supercell of MoS 2 with one passivated sulfur vacancy (a) Top view and (b) Side view.
P ex .
exPrecession of the exciton total angular momentum about Ω(P ex ) can cause valley de-polarization due to inhomogeneous broadening. Exciton momentum scattering rate can influence its spin scattering rate through Maialle-Silva-Sham (MSS) mechanism, which has similar characteristics as the DY process for the electrons and holes. In the weak scattering regime, the spin scattering rate is proportional to the momentum scattering rate, while the two rates follow the inverse relationship in the strong scattering regime. Presence of defects can thus influence the momentum relaxation rate of the excitons and hence can affect thevalley de-polarization. Here we explore the influence of sulfur vacancy related defects on the valley polarization property of CVD grown 1L-MoS 2 . Our study reveals that momentum scattering of the excitons due to the sulfur-vacancies, which are physisorbed with air-molecules, influence the intervalley spin-flip transition rate of the excitons and hence the valley de-polarization process. Both weak and strong scattering regimes of the intervalley excitonic transfer processes could indeed be identified from the dependence of the degree of valley polarization on the defect concentration and the temperature, validating the MSS mechanism. It has been found that in the presence of biaxial tensile strain, high defect densities and/or at sufficiently high temperatures, (LA) phonon assisted intervalley scattering via Γ valley becomes important. This has been corroborated by ab-initio band-structural calculations. Three types of samples were used for the study; CVD-grown 1L-MoS 2 films on sapphire substrates [sample M1], 1L-MoS 2 films-on-sapphire capped by the deposits from hBN pellets using pulsed laser deposition(PLD) [sample M2], and CVD grown 1L-MoS 2 filmstransferred onto SiO 2 /Si substrates using a polystyrene(PS) based surface energy-assisted
transfer procedure [sample M3]. More details about the growth, transfer process and char-
acterizations of these samples can be found in the supplementary. Photoluminescence (PL)
and polarization-resolved PL studies were conducted keeping the samples in a liquid ni-
trogen cryostat. Measurements were carried out in backscattering configuration within a
microscope set-up equipped with a 50X long working distance objective (NA= 0.5). For
PL, a 532 nm diode laser was used as excitation source. For polarization-resolved PL, an
achromatic quarter wave-plate was used to produce circularly polarized lights (σ −/+ ) from
the linearly polarized HeNe (632.8 nm) laser. A combination of a separate achromatic quar-
ter wave-plate and a Glan-Tylor analyzer was placed before spectrometer entrance slit to
select between σ − and σ + emitted photons. The spectra were recorded using a 0.55 m focal
length monochromator equipped with Peltier cooled CCD detector. To avoid Joule heating,
excitation intensity was kept at 150 µw on a spot diameter of ∼5 µm.
FIG. 1. Normalized (with respect to the A excitonic feature) photoluminescence spectra recorded
with 532 nm laser excitation for different samples at (a) room temperature and (b) 85 K.
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arxiv |
Mathematics Subject Classification: Primary:32B20,14P15; Secondary: 32C05, 32C09
2010
Daniel Barlet [email protected]
Teresa Monteiro Fernandes
D Barlet
T Monteiro Fernandes
Institut Elie Cartan
Laboratoire de Mathématiques
Université de Nancy
B.P. 23954506Vandoeuvre lès Nancy CedexFrance
Centro de Matemática e Aplicações Fundamentais Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa Edifício
C6, P.2, Campo Grande1749-16LisboaPortugal
Mathematics Subject Classification: Primary:32B20,14P15; Secondary: 32C05, 32C09
20101 2Grauert's theoremsubanalytic setsStein open sets
By open neighbourhood of an open subset Ω of R n we mean an open subset Ω ′ of C n such that R n ∩ Ω ′ = Ω. A well known result of H. Grauert implies that any open subset of R n admits a fundamental system of Stein open neighbourhoods in C n . Another way to state this property is to say that each open subset of R n is Stein. We shall prove a similar result in the subanalytic category, so, under the assumption that Ω is a subanalytic open subset in a paracompact real analytic manifold, we show that Ω admits a fundamental system of subanalytic Stein open neighbourhoods in any of its complexifications.
Introduction.
A classical result of H. Grauert gives that an open set in a real analytic manifold M R is locally the trace on M R of a Stein open set in any given complexification M C of M R . The analogous result in the semi-analytic setting is easy to obtain because when f is a real analytic function, say near 0 in R n , the set {f > 0} is near 0 the trace on R n on the Stein open set {ℜ(f ) > 0} cut with a small open ball in C n . We solve the subanalytic case of this problem using the rather deep following result (Theorem 2.1 below):
• a compact subanalytic set in R n may be defined as the zero set of a C 2 subanalytic function on R n .
The construction of the subanalytic Stein open subset we are looking for is then an easy consequence of H. Grauert's idea. Let us mention without technical details that applications of our result arise naturally in the theory of sheaves on subanalytic sites, as it has been developped by L. Prelli in [14] (cf. [11] for the foundations of the theory of ind-sheaves). It entails, for instance, that the subanalytic sheaf of tempered analytic functions on a real analytic manifold is concentrated in degree zero as in the classical case. We conclude this article by computing one very simple example which is not semi-analytic in order to show that the subanalytic case is much more involved and also to explain to non specialists of subanalytic geometry (as we are) what are the ideas and tools hidden behind this construction.
We wish to thank Adam Parusinski for having pointed out to us a precise reference of Theorem 2.1, and the referee for asking us about the unbounded case.
Main results and proofs
We refer to [1], [3], [10] and [13] for the basic material on subanalytic geometry.
The following result is due to Bierstone, Milman and Pawlucki in a private letter to W. Schmid and K.Vilonen in 1995 (cf. [16]). We refer [4], C.11, for a proof in the more general setting of o-minimal structures.
Theorem 2.1. Let A be a compact subanalytic set of R n and let p ∈ N be given. Then there exists a C p subanalytic function f in R n such that A = f −1 (0).
Remark 2.2. Let
U be a open ball in R n and Z a relatively compact subanalytic open set in U. Then there exists a C 2 subanalytic function g : R n → R + with compact support in U such that
Z = {x ∈ R n ; g(x) > 0}.
Apply the previous theorem toŪ \ Z and define g to be f on U and 0 on R n \ U. As U is subanalytic and f identically zero around ∂U, this function g satifies the required properties. Moreover, we can divide this function g by any given positive constant without changing the set Z, so for each ε > 0 we may assume that the Levi form of g is uniformely bounded on R n by ε.||z|| 2 . ǫ such that ϕ(V ) is the closed ballB C ǫ/2 , and ϕ is real on W ∩ M R . In particular,V ∩ M R ⊂ U is a compact subanalytic subset, andŪ is a compact subanalytic subset of W . As M R is paracompact, we get a locally finite countable cover (U i ) i∈N * of Ω such that the conditions above are satified. On each U i , by the remark following the Theorem 2.1, we may choose a C 2 non negative subanalytic function f i on M C with compact support in U i whose non zero set is exactly V i ∩ Ω, and such that its Levi form is bounded by h/2 i for any given hermitian metric h on M C . Then define f := ∞ i=1 f i . As this sum is locally finite, it clearly satisfies our requirements. The last assertion follows by applying this construction in any open neighbourhood W ofΩ in M C regarded as a complexification of W ∩ M R . q.e.d.
(2.1) Ω = Ω C ∩ M R
Proof. Let n be the dimension of M R . By Grauert's Theorem 3, page 470 of [5], there exist a natural number N ∈ N and a real analytic regular proper embedding ϕ of M R in the euclidean space R N . By complexification, one defines a holomorphic map ϕ C in a neighborhood V of M R in M C taking values in C N , such that ϕ C | M R = ϕ and such that the rank of ϕ C is everywhere equal to n. Note that the Levi form of the real analytic function
g(z 1 , ..., z N ) = N j=1 (ℑ z j ) 2
is half the square norm in C N , hence g is strictly plurisubharmonic on C N . By the maximality of the rank of ϕ C , the function ϕ * C (g) is also strictly plurisubharmonic on V and subanalytic (in fact analytic). Fix now a smooth hermitian metric 1 h on T C V such that the Levi form of ϕ * C (g) is bigger at each point than 2.h. By Proposition 2.3, there exists a subanalytic C 2 non negative function f with support on V such that is (strongly 1-complete) Stein by Grauert's famous result and subanalytic in M C by construction. Moreover, as we have ϕ *
C (g) = 0 in M R , it follows that Ω C ∩ M R = Ω. q.e.d.
Example: A strange four-leaved trefoil
Our aim is now to give an explicit construction of the function f in Theorem 2.1 in the case of one of the simplest example which is not semi-analytic.
For that purpose we shall only use Lojasiewicz inequalities and Theorem 3.2 which are basic tools in subanalytic geometry. We think that this analysis will convince the reader of the strength and usefulness of Theorem 2.1 and that this tool is far from being elementary.
We shall need the following refinement of subanalyticity.
Strong subanalyticity
For a continuous function f : R n → R to be subanalytic simply means that its graph is a subanalytic set in R n × R, but in the non continuous case we shall use a stronger assumption, in order to control the behaviour of the graph near points where f is not locally bounded. We restrict ourself to the context of the situation we need here. f : Ω → R be a continuous function. We shall say that f is strongly subanalytic if the functionf : R n → R defined by extending f by 0 on R n \ Ω has a subanalytic graph in
R n × P 1 , where P 1 is the 1−dimensional projective space R ∪ {∞}.
It is easy to see that such a condition implies that the growth of f near a boundary point in ∂Ω has to be bounded by some power of the function d(x, ∂Ω) thanks to Lojasiewicz inequalities ( [1]). Remark that iff is continuous this condition reduces to the usual subanalyticity of the graph off in R n × R.
We shall need also the following theorem (cf. [10], Theorem (2.4)).
Theorem 3.2.
Let Ω ⊂⊂ R n a relatively compact subanalytic open set, and let f : Ω → R be a C 1 function which is strongly subanalytic. Then any partial derivative of f in Ω is also strongly subanalytic.
Since, in Definition 3.1, the continuity off just means that f (x) goes to 0 when x ∈ Ω goes to the boundary ∂Ω, using Lojasiewicz inequalities we easily obtain the following corollary:
Corollary 3.3.
In the situation of the previous theorem, assume thatf is continuous. Then there exists an integer N 1 such thatf N 1 is C 1 on R n and subanalytic.
Now applying again the ideas of the previous corollary we finally obtain:
Corollary 3.4. In the situation of the previous corollary there exists an integer N 2 such thatf N 2 is C 2 on R n and subanalytic.
Remark 3.5. As the reader can see in view of the preceding results, the remaining and non trivial step to prove the existence of a subanalytic C 2 function which vanishes exactly on R n \ Ω as stated in Theorem 2.1, is to show the existence of a C 2 strictly positive (strongly) subanalytic function f on Ω which vanishes at the boundary. The natural candidate is, of course, the function x → d(x, ∂Ω). But all conditions are satisfied excepted smoothness. And the non smoothness points may go to the boundary. If one tries to use the "desingularization theorem" of H. Hironaka to solve this problem, a new difficulty comes then because the jacobian of the modification may vanish inside Ω and not only on some points in ∂Ω.
Example
Let us consider the analytic map F :
R 3 → R 3 defined by F (x, y, z) = y.(e x − 1) + x 2 + y 2 + z 2 − ε 2 , y.(e x. √ 2 − 1), y.(e x. √ 3 − 1) .
Denote Ω the interior of the imageΩ of the compact ballB 3 (0, ε). Let us start by showing that the image by F of the sphere S ε (the boundary of B(0, ε)), is a subanalytic compact subset of R 3 which is not semi-analytic in the neighborhood of (0, 0, 0). This example is extracted from [8]( example I.6).
Lemma 3.6. The compact F (S ε ) is not semi-analytic in the neighbourhood of the origin.
Proof. Since this compact set has an empty interior, if it is semi-analytic in a neighbourhood of the origin, there shall exist an analytic function f : U → R on a ball U centered in 0, non identically zero, such that f −1 (0) contains U ∩ F (S ε ). Let f = m≥m 0 P m be the Taylor series of f at the origin, which we may assume to be convergent in U provided that U is small enough. We shall assume that the homogeneous polynomial P m 0 is not identically zero. Hence, considering (x, y, z) ∈ S ε close enough to (0, 0, ε), the definition of F entails the equality 0 ≡ which holds for (x, y) ∈ R 2 close enough to (0, 0). We conclude that P m 0 ((e x − 1), (e x. The behaviour at infinity of this function easily entails 2 that we must have P m 0 ≡ 0, which gives a contradiction. q.e.d.
We shall now describe the open set Ω. Let us remark that the jacobian of F is given by (e x.
J(F )(x, y, z) = 2yz. ( √ 2 − √ 3).e x.( √ 2+ √ 3) − √ 2.e x. √ 2 + √ 3.e x.√ 3 − 1) = v w = √ 2 √ 3 .h(x)
whenever h ∈ C{x} converges for |x| < 2π/ √ 3 et verifies h(0) = 1 and h ′ (0) = ( √ 2 − √ 3)/2; these equations determine a unique x ∈ [−ε, ε], for ε ≪ 1, and hence a unique y. Remark that for x in a neighbourhood of 0, we have v/w close to √ 2/ √ 3. Therefore Γ doesn't approach (0, 0) other than along that direction. The fiber in (0, 0) of G is the curve {x.y = 0} ∩B 2 (0, ε). Remark that the points in the sphere {x 2 + y 2 = ε 2 } are mapped on the boundary of Γ. Indeed, for those who lie on {x.y = 0} their image is the origin. Otherwise, for each of such points not mapped on the origin, the jacobian of G would vanish and the boundary ofB 2 (0, ε) would be mapped on the boundary of Γ in its neighbourhood. Hence, any point of the interior Γ ′ of Γ is the image by G of some point in B 2 (0, ε) \ {x.y = 0}.
We shall denote by ϕ : Γ \ {(0, 0)} → R the subanalytic function 4 given by ϕ(v, w) = ||G −1 (v, w)|| 2 , in other words, the composition of G −1 with the square of the euclidean norm in R 2 . We shall denote by ψ : Γ \ {(0, 0)} → R the subanalytic function defined by setting ψ(v, w) = y.(e x − 1) where G −1 (v, w) = (x, y), and we set
∆ + := (ψ(v, w), v, w), for (v, w) ∈ Γ \ {(0, 0)} ∆ − := (ψ(v, w) + ϕ(v, w) − ε 2 , v, w), for (v, w) ∈ Γ \ {(0, 0)} ∆ 0 := [−ε 2 , 0] × {(0, 0)} Note that ∆ + ∩∆ − = (u, v, w) ∈ R×(Γ\{(0, 0)}) / u = ψ(v, w) and ϕ(v, w) = ε 2
is the graph of the restriction of ψ to ∂Γ \ {(0, 0)}.
We have now the following description ofΩ and of its interior Ω.
Lemma 3.7. One has ∂Ω = ∆ + ∪ ∆ − ∪ ∆ 0 . The interior Ω is the open set Ω = (u, v, w) ∈ R × Γ ′ / ψ(v, w) + ϕ(v, w) − ε 2 < u < ψ(v, w)
where Γ ′ denotes the interior of Γ.
Proof. Let (u, v, w) ∈Ω. If v.w = 0 then x.y = 0 and v = w = 0, and u = x 2 + y 2 + z 2 − ε 2 belongs to [−ε 2 , 0] which is contained in ∆ 0 . Since the projection of Ω on R 2 is an open set contained in Γ, hence in Γ ′ , the point (u, v, w) does not belong to Ω. Let us now exclude this case.
We have a point (x, y, z) ∈B 3 (0, ε) such that F (x, y, z) = (u, v, w), with x.y = 0. Then (x, y) ∈B 2 (0, ε) \ {x.y = 0} and G(x, y) = (v, w) is not (0, 0). Since ϕ(v, w) = x 2 + y 2 we have
u = ψ(v, w) + ϕ(v, w) + z 2 − ε 2
where z ∈ [−ε, ε] is, up to a sign, determined by this equation. We conclude that the inequalities
(3.1) ψ(v, w) + ϕ(v, w) − ε 2 ≤ u ≤ ψ(v, w)
hold onΩ. We have to check that ∂Ω \ ∆ 0 is exactly described by the equality
(3.2) (u − ψ(v, w) − ϕ(v, w) + ε 2 )(ψ(v, w) − u) = 0.
Since the projection on R 2 is open, if (v, w) ∈ Γ ′ then it must lie in the boundary of Ω. It suffices to prove that for (v, w) ∈ Γ ′ the equality above implies that (v, w) is in the boundary. This is clear because near any (u, v, w) of Ω one can find δ > 0 such that ]u − δ, u + δ[×(v, w) is contained in Ω, which is not possible by the inequalities (3.1) in a point where the equality (3.2) is satisfied. Hence it is sufficient to prove thatΩ \ ∆ 0 is the set of points (u, v, w) in R × (Γ \ {(0, 0)}) satisfying the inequalities (3.1). Indeed, any choice of (v, w) ∈ Γ \ {(0, 0)} gives a unique point (x, y) ∈ B 2 (0, ε) such that G(x, y) = (v, w) and the inequalities (3.1) entail that we can find at least a z ∈ R such that z 2 = u − ψ(v, w) − ϕ(v, w) + ε 2 and that ϕ(v, w) + z 2 ≤ ε 2 . Note that if u = ψ(v, w) + ϕ(v.w) − ε 2 we will have z = 0. Therefore, the boundary ∆ − corresponds to the image ofB 3 (0, ε) ∩ {z = 0} \ ∆ 0 . Similarly the equality u = ψ(v, w) corresponds to the image of the sphere {x 2 + y 2 + z 2 = ε 2 } deprived of ∆ 0 . q.e.d.
Let us now consider the function f : R 3 → R + defined as follows:
• For (u, v, w) ∈ Ω one sets f (u, v, w) = (ψ(v, w) − u)(u − ψ(v, w) − ϕ(v, w) + ε 2 )
• For (u, v, w) ∈ Ω one sets f (u, v, w) = 0.
Note that f est strictly positive on Ω by Lemma 3.7, and that it is analytic on the complement of ∂Ω, since the functions ϕ and ψ are analytic on Γ ′ . Moreover f is bounded.
Let us now definef (u, v, w) = f (u, v, w).v 2 .w 2 .
Lemma 3.8. The functionf : R 3 → R + is subanalytic and continuous, it satisfies
Ω = {(u, v, w) ∈ R 3 /f (u, v, w) > 0}
and it is C ∞ on R 3 \ ∂Ω.
Proof. First we prove that f is subanalytic 5 . Since its graph is the union of the graph of its restriction to Ω and the set (R 3 \ Ω) × {0} which is subanalytic, Ω being an open subanalytic set of R 3 , it is sufficient to prove that the graph of the restriction of f to Ω is subanalytic. Let us consider the polynomial morphism h : R 3 → R given by h(x, y, z) = (ε 2 − (x 2 + y 2 + z 2 )).z 2 and denote by X, X 1 , X 2 the graph of the restriction of h respectively tō B 3 (0, ε), ∂B 3 (0, ε),B 3 (0, ε) ∩ {x.y = 0} and Y, Y 1 , Y 2 the respective images of these graphs by the morphism F × id :
R 3 × R → R 3 × R.
Let us prove that the graph of the restriction of f to Ω is equal to Y \ (Y 1 ∪ Y 2 ). Indeed, for (u, v, w) ∈ Ω, if (x, y, z) ∈B 3 (0, ε) verifies F (x, y, z) = (u, v, w), we get ϕ(v, w) = x 2 + y 2 , ψ(v, w) = y.(e x − 1) and u = ψ(v, w) + ϕ(v, w) + z 2 − ε 2 .
One sees that f (u, v, w) = (ε 2 − (x 2 + y 2 + z 2 )).z 2 . To finish, it is enough to 5 As pointed by the referee, this fact is consequence of basic stability properties of subanalytic functions. We give a direct proof for non specialists.
Let us finally show that Ω is the set wheref is strictly positive. It is sufficient to check that v.w = 0 on Ω. But v.w = 0 entails x.y = 0 and so v = w = 0 and u = x 2 + y 2 + z 2 − ε 2 , in other words, (u, v, w) ∈ [−ε 2 , 0] × (0, 0) = ∆ 0 . Hence such (v, w) belongs to ∂Ω.
We have now constructed a subanalytic functionf on R 3 which is continuous and strictly positive exactly on Ω ⊂⊂ R 3 . By Corollary 3.4 there exists a positive integer N such thatf N is of class C 2 . Then one gets a Stein open subanalytic set of C 3 which cuts R 3 exactly on Ω as in the general proof of Theorem 2.4.
q.e.d.
Corollary 2. 3 .
3Let Ω be a subanalytic open set in a real paracompact analytic manifold M R . Then, for any complexification M C of M R , and for any given smooth hermitian metric on the complex tangent bundle on M C there exists a subanalytic non negative real function f on M C of class C 2 such that {f > 0} ∩ M R = Ω and such that the Levi form of f is bounded by the given hermitian metric. Moreover, f can be chosen so that Suppf is contained in any given open set in M C containing the closed setΩ. Proof. For ǫ > 0, let us denote B ǫ an open ball of R n of radius ǫ and by B C ǫ the corresponding ball in C n . For each p ∈Ω (the closure of Ω) there exists two relatively compact open subanalytic neighbourhoods V ⊂⊂ U of p in M C and a complex analytic isomorphism ϕ defined in an open neighbourhood W ofŪ to an open ball B C
Theorem 2 . 4 .
24Let Ω be a subanalytic open set of a real paracompact analytic manifold M R . Then, given a complexification M C of M R , there exists a subanalytic Stein open subset Ω C of M C such that
{f > 0} ∩ M R = Ω and such that the Levi form of f is bounded by h. So the Levi form of the C 2 subanalytic function ψ := ϕ * C (g) − f is positive definite at each point of V . It follows that the open set Ω C = {ψ < 0} ∩ V 1 for instance 1/2 of the Levi form of ϕ * C (g) may be choose as Kähler form on V .
Definition 3. 1 .
1Let Ω ⊂⊂ R n a relatively compact subanalytic open set, and let
m≥m 0 y
0m .P m ((e x − 1), (e x.
√3
− 1)) is identically zero for x in a neighbourhood of 0. Hence this analytic function vanishes identically on R.
ε small enough, it doesn't vanish on {x.y.z = 0} within the ball B 3 (0, ε). Indeed, the brackets give an analytic function of a single variable x; hence it has an isolated zero inx = 0. The image of {x.y = 0} ∩B 3 (0, ε) by F is [−ε 2 , 0] × {(0, 0)} which is contained in 3 the boundary ofΩ.The image of {z = 0} is more complicated to describe.Let us now consider the analytic morphism G :R 2 → R 2 defined by G(x, y) := y.(e x. √ 2 − 1), y.(e x. √ 3 − 1) .Let us denote by Γ the image by G of the ballB 2 (0, ε) of R 2 . If (v, w) ∈ Γ \ {(0, 0)} then the fiber G −1 (v, w) is reduced to a single point (for ε small enough). In fact we must have v.w
2
This is equivalent to prove the algebraic independency of the functions (e x − 1), (e x.
See the description of Γ near (0, 0) given below
The graph of G −1 : Γ \ {(0, 0)} →B 2 (0, ε \ {x.y = 0} is the same as that the graph of G :B 2 (0, ε) \ {x.y = 0} → Γ \ {(0, 0)}.
AcknowledgementsThe authors gratefully acknowledge the support of FCT and FEDER, within the project ISFL-1-143 of the Centro deÁlgebra da Universidade de Lisboa.
Let us show that it is continuous along ∂Ω, since it is C ∞ on R 3 \ ∂Ω. Let (u 0 , v 0 , w 0 ) ∈ ∂Ω. ) ∩ {x.y = 0}) and of F (∂B 3 (0, ε)) are never in Ω. Hencef is subanalytic. that the points of F (B 3 (0, ε. First assume that (u 0 , v 0 , w 0 ) belongs to ∆ + . Then u 0 = ψ(v 0 , w 0 ), in other words, we get the image by F of a point (x, y, z) ∈ ∂B 3 (0, ε) \ {x.y = 0}. Hence the limit of (u − ψ(v, w)) when (u, v, w) ∈ Ω tends to (u 0 , v 0 , w 0 ) is zero. As the functions ψ and ϕ are bounded on Ω, the limit off is zero in such a pointthat the points of F (B 3 (0, ε) ∩ {x.y = 0}) and of F (∂B 3 (0, ε)) are never in Ω. Hencef is subanalytic. Let us show that it is continuous along ∂Ω, since it is C ∞ on R 3 \ ∂Ω. Let (u 0 , v 0 , w 0 ) ∈ ∂Ω. First assume that (u 0 , v 0 , w 0 ) belongs to ∆ + . Then u 0 = ψ(v 0 , w 0 ), in other words, we get the image by F of a point (x, y, z) ∈ ∂B 3 (0, ε) \ {x.y = 0}. Hence the limit of (u − ψ(v, w)) when (u, v, w) ∈ Ω tends to (u 0 , v 0 , w 0 ) is zero. As the functions ψ and ϕ are bounded on Ω, the limit off is zero in such a point.
0 ) ∈ ∆ − , then we have the image of a point in (B 3 (0, ε) ∩ {z = 0}) \ {x. y = 0}If (u 0 , v 0 , w 0 ) ∈ ∆ − , then we have the image of a point in (B 3 (0, ε) ∩ {z = 0}) \ {x.y = 0}.
Since the function ψ is bounded on Γ the limit of f in such a point is zero, and so it is forf. Since the function ψ is bounded on Γ the limit of f in such a point is zero, and so it is forf .
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| {'fraction_non_alphanumeric': 0.08250207813798836, 'fraction_numerical': 0.029551122194513717, 'mean_word_length': 3.251060070671378, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 9, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, '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': 'By open neighbourhood of an open subset Ω of R n we mean an open subset Ω ′ of C n such that R n ∩ Ω ′ = Ω. A well known result of H. Grauert implies that any open subset of R n admits a fundamental system of Stein open neighbourhoods in C n . Another way to state this property is to say that each open subset of R n is Stein. We shall prove a similar result in the subanalytic category, so, under the assumption that Ω is a subanalytic open subset in a paracompact real analytic manifold, we show that Ω admits a fundamental system of subanalytic Stein open neighbourhoods in any of its complexifications.', 'arxivid': '1011.4208', 'author': ['Daniel Barlet [email protected] ', 'Teresa Monteiro Fernandes ', 'D Barlet ', 'T Monteiro Fernandes ', '\nInstitut Elie Cartan\nLaboratoire de Mathématiques\nUniversité de Nancy\nB.P. 23954506Vandoeuvre lès Nancy CedexFrance\n', '\nCentro de Matemática e Aplicações Fundamentais Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa Edifício\nC6, P.2, Campo Grande1749-16LisboaPortugal\n'], 'authoraffiliation': ['Institut Elie Cartan\nLaboratoire de Mathématiques\nUniversité de Nancy\nB.P. 23954506Vandoeuvre lès Nancy CedexFrance', 'Centro de Matemática e Aplicações Fundamentais Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa Edifício\nC6, P.2, Campo Grande1749-16LisboaPortugal'], 'corpusid': 119284332, 'doi': '10.4064/sm204-3-5', 'github_urls': [], 'n_tokens_mistral': 8961, 'n_tokens_neox': 7774, 'n_words': 4789, 'pdfsha': 'fc0d39730036a69fa372d1aec4c68d066605b26f', 'pdfurls': ['https://arxiv.org/pdf/1011.4208v2.pdf'], 'title': ['Mathematics Subject Classification: Primary:32B20,14P15; Secondary: 32C05, 32C09', 'Mathematics Subject Classification: Primary:32B20,14P15; Secondary: 32C05, 32C09'], 'venue': []} |
arxiv |
Multi-wavelength Intra-day Variability and Quasi-periodic Oscillation in Blazars
Alok C Gupta *correspondence:[email protected]
Aryabhatta Research Institute of Observational Sciences (ARIES)
Manora PeakNainital -263002India
Multi-wavelength Intra-day Variability and Quasi-periodic Oscillation in Blazars
Article Academic Editor: Version December 8, 2017 submitted to Galaxies; Typeset by L A T E X using class file mdpi.clsactive galaxiesBL Lacertae object: BL LacQuasars: Flat Spectrum Radio Quasarsjets, accretion disk
We reviewed multi-wavelength blazars variability and detection of quasi-periodic oscillations on intra-day timescales. The variability timescale from few minutes to up to less than a days is commonly known as intra-day variability. These fast variations are extremely useful to constrain the size of emitting region, black hole mass estimation, etc. It is noticed that in general blazars show intra-day variability in the complete electromagnetic spectrum. But some class of blazars either do not show or show very little intra-day variability in a specific band of electromagnetic spectrum. Blazars show rarely quasi-periodic oscillations in time series data in optical and X-ray bands. Other properties and emission mechanism of blazars are also briefly discussed.(Ghisellini et al. 1997;Fossati et al. 1998): A low-frequency component from radio to the UV or X-rays, generally agreed to be due to synchrotron radiation from relativistic electrons in the jet, and a high-frequency component from X-rays to γ−rays, which can be either due to Compton scattering of lower-frequency radiation by the same relativistic electrons (leptonic models i.e. Krawczynski 2004) or due to interactions of ultra-relativistic protons in the jet (hadronic models), either via proton synchrotron radiation (Mucke et al. 2003) or via secondary emission from photo-pion and photo-pair production process (Böttcher 2007; and references therein).Blazars can be classified into three sub-classes, depending on the peak frequency of their synchrotron emission: LSPs (low-synchrotron-peaked blazars), consisting predominantly of LBLs (red or low energy or radio selected blazars) and defined by a peak of their synchrotron component at ν sy < 10 14 Hz, ISPs (intermediate-synchrotron-peaked blazars) or IBLs, consisting mostly of intermediate blazars, defined by 10 14 Hz < ν sy < 10 15 Hz, and HSPs (high-synchrotron-peaked blazars), all of which are HBLs (blue or high energy or X-ray selected blazars) and which are defined through ν sy > 10 15 Hz (Abdo et al. 2010). The high-energy component of the spectral energy distribution (SED) of blazars extends up to γ−rays, peaking at GeV energies in LSPs and at TeV energies in HSPs. Blazar properties are consistent with relativistic beaming, i.e. bulk relativistic motion of the jet plasma at small angles to the line of sight, which gives rise to a strong amplification and rapid variability in the observer's frame.
Introduction
It is commonly accepted that super massive black holes (SMBHs, with masses between 10 6 − 10 10 M ) are present in the nuclei of all galaxies with stellar bulges. At any given time a few percent of these SMBHs are fed with a sufficient amount of gas that they will possess significant accretion discs. The emission of these discs is comparable to the total emission of stars in the entire host galaxy because of a very high efficiency for the conversion of matter into radiation as it spirals into a BH. This is the fundamental mechanism underlying "active galactic nucleus", or AGN.
Roughly 85−90 % of AGNs have very little radio emission (F 5GHz /F B ≤ 10, here F 5GHz = flux at radio 5 GHz and F B = flux at optical B band 4400Å) and are therefore called radio-quiet AGNs (RQAGNs). The remaining ∼ 10-15 % of AGNs are radio-loud AGNs (RLAGNs). It has been proposed that different types of AGNs can be explained by the idea that different line of sight (LOS) angles can play an important role in understanding their different properties (Antonucci 1993;Urry & Padovani 1995).
Blazars belongs to the RLAGN class and its LOS is pointed close to the observer. Blazars show rapid variability at almost all wavelengths of the electromagnetic (EM) spectrum with the emission being strongly polarized (optical linear polarization ≥ 3%). Due to their strong and large amplitude variability nature in the complete EM spectrum, they are considered as transient astronomical objects. BL Lacertae objects (BLLs) and flat spectrum radio quasars (FSRQs) are collectively known as blazars. BLLs show featureless optical continua (no prominent emission or absorption lines) while FSRQs show prominent emission lines in their optical spectra. The radiation from blazars is dominated by non-thermal emission at all wavelengths, consisting of two broad spectral bumps
The study of variability is one of the most powerful tools for understanding the nature and processes occurring in blazars. Variability of blazars can be broadly divided into three classes. Significant variations in flux may occur over few tens of minutes to the course of less than a day, often called micro-variability, intra-night variability or intra-day variability (IDV) (Wagner & Witzel 1995). Short term variability (STV) can range time scales from days to few months and long term variability (LTV) can have time scales of several months to several years (Gupta et al. 2004).
Variability observations on IDV timescales in AGN is the most puzzling. Variability of blazars on IDV time scale can provide important clues to the physics of the innermost nuclear regions in these objects. Blazar properties are consistent with relativistic beaming caused by bulk relativistic motion of the jet plasma at small angles to the line of sight, which gives rise to a strong amplification and rapid variability in the observer's frame. With simultaneous multi-wavelength observations of blazars in the entire EM spectrum is an important tool to test several The possible models for IDV: shock-in-jet models, accretion-disk based models, and models based on plasma instabilities in shear layers, etc.
Intra-day Variability in different EM Bands
There have been several dedicated monitoring campaigns in which the IDV of blazars has been studied over the entire EM spectrum (e.g. Miller et al. 1989;Carini et al. 1990Carini et al. , 1991Quirrenbach, et al. 1991;Heidt & Wagner 1996Sagar et al. 1999 There are several methods which can be used to find the genuine IDV and variability parameters in time series data. These methods are described by different groups and relevant source references are cited in their papers. Romero et al. (1999) . This method takes care of large brightness difference in blazar and comparison / standard stars or large brightness difference in comparison / standard stars. It is used by Gaur et al. (2015). χ 2 −Test is explained by (Gaur et IDV in blazars can be intrinsic to the source or due to extrinsic origin. Interstellar scintillation and gravitational microlensing are the main extrinsic cause of IDV. Interstellar scintillation is only relevant in low-frequency radio observations. Gravitational microlensing is only applicable in a few blazars which are lensed system e.g. the blazar AO 0235+135 at z = 0.94 have revealed foreground-absorbing systems at z = 0.524 and z = 0.851 (Cohen et al. 1987;Nilsson et al. 1996). In the blazars where IDV is detected in low-frequency radio observations or gravitationally lensed sources can also have some intrinsic origin, and to find that we need to do simultaneous multi-wavelength of the blazar, and co-related variability in different EM bands will be helpful to find the nature of variability.
IDV in the high state (pre/post to outburst state) of blazars can be explained by jet based models e.g. helical instabilities in the jet (Marscher & Gear 1985;Qian et al. 1991) or turbulence behind the shock in the jet (Marscher, Gear, Travis 1992). Jet based models can explain IDV over the entire range of EM wavelengths. Other theoretical models seek to explain IDV in blazars (mainly in their low-state) involving accretion-disk based models. These models include pulsations of the gravitational modes of the gaseous disk (Kato & Fukue 1980;Nowak & Wagoner 1992) or orbital signatures from "hot-spots" in the gas surrounding the black hole, either from the disk itself or the corona above it (Zhang & Bao 1991;Mangalam & Wiita 1993). Accretion-disk based models can explain the variations in optical, UV and X-ray bands, but are difficult to connect to the observed rapid variability in γ−rays. Plasma instabilities in the jet (e.g., Kelvin-Helmholtz type instabilities due to the interaction of a fast inner spine of the jet with a slower, outer layer) could play an important role in the production of IDV at a variety of wavelengths.
IDV In Gamma-rays
IDV in Optical and infrared (IR) bands
The first evidence of optical micro-variability is reported in BL Lacertae by Miller et al. (1989). The result of this paper motivated several groups around the globe to start dedicated project to search for optical micro-variability in blazars. Optical IDV in blazars is pioneered by the USA group in which they studied optical micro-variability in five blazars (e.g. Miller et al. 1989;Carini et al. 1990Carini et al. , 1991, and reported that the probability of finding genuine micro-variability is about 80% for the blazar continuously monitored for < 8 hours. An extensive search for optical IDV in a sample of 34 BLLs from 1 Jy catalog was done by (Heidt & Wagner 1996). IDV was detected in 28 out of 34 BLLs (82%), and 75% of the variable BLLs changed significantly over a time span < 6 hours. But this data lacks continuity in the LCs. Gupta Wagner et al. (1990) did simultaneous optical and radio monitoring of blazars. reported for the first time the correlated optical and radio IDV in the blazar S5 0716+714. To test the inverse-Compton (IC) catastrophe scenario in the blazar S5 0716+714 extensive observational campaign in radio and mm wavelengths were coordinated (Ostorero, et al. 2006;Agudo et al. 2006;Fuhrmann et al. 2008), and the lower limits to brightness temperature was derived from the inter-day variations exceed the 10 12 K IC-limit by up to 2-4 orders of magnitude. Gabányi et al. (2007) reported radio IDV of the blazar J 1128+5925 in three frequencies i.e. 2.7 and 10.45 GHz observations using 100m Effelsberg radio telescope in Germany, and 4.8 GHz observations using 25m radio telescope in Urumqi, China. The observed frequency dependent IDV in the source was in good agreement with prediction from interstellar scintillation. VLBA observation of the blazar J1128+592 is reported by Gabányi et al. (2009), and with VLBA observations they detected an east-west oriented core-jet structure with no significant motion in its jet. Radio IDV in blazars are studied where variability characteristics have changed abruptly by interstellar scintillation (Marchili et al. 2011(Marchili et al. , 2013. Radio IDV at 4.8 GHz using 25m Urumqi, China telescope for the blazars S5 0716+714 and 1156+295 are reported by Liu et al. 2013). Simultaneous IDV in X-ray, optical four bands and three frequencies in radio are reported by Gupta et al. (2012). IDV detected in all three radio frequencies and also noticed that low and high frequencies correlation does not peak at zero lag which show that low frequency radio observation is combined effect of intrinsic and extrinsic mechanism. Optical and radio IDV observations were carried out by Liu et al. (2017). IDV observation along with VLBI analysis is carried out for the blazar S4 0917+624 (Liu et al. 2015).
Quasi Periodic Oscillations in Intra-day Time Series Multi-wavelength Data
Detection of quasi-periodic oscillations (QPOs) in time series data are very rare in AGN. In last one decade, several detection of QPOs in AGN on diverse timescales ranging as short as few minutes and as long as few years using γ−ray, X-ray, optical and radio time series data are made (e.g. QPOs in blazars on IDV timescales can be explained by several standard models of AGN. One of the simplest models in which the central BHs of AGN would attribute the QPOs can be explained by presence of a single dominating hot-spot on the accretion disk (e.g., Mangalam & Wiita 1993;Chakrabarti & Wiita 1993). Using QPO or nearly periodic signal, the period can be used to estimate the BH mass for non-rotating (Schwarzschild) BH, and maximally rotating (Kerr) BH. The detailed explanation is given in Gupta et al. (2009). Other alternative possible mechanisms for QPOs in blazars on IDV timescales can also have a disk origin or can arise from relativistic jets. The former class includes small epicyclic deviations in both radial and vertical directions from exact planar motions within a thin accretion disk (e.g., Abramowicz 2005), and trapped pulsational modes within a disk (e.g., Perez et al. 1997;Espaillat et al. 2008). Using detailed explanation of Perez et al. (1997), one can also get the BH mass of the blazar. There are various jet models which also can explain the QPO detection in blazars on IDV timescales e.g. a shock propagating down a jet in which jet structure is quasi-helical and change in electron density or magnetic field can produce QPO, a short lived QPO can be due to turbulence behind the shock in the relativistic jet (e.g. Camenzind (Montagni et al. 2006). They used wavelet analysis along with randomization test and found strong evidence for nearly periodic variations on 5 light curves with probability > 99%. The period for these five light curves are found in the range of 25 minutes to 73 minutes which lead to BH mass ranging 2.47-7.35 × 10 6 M and 1.57-4.67 × 10 7 M for non-rotating BH and maximally rotating BH, respectively. Another evidence of QPO detection in optical band on the same blazar S5 0716+714 is reported by Rani et al. (2010). They found QPO period of ∼ 15 minutes using various techniques (e.g. SF, LSP, PSD, data folding). This period yield the BH mass 1.5 × 10 6 M and 9.6 × 10 6 M for non-rotating BH and maximally rotating BH, respectively.
In Optical
In X-rays
Using wavelet technique, (Espaillat et al. 2008) analyzed 19 observations of 10 AGN observed with EPIC/pn detector on board to XMM-Newton, and detected QPO period of 3.3 ks in one light curve of the blazar 3C 273. The QPO period is used to get the black hole (BH) mass of the blazar. They estimated the BH mass of the blazar is 7.3 × 10 6 M and 8.1 × 10 7 M for non-rotating BH and maximally rotating BH, respectively. In another observation of EPIC/pn of XMM-Newton for the blazar PKS 2155-304, (Lachowicz et al. 2009) detected QPO period of 4.6 h in which period was present for ∼ 3.8 cycles. This QPO detection was verified by various techniques (e.g. SF, PSD, MHAoV, data folding and wavelet). The BH mass of the blazar is estimated to be 3.29 × 10 7 M and 2.09 × 10 8 M for non-rotating BH and maximally rotating BH, respectively.
Conclusions
With the extensive studies of IDV in radio to optical bands in last about three decades and in high energies (X-ray and γ−rays) in last one decade, we reach on the following conclusion:
• Blazars show large amplitude IDV in radio bands which is basically the mixture of extrinsic and intrinsic nature. • LBLs and IBLs show large amplitude IDV in optical/IR bands with high duty cycle. • HBLs either don't show optical/IR IDV or if show the amplitude is low and the duty cycle is very less compare to LBLs/IBLs. • In general blazars don't show color variation on IDV timescales. But occasionally it is seen. • Optical inter-band cross-correlation show that in general there is no time lag in different optical bands. On some occasions the time lag of a few minutes are reported in different optical bands. • Optical/IR SEDs are well fitted with single power law. • Sometimes optical/IR SEDs show big blue bump which is a signature of emission from accretion disk.
• In X-rays, HBLs show large amplitude IDV, and the duty cycle is high.
• In X-rays, LBLs/IBLs either don't show IDV or if show the amplitude is less. The duty cycle of X-ray IDV for LBLs/IBLs are much less compare to duty cycles for HBLs. • In general hard and soft X-ray LCs of blazars show strong cross-correlation with zero lag which imply the emission in hard and soft bands are co-spatial. • IDV timescales in different EM bands are used to get black hole mass of the blazars, size of emitting region, and the Doppler factor δ. • In general sensitivity of very high energy γ−ray facilities are poor but occasionally blazars are observed on time resolutions of a few minutes to few hours. The best time resolution γ−ray light curves give high Doppler factor ∼ 100 for the blazar PKS 2155-304. • QPOs in blazars on IDV timescales are rare. • Occasionally QPOs on IDV timescales are detected in a few blazars in X-ray and optical bands.
Author Contributions: The author was asked by the journal to write the review article. The author planned and wrote the article.
Conflicts of Interest:
The author declares no conflict of interest.
; Montagni et al. 2006; Aharonian et al. 2007; Gupta et al. 2008a, 2008b, 2012, 2016, 2017; Gaur et al. 2010, 2012a, 2012b, 2015; Foschini et al. 2011; Kushwaha et al. 2014; Paliya et al. 2015; Paliya 2015; Kalita et al. 2015; Agarwal & Gupta 2015; Agarwal et al. 2015, 2016; Ackermann et al. 2016; Pandey et al. 2017; Aggrawal et al. 2017; and references therein).
al. 2012b; Agarwal & Gupta 2015) also provide the presence of IDV in time series data. The analysis of variance (ANOVA) test is a very robust test to investigate the variability in the light curves (LCs) (e.g. de Diego et al. 1998; Gaur et al. 2012b; Agarwal & Gupta 2015; Gupta et al. 2017; and references therein). Percentage amplitude variation in the time series data is introduced by Heidt & Wagner (1996) and used in several papers (e.g. Gupta et al. 2008a; Gaur et al. 2012b; Agarwal & Gupta 2015; Gupta et al. 2017; and references therein). One another method called excess variance and fractional rms variability amplitude are introduced by (Edelson et al. 2002; Vaughan et al. 2003) and extensively used for getting the variability in AGN LCs (e.g. Kalita et al. 2015; Bhagwan et al. 2016; Gupta et al. 2016; Pandey et al. 2017; and references therein).
Most of the ground and space based gamma-ray experiments are not sensitive enough to observe blazars with few minutes time resolution. So, it is extremely difficult to do IDV studies of blazars. But there is an excellent IDV observation of the blazar PKS 2155-304 in an exceptional very high energy gamma-ray flare observed on July 28, 2006 using HESS (High Energy Stereoscopic System). The IDV LC obtained with time resolution of 1 minute and IDV is seen up to 10 minutes which gives Doppler factor more than 100. The average flux at > 200 GeV during outburst was ∼ 7 times of flux observed from Crab Nebula(Aharonian et al. 2007). Thanks to the LAT (large area telescope) on board to Fermi gamma-ray space telescope (hereafterFermi-LAT; Atwood et al. 2009) which is observing sky in gamma-ray since its launch in June 2008. Fermi-LAT covers 20 MeV -300 GeV energies and made a revolution in the discovery of blazars. It has observed several blazars in flaring state and detected strong IDV (Foschini et al. 2011; Kushwaha et al. 2014; Paliya et al. 2015; Paliya 2015; Ackermann et al. 2016; and references therein). Fermi-LAT LC with time resolution of minutes of blazars were presented by (Foschini et al. 2011; Ackermann et al. 2016) while (Kushwaha et al. 2014; Paliya et al. 2015; Paliya 2015) presented Fermi-LAT LCs of blazars with few hours time bin. Kushwaha et al. (2014) reported Fermi-LAT LC of the blazar PKS 1222+216 with 6h time bin show asymmetric rise profiles but rapid decline during the April 2010 flare. 2.2. IDV in X-rays A pilot project on searching for IDV in blazars are initiated (Gaur et al. 2010; Kalita et al. 2015; Gupta et al. 2016; Pandey et al. 2017; Aggrawal et al. 2017; and references therein). A sample of four HBLs observed on 23 occasions by XMM-Newton are studied, IDV timescales ranging from 15.7 to 46.8 ks were found on 8 occasions, in 13 cases IDV timescales were longer than the data length, and hint of weak quasi periodic oscillations (QPOs) were observed on one LC each of blazars ON 231 and PKS 2155 -304 (Gaur et al. 2010). González-Martín & Vaughan (2012) took the sample of 104 nearby AGN (z < 0.4) observed with XMM-Newton. The AGN sample also include several blazars. They did power-spectrum density (PSD) analysis of all those LCs and reported IDV variability parameters. The LBL 3C 273 observed on 24 occasions during 2000 -2012 with XMM-Newton, and on IDV time-scales 3C 273 have shown occasionally small amplitude variability (Kalita et al. 2015). A complete sample of 12 LBL and IBL with 50 IDV LCs taken with XMM-Newton are compiled (Kalita et al. 2015; Gupta et al. 2016b). It is noticed that the duty cycle of genuine IDV detection in LBL and IBL in X-ray band is only 4% (2 out of 50 LCs) (Gupta et al. 2016). It is concluded that probably peak of the spectral energy distribution seems to be responsible for IDV properties (Gupta et al. 2016b). In a recent study (Pandey et al. 2017) reported search for X-ray IDV in five TeV blazars using NuStar. Four TeV blazars have shown large amplitude IDV, using auto correlation function (ACF), IDV timescale in the range of 2.5 to 32.8 ks is reported in eight LCs of Mrk 421; a timescale of about 8.0 ks for one LC in Mrk 501; and timescales of 29.6 to 57.4 ks in two LCs of PKS 2155-304. In general soft (3 -10 keV) and hard (10 -79 keV) LCs were well correlated which indicate the same population is emitting soft and hard X-rays. IDV timescales are used to calculated δ the Doppler factor, B the magnetic field, γ the Lorentz factor, and R the size of emitting region (Pandey et al. 2017). Recently in another search for X-ray IDV, 83 LCs of the TeV blazar Mrk 421 taken during 1999 -2015 with Chandra are studied (Aggrawal et al. 2017). IDV timescale ranging 2.4 to 30.0 ks, IDV duty cycle ∼ 77%, soft (0.3 -2.0 keV) and hard (2.0 -10.0 keV) LCs were well correlated with zero lag are found. IDV timescales are also used to calculate δ the Doppler factor, B the magnetic field, γ the Lorentz factor, and R the size of emitting region for the blazar Mrk 421 (Aggrawal et al. 2017).
& Joshi (2005) compiled the optical IDV studies of blazars till ∼ 2004 and noticed that the occurrence of IDV on blazars if observed for less than 6 h is about 60-65%. If the blazar is observed for more than 6h then the possibility of IDV detection is about 80-85% Several hundred nights of optical IDV search in blazars were done by different Chinese groups (e.g. Bai et al. 1998, 1999; Dai et al. 2001; Xie et al. 1999, 2001, 2002, 2004; Fan et al. 2001, 2004; Qian & Tao 2004; Wu et al. 2005, 2007; Poon et al. 2009; Hu et al. 2014; Li et al. 2017; Xiong et al. 2016; Guo et al. 2017; Feng et al. 2017; and references therein). Significant IDV is detected in most of these observations but in several papers data lacks continuity in the LCs. Indian groups also did extensive search for optical/IR IDV in blazars with collaborators around the globe (e.g. Sagar et al. 1999; Ghosh et al. 2000, 2001; Gupta et al. 2004, 2008a, 2008b, 2016a, 2017; Stalin et al. 2005, 2006; Goyal et al. 2009; Rani et al. 2011; Gaur et al. 2012a, 2012b, 2012c, 2015; Agarwal & Gupta 2015; Agarwal et al. 2015, 2016; and references therein). There are some other collaborations world wide which are doing search for optical/NIR IDV of blazars (e.g. Heidt & Wagner 1998; Villata et al. 2000, 2002, 2004, 2008; Romero et al. 2002; Papadakis et al. 2003, 2004; Ostorero et al. 2006; Xilouris et al. 2006; Bachev et al. 2011, 2012, 2017; Bachev 2015; Bhatta et al. 2013, 2016a; and references therein).The important results reported in these papers are: i) LBLs and IBLs show large amplitude IDV with high duty cycles; ii) HBLs either don't show IDV or if show, the amplitude of variability and duty cycle of IDV detection are less compare to LBLs and IBLs; iii) Cross-correlation analysis in different optical bands on IDV timescales in general show strong correlation with zero lag which imply that the emission in different optical bands are co-spatial, but occasionally a few minutes time lags are also reported; iv) optical color variation on IDV timescales show a range of nature e.g. sometimes there is no color variation is seen, on some occasions BLLs show bluer when brighter (BWB) and FSRQs show redder when brighter trend (RWB), and rarely an opposite trend is also noticed; v) IR/optical SED in general fitted with a power-law but occasionally show big blue bump (BBB) which show the signature of thermal emission from accretion disk; vi) IDV LCs are used to get variability timescale which give clue of size of emitting region, black hole mass estimation of the blazar; etc.2.4. IDV in Radio bandsIDV in radio bands are mixture of intrinsic and extrinsic origin. Mostly observations in centimeter and meter wavelengths have dominant extrinsic variability which is due to interstellar scintillation of radio waves caused by the turbulent interstellar medium of the Milky Way while IDV in millimeter wavelength of intrinsic origin. Radio IDV of blazars are pioneered by 100m Effelsberg radio telescope in Germany, and other radio and mm wavelengths telescopes (e.g.Wagner et al. 1990; Quirrenbach et al. 1991; Ostorero et al. 2006; Agudo et al. 2006; Gabányi et al. 2007, 2009; Fuhrmann, et al. 2008; Marchili et al. 2011, 2012; Gupta et al. 2012; Liu et al. 2013; Liu et al. 2012, 2015, 2017; and references therein).
Gierliński et al. 2008; Espaillat et al. 2008; Gupta et al. 2009; Lachowicz et al. 2009; Rani et al. 2009, 2010; Lin et al. 2013; Sandrinelli et al. 2014, 2016a, 2016b, 2017; Graham et al. 2015; Ackermann et al. 2015; Alston et al. 2014, 2015; Pan et al. 2016; Bhatta et al. 2016b; Bhatta 2017; and references therein). But only a few claims of QPO detection on IDV timescales using X-ray and optical monitoring data of a few blazars are reported (Espaillat et al. 2008; Gupta et al. 2009; Lachowicz et al. 2009; Rani et al. 2010). There are several methods which can be used to search for QPO or periodic signal in time series data. These methods are described by different groups and relevant source references are cited in their papers. Gierliński et al. (2008) used power spectral density (PSD) and data folding. Espaillat et al. (2008) used wavelet analysis, whereas Gupta et al. (2009) used wavelet plus randomization technique. Weighted wavelet z-transform (WWZ) and Lomb Scargle periodogram (LSP) are used by Bhatta (2017). Multiple analysis techniques e.g. structure function (SF), wavelet analysis, data folding, PSD, multi-harmonic AoV periodogram (mhAoV) are used by (Lachowicz et al. 2009). SF and auto correlation function (ACF) are used to get variability timescale and QPOs (Gaur et al. 2010; Pandey et al. 2017). One or multiple methods used in these papers are used in other searches for QPOs.
Gupta et al. (2009) selected 20 optical IDV light curves of the blazar S5 0716+714 from a database of 102 lights curves taken in three years span
introduced C-Test which is used by several groups (e.g. Jang & Miller 1997; Stalin et al. 2004; Gupta et al. 2008a; Gaur et al. 2012b; Agarwal & Gupta 2015; and references therein). But later de Diego (2010) explained that the C-Test statistic is too conservative approach for IDV detection. IDV detection results can be tested using F-test which is a distributed statistic (de Diego 2010). F-test is also used by several groups (e.g. Gaur et al. 2012b; Agarwal & Gupta 2015; and references therein). A better method called power enhanced F-test is introduced by (de Diego 2014; de Diego et al. 2015)
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Submitted to Galaxies for possible open access publication under the terms and conditions of the Creative Commons Attribution license. c 2017 by the authors. Submitted to Galaxies for possible open access publication under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/4.0/).
| {'fraction_non_alphanumeric': 0.08309932826781191, 'fraction_numerical': 0.061694747274529234, 'mean_word_length': 3.8007004559571795, 'pattern_counts': {'":': 0, '<': 6, '<?xml version=': 0, '>': 5, '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': "We reviewed multi-wavelength blazars variability and detection of quasi-periodic oscillations on intra-day timescales. The variability timescale from few minutes to up to less than a days is commonly known as intra-day variability. These fast variations are extremely useful to constrain the size of emitting region, black hole mass estimation, etc. It is noticed that in general blazars show intra-day variability in the complete electromagnetic spectrum. But some class of blazars either do not show or show very little intra-day variability in a specific band of electromagnetic spectrum. Blazars show rarely quasi-periodic oscillations in time series data in optical and X-ray bands. Other properties and emission mechanism of blazars are also briefly discussed.(Ghisellini et al. 1997;Fossati et al. 1998): A low-frequency component from radio to the UV or X-rays, generally agreed to be due to synchrotron radiation from relativistic electrons in the jet, and a high-frequency component from X-rays to γ−rays, which can be either due to Compton scattering of lower-frequency radiation by the same relativistic electrons (leptonic models i.e. Krawczynski 2004) or due to interactions of ultra-relativistic protons in the jet (hadronic models), either via proton synchrotron radiation (Mucke et al. 2003) or via secondary emission from photo-pion and photo-pair production process (Böttcher 2007; and references therein).Blazars can be classified into three sub-classes, depending on the peak frequency of their synchrotron emission: LSPs (low-synchrotron-peaked blazars), consisting predominantly of LBLs (red or low energy or radio selected blazars) and defined by a peak of their synchrotron component at ν sy < 10 14 Hz, ISPs (intermediate-synchrotron-peaked blazars) or IBLs, consisting mostly of intermediate blazars, defined by 10 14 Hz < ν sy < 10 15 Hz, and HSPs (high-synchrotron-peaked blazars), all of which are HBLs (blue or high energy or X-ray selected blazars) and which are defined through ν sy > 10 15 Hz (Abdo et al. 2010). The high-energy component of the spectral energy distribution (SED) of blazars extends up to γ−rays, peaking at GeV energies in LSPs and at TeV energies in HSPs. Blazar properties are consistent with relativistic beaming, i.e. bulk relativistic motion of the jet plasma at small angles to the line of sight, which gives rise to a strong amplification and rapid variability in the observer's frame.", 'arxivid': '1712.02516', 'author': ['Alok C Gupta *correspondence:[email protected] \nAryabhatta Research Institute of Observational Sciences (ARIES)\nManora PeakNainital -263002India\n'], 'authoraffiliation': ['Aryabhatta Research Institute of Observational Sciences (ARIES)\nManora PeakNainital -263002India'], 'corpusid': 46796114, 'doi': '10.3390/galaxies6010001', 'github_urls': [], 'n_tokens_mistral': 28437, 'n_tokens_neox': 22732, 'n_words': 11737, 'pdfsha': '856509eea18e1a7f97262089f096bf68b60787a8', 'pdfurls': ['https://arxiv.org/pdf/1712.02516v1.pdf'], 'title': ['Multi-wavelength Intra-day Variability and Quasi-periodic Oscillation in Blazars', 'Multi-wavelength Intra-day Variability and Quasi-periodic Oscillation in Blazars'], 'venue': []} |
arxiv |
DiffULD: Diffusive Universal Lesion Detection
Peiang Zhao
School of Biomedical Engineering
Center for Medical Imaging
Analytic Computing & LEarning (MIRACLE)
Suzhou Institute for Advanced Research
Robotics
University of Science and Technology of China
215123SuzhouChina
Han Li
School of Biomedical Engineering
Center for Medical Imaging
Analytic Computing & LEarning (MIRACLE)
Suzhou Institute for Advanced Research
Robotics
University of Science and Technology of China
215123SuzhouChina
Ruiyang Jin
School of Biomedical Engineering
Center for Medical Imaging
Analytic Computing & LEarning (MIRACLE)
Suzhou Institute for Advanced Research
Robotics
University of Science and Technology of China
215123SuzhouChina
S Kevin Zhou
School of Biomedical Engineering
Center for Medical Imaging
Analytic Computing & LEarning (MIRACLE)
Suzhou Institute for Advanced Research
Robotics
University of Science and Technology of China
215123SuzhouChina
Key Lab of Intelligent Information Processing of Chinese Academy of Sciences (CAS)
Institute of Computing Technology
CAS
100190BeijingChina
DiffULD: Diffusive Universal Lesion Detection
Universal Lesion Detection · Diffusion Model
Universal Lesion Detection (ULD) in computed tomography (CT) plays an essential role in computer-aided diagnosis. Promising ULD results have been reported by anchor-based detection designs, but they have inherent drawbacks due to the use of anchors: i) Insufficient training target and ii) Difficulties in anchor design. Diffusion probability models (DPM) have demonstrated outstanding capabilities in many vision tasks. Many DPM-based approaches achieve great success in natural image object detection without using anchors. But they are still ineffective for ULD due to the insufficient training targets. In this paper, we propose a novel ULD method, DiffULD, which utilizes DPM for lesion detection. To tackle the negative effect triggered by insufficient targets, we introduce a novel center-aligned bounding box padding strategy that provides additional high-quality training targets yet avoids significant performance deterioration. DiffULD is inherently advanced in locating lesions with diverse sizes and shapes since it can predict with arbitrary boxes. Experiments on the benchmark dataset DeepLesion[1]show the superiority of DiffULD when compared to state-of-the-art ULD approaches.
Introduction
Universal Lesion Detection (ULD) in computed tomography (CT) [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] plays an important role in computer-aided diagnosis (CAD) [19,20]. The design of detection-only instead of identifying the lesion types in ULD [21][22][23][24][25][26][27][28] prominently decreases the difficulty of this task for a specific organ (e.g., lung, liver), but it is still challenging for lesions vary in shapes and sizes among whole human body.
Previous arts in ULD are mainly motivated by the anchor-based detection framework, e.g., Faster-RCNN [29]. These studies focus on adapting the detection backbone to universally locate lesions in CT scans. For instance, Li et al. [8] propose the so-called MVP-Net, a multi-view FPN with a position-aware attention mechanism to assist ULD training. Yang et al. [10,16,30] propose a series of 3D feature fusion operators to incorporate context information from several adjacent CT slices for better performance. Li et al. [31] introduce a plugand-play transformer block to form hybrid backbones which can better model long-distance feature dependency. While achieving success, these anchor-based methods have inherent drawbacks: (i) Insufficient training target.
In stage-1, anchor-based methods identify the positive (lesion) anchors and label them as the region of interest (RoI) based on the IoU between anchors and ground-truth (GT) bounding boxes (BBoxes). An anchor is considered positive if its IoU with any GT BBox is greater than the IoU threshold and negative otherwise [29]. The positive anchors are sufficient in natural images as they usually have many targets per image [12]. However, the number of lesions per CT scan is limited, most CT slices only contain one or two lesions (i.e., detection targets in ULD) per CT slice [17]. Still applying the IoU-based anchor matching mechanism with such limited targets can lead to severe data imbalance and further hinders network convergence. Simply lowering the positive IoU threshold in the anchor-selecting mechanism can alleviate the shortage of positive anchors to some degree, but it leads to a higher false positive (FP) rate by labeling more low-IoU anchors as positive. (ii) Difficulties in anchor design. In anchorbased methods, the size, ratio and number of anchors are pre-defined hyperparameters that significantly influence the detection performance [32]. Thus a proper design of anchor hyper-parameters is of great importance. However, tuning anchor hyper-parameters is a challenging task in ULD because of the variety of lesions (target) with diverse diameters (from 0.21 to 342.5mm). Even with a careful design, the fixed rectangle anchor boxes can be a kind of obstruction in capturing heterogeneous lesions, which have irregular shapes and vary in size.
To get rid of the drawbacks caused by the anchor mechanism, researchers resort to anchor-free detection, e.g., FCOS [33] and DETR [34]. But these methods experience difficulties in achieving state-of-the-art results in ULD, as they lack the initialization of position prior provided by anchors.
Recently, the diffusion probabilistic model (DPM) [35][36][37][38][39] has demonstrated its outstanding capabilities in various vision tasks. Chen et. al. follow the key idea of DPM and propose a noise-to-box pipeline, DiffusionDet [40], for natural image object detection. They achieved success on natural images with sufficient training targets, but still experience difficulties in dealing with tasks with insufficient training targets like ULD. This is because the DPM's denoising is a dense distribution-to-distribution forecasting procedure that heavily relied on a large number of high-quality training targets to learn targets' distribution accurately.
To address this issue, we hereby introduce a novel center-aligned BBox padding strategy in DPM detection to form a diffusion-based detector for Universal Lesion Detection, termed DiffULD.
As shown in Fig. 1, DiffULD also formulates lesion detection as a denoising diffusion process from noisy boxes to prediction boxes similar to [40], but we further introduce the center-aligned BBox padding before DiffULD's forward diffusion process to generate perturbated GT. Specifically, we add random perturbations to both scales and center coordinates of the original GT BBox, resulting in perturbated boxes whose centers remain aligned with the corresponding original GT BBox. Next, original GT boxes paired with these perturbated boxes are called perturbated GT boxes for simplicity. Finally, we feed the perturbated GT boxes to the model as the training objective during training. Compared with other training target padding methods (e.g., padding with random boxes), our strategy can provide additional targets of higher quality, i.e., center aligned with the original GT BBox. This approach effectively expands the insufficient training targets on CT scans, enhancing DPM's detection performance and avoiding deterioration triggered by adding random targets.
The following DPM training procedure contains two diffusion processes. i) In the forward training process, DiffULD corrupts the perturbated GT with Gaussian noise gradually to generate noisy boxes step by step. Then the model is trained to remove the noise and reconstruct the original perturbated GT boxes. ii) In the reverse inference process, the trained DiffULD can refine a set of randomly generated boxes iteratively to obtain the final detect predictions.
Our method gets rid of the drawbacks of pre-defined anchors and the deterioration of training DPM with insufficient GT targets. Besides, DiffULD is inherently advanced in locating targets with diverse sizes since it can predict with arbitrary boxes, which is an advantageous feature for detecting lesions of irregular shapes and various sizes. To validate the effectiveness of our method, we conduct experiments against seven state-of-the-art ULD methods on the public dataset DeepLesion [1]. The results demonstrate that our method achieves competitive performance compared to state-of-the-art ULD approaches.
Method
In this section, we first formulate our overall detection process for DiffULD and then specify the training manner, inference process and backbone design.
Diffusion-based detector for lesion detection
Universal Lesion Detection can be formulated as locating lesions in input CT scan I ct with a set of boxes predictions z 0 . For a particular box z , it can be denoted as z = [x 1 , y 1 , x 2 , y 2 ], where x 1 , y 1 and x 2 , y 2 are the coordinates of the top-left and bottom-right corners, respectively. We design our model based on a diffusion model mentioned in [40]. As shown in Fig. 2, our method consists of two stages, a forward diffusion (or training) process and a reverse refinement (or inference) process. In the forward process, We denote GT bounding boxes as z 0 and generate corrupted training samples z 1 , z 2 , ..., z T for latter DiffULD training by adding Gaussian noise iteratively, which can be defined as:
(z t | z 0 ) = N z t | √ᾱ t z 0 , (1 −ᾱ t ) I(1)
whereᾱ t represents the noise variance schedule and t ∈ {0, 1, ..., T }. Subsequently, a neural network f θ (z t , t, I ct ) conditioned on the corresponding CT scan I ct is trained to predict z 0 from a noisy box z T by reversing the noising process step by step. During inference, for an input CT scan I ct with a set of random boxes, the model is able to refine the random boxes to get a lesion detection prediction box z 0 , iteratively.
Training
In this section, we specify the training process with our novelty introduced 'Center-aligned BBox padding'. Center-aligned BBox padding. As shown in Fig. 1, we utilize Centeraligned BBox padding to generate perturbated boxes. Then, the perturbated boxes are paired with original GT BBoxes, forming perturbated GT boxes which are used to generate corrupted training samples z 1 , z 2 , ..., z T for the latter Dif-fULD training by adding Gaussian noise iteratively.
For clarity, we denote [x i c , y i c ] as center coordinates of original GT BBox, where z i , [w i , h i ] are the width and height of z i .
We consider the generation in two parts: box scaling and center sampling. (i) Box scaling: We set a hyper-parameter λ scale ∈ (0, 1) for scaling. For z i , The width and height of the corresponding perturbated boxes are randomly sampled in uniform distributions on ] of perturbated boxes from a 2D Gaussian distribution N whose probability density function can be denoted as:
U w ∼ [(1 − λ scale )w i , (1 + λ scale )w i ] and U h ∼ [(1 − λ scale )h i , (1 + λ scale )h i ]. (f(x, y) = exp − x − x i c 2 + y − y i c 2 2σ 2 , (x, y) ∈ z i(2)
where σ is a size-adaptive parameter, which can be calculated according to the z i 's width and height:
σ = 1 6 (w i + h i ).(3)
Besides, for each input CT scan I ct , we collect all GT BBoxes in z and add random perturbations to them and generate multiple perturbated boxes for each of them. Thus the number of perturbated boxes in an image varies with its number of GT BBoxes. For better training, we fix the number of perturbated boxes as N for all training images. As shown in Fig 1., the perturbated boxes cluster together and their centers are still aligned with the corresponding original GT BBox. Subsequently, perturbated GT boxes z 0 are sent for corruption as the training objective.
Box corruption. As shown in Fig. 1, we corrupt the parameters of z 0 with Gaussian noises. The noise scale is controlled byᾱ t (in Eq. 1), which adopts the decreasing cosine scheduler in the different time step t.
Loss function. As the model generates the same number of (N ) predictions for the input image, termed as a prediction set, the loss function should be setwised [34]. Specifically, each GT is matched with the prediction by the least matching cost, and the overall training loss [34] can be represented as:
L = λ L1box · L L1box + λ giou · L giou(4)
where L L1box and L giou are the pairwise box loss. We adopt λ L1box = 2.0 and λ giou = 5.0.
Inference
At the inference stage, with a set of random boxes sampled from Gaussian distribution, the model does refinement step by step to obtain the final predictions z 0 . For better performance, two key components are used: Box filtering. In each refinement step, the model receives a set of box proposals from the last step. As the prediction starts from arbitrary boxes and the lack of GT (lesion), most of these proposals are very far from lesions. Keeping refining them in the following steps will hinder network training. Toward efficient detection, we send the proposals to the detection head and remove the boxes whose confidential scores are lower than a particular threshold λ conf . The remaining high-quality proposals are sent for followed DDIM sampling.
Box update with DDIM sampling. DDIM [41] is utilized to further refine the received box proposals by denoising. Next, these refined boxes are sent to the next step and start a new round of refinement. After multiple steps, final predictions are obtained.
However, we observe that if we just filter out boxes with low scores during iterative refinement, the model runs out of usable box proposals rapidly, which also leads to a deterioration in performance. Therefore, after the box updating, we add new boxes sampled from a Gaussian Distribution to the set of remaining boxes with. The number of box proposals per image is padded to the fixed number N before they are sent to the next refinement step.
Backbone Design
Multi-window input. Most prior arts in ULD use a single and fixed window (e.g., a wide window of [1024,4096]) to render the input CT scan, which suppresses organ-specific information and makes it hard for the network to focus on the various lesions. Therefore, taking cues from [8], we introduce 3 organ-specific HU windows to highlight multiple organs of interest. Their window widths and window levels are: W 1 = [1200, −600] for chest organs, W 2 = [400, 50] for soft tissues and W 3 = [200, 30] for abdomenal organs. Multi-window features are extracted with a ConvNeXt-T shared network.
3D context feature fusion. We modify the original A3D [16] DenseNet backbone for context fusion. We remove the first Conv3D Block and use the truncated network as our 3D context fusion module, which fuses the multi-view features from the last module. Multi-window features are fused with this module and sent to the detector subsequently for lesion detection.
Experiments
Settings
Our experiments are conducted on the standard ULD dataset DeepLesion [1]. The dataset contains 32,735 lesions on 32,120 axial slices from 10,594 CT studies of 4,427 unique patients. We use the official data split of DeepLesion which Table 1. Sensitivity (%) at various FPPI on the standard test set of DeepLesion. DKA-ULD [42] and SATr [31] are up-to-date SOTA ULD methods under the settings of 3 slices and 7 slices, respectively. consists of 70%, 15%, 15% for training, validation, and test, respectively. Besides, we also evaluate the performance of 3 methods based on a revised test set from [].
Methods
Training details. DiffULD is trained on CT scans of size 512 × 512 with a batch size of 4 on 4 NVIDIA RTX Titan GPUs with 24GB memory. For hyperparameters, the threshold N for box padding is set to 300. λ scale for box scaling is set to 0.4. λ conf for box filtering is set to 0.5. We use the Adam optimizer with an initial learning rate of 2e − 4 and the weight decay as 1e − 4. The default training schedule is 120K iterations, with the learning rate divided by 10 at 60K and 100K iterations. Data augmentation strategies contain random horizontal flipping, rotation, and random brightness adjustment.
Evaluation metrics. The lesion detection is classified as true positive (TP) when the IoU between the predicted and the GT bounding box is larger than 0.5. Average sensitivities computed at 0.5, 1, 2, and 4 false-positives (FP) per image are reported as the evaluation metrics on the test set for a fair comparison.
Lesion detection performance
We evaluate the effectiveness of DiffULD against anchor-based ULD approaches such as 3DCE [7], MVP-Net [8], A3D [16] and SATr [31] on DeepLesion. Several anchor-free natural image detection methods such as FCOS [33] and DN-DETR [44] are also introduced for comparison. In addition, we conduct an ex- tensive experiment to explore DiffULD's potential on an augmented test set of completely annotated DeepLesion volumes, introduced by Lesion-Harvester [45]. Table 1. demonstrates that our proposed DiffULD achieves favorable performances when compared to recent state-of-the-art anchor-based ULD approaches such as SATr on both 3 slices and 7 slices. It outperforms prior well-established methods such as A3D and MULAN by a non-trivial margin. This validates that with our padding strategy, the concise DPM can be utilized in general medical object detection tasks such as ULD and attain impressive performance.
Ablation study
We provide an ablation study about our proposed approach: center-aligned BBox padding. As shown in Table 3., we compared it with various other padding strategies, including: (i) duplicating original GT boxes; (ii) padding random boxes sampled from a uniform distribution; (iii) padding random boxes sampled from a Gaussian distribution; (iv) padding with center-aligned strategy.
Our baseline method is training the diffusion model directly with no box padding, using our proposed backbone in 2.4. The performance is increased by 0.30% by simply duplicating the original GT boxes. Padding random boxes following uniform and Gaussian distributions brings 0.51% and 0.91% improvement respectively. Our center-aligned padding strategy accounts for 1.13% improvement from the baseline. We attribute this performance boost to center-aligned padding's ability to provide high-quality additional training targets. It effectively expands the insufficient training targets on CT scans and enhances DPM's detection performance while avoiding deterioration triggered by adding random targets. This property is favorable for utilizing DPMs on a limited amount of GT like ULD.
Conclusion
In this paper, we propose a novel ULD method termed DiffULD by introducing the diffusion probability model (DPM) to Universal Lesion Detection. We present a novel center-aligned BBox padding strategy to tackle the performance deterioration caused by directly utilizing DPM on CT scans with sparse lesion bounding boxes. Compared with other training target padding methods (e.g., padding with random boxes), our strategy can provide additional training targets of higher quality and boost detection performance while avoiding significant deterioration. DiffULD is inherently advanced in locating lesions with diverse sizes and shapes since it can predict with arbitrary boxes, making it a promising method for ULD. Experiments on both standard and augmented DeepLesion datasets show that our proposed method can achieve competitive performance compared to state-of-the-art ULD approaches.
Fig. 1 .
1Overview of the proposed center-aligned BBox padding strategy.
Fig. 2 .
2Overview of DiffULD. The backbone extracts feature representation from an input CT scan. Then, the model takes the feature map and a set of noisy boxes as input and makes predictions.
Table 3 .
3Ablation study of padding strategies at various FPs per image (FPPI).Baseline Duplicate Gaussian Uniform Center-Aligned
FPPI = 0.5
FPPI = 1
76.71
83.49
77.01
83.61
77.68
83.98
77.22
83.75
77.84 (1.13↑) 84.57 (1.08↑)
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| {'fraction_non_alphanumeric': 0.04437446321213856, 'fraction_numerical': 0.030918980818780417, 'mean_word_length': 4.734768332725283, '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': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Universal Lesion Detection (ULD) in computed tomography (CT) plays an essential role in computer-aided diagnosis. Promising ULD results have been reported by anchor-based detection designs, but they have inherent drawbacks due to the use of anchors: i) Insufficient training target and ii) Difficulties in anchor design. Diffusion probability models (DPM) have demonstrated outstanding capabilities in many vision tasks. Many DPM-based approaches achieve great success in natural image object detection without using anchors. But they are still ineffective for ULD due to the insufficient training targets. In this paper, we propose a novel ULD method, DiffULD, which utilizes DPM for lesion detection. To tackle the negative effect triggered by insufficient targets, we introduce a novel center-aligned bounding box padding strategy that provides additional high-quality training targets yet avoids significant performance deterioration. DiffULD is inherently advanced in locating lesions with diverse sizes and shapes since it can predict with arbitrary boxes. Experiments on the benchmark dataset DeepLesion[1]show the superiority of DiffULD when compared to state-of-the-art ULD approaches.', 'arxivid': '2303.15728', 'author': ['Peiang Zhao \nSchool of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics\n\nUniversity of Science and Technology of China\n215123SuzhouChina\n', 'Han Li \nSchool of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics\n\nUniversity of Science and Technology of China\n215123SuzhouChina\n', 'Ruiyang Jin \nSchool of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics\n\nUniversity of Science and Technology of China\n215123SuzhouChina\n', 'S Kevin Zhou \nSchool of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics\n\nUniversity of Science and Technology of China\n215123SuzhouChina\n\nKey Lab of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina\n'], 'authoraffiliation': ['School of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics', 'University of Science and Technology of China\n215123SuzhouChina', 'School of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics', 'University of Science and Technology of China\n215123SuzhouChina', 'School of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics', 'University of Science and Technology of China\n215123SuzhouChina', 'School of Biomedical Engineering\nCenter for Medical Imaging\nAnalytic Computing & LEarning (MIRACLE)\nSuzhou Institute for Advanced Research\nRobotics', 'University of Science and Technology of China\n215123SuzhouChina', 'Key Lab of Intelligent Information Processing of Chinese Academy of Sciences (CAS)\nInstitute of Computing Technology\nCAS\n100190BeijingChina'], 'corpusid': 257771843, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8970, 'n_tokens_neox': 7447, 'n_words': 4606, 'pdfsha': 'fcc316197474129fa5f55665652200d418d3703a', 'pdfurls': ['https://export.arxiv.org/pdf/2303.15728v1.pdf'], 'title': ['DiffULD: Diffusive Universal Lesion Detection', 'DiffULD: Diffusive Universal Lesion Detection'], 'venue': []} |
arxiv |
DISCOVERING LATENT NETWORK STRUCTURE IN POINT PROCESS DATA
Scott W Linderman
Harvard University
Ryan P Adams
Harvard University
DISCOVERING LATENT NETWORK STRUCTURE IN POINT PROCESS DATA
Networks play a central role in modern data analysis, enabling us to reason about systems by studying the relationships between their parts. Most often in network analysis, the edges are given. However, in many systems it is difficult or impossible to measure the network directly. Examples of latent networks include economic interactions linking financial instruments and patterns of reciprocity in gang violence. In these cases, we are limited to noisy observations of events associated with each node. To enable analysis of these implicit networks, we develop a probabilistic model that combines mutuallyexciting point processes with random graph models. We show how the Poisson superposition principle enables an elegant auxiliary variable formulation and a fully-Bayesian, parallel inference algorithm. We evaluate this new model empirically on several datasets.
1. Introduction. Many types of modern data are characterized via relationships on a network. Social network analysis is the most commonly considered example, where the properties of individuals (vertices) can be inferred from "friendship" type connections (edges). Such analyses are also critical to understanding regulatory biological pathways, trade relationships between nations, and propagation of disease. The tasks associated with such data may be unsupervised (e.g., identifying low-dimensional representations of edges or vertices) or supervised (e.g., predicting unobserved links in the graph). Traditionally, network analysis has focused on explicit network problems in which the graph itself is considered to be the observed data. That is, the vertices are considered known and the data are the entries in the associated adjacency matrix. A rich literature has arisen in recent years for applying statistical machine learning models to this type of problem, e.g., Liben-Nowell & Kleinberg (2007); Hoff (2008); Goldenberg et al. (2010).
In this paper we are concerned with implicit networks that cannot be observed directly, but about which we wish to perform analysis. In an implicit network, the vertices or edges of the graph may not be directly observed, but the graph structure may be inferred from noisy emissions. These noisy observations are assumed to have been generated according to underlying dynamics that respect the latent network structure.
For example, trades on financial stock markets are executed thousands of times per second. Trades of one stock are likely to cause subsequent activity on stocks in related industries. How can we infer such interactions and disentangle them from market-wide fluctuations that occur throughout the day? Discovering latent structure underlying financial markets not only reveals interpretable patterns of interaction, but also provides insight into the stability of the market. In Section 4 we will analyze the stability of mutually-excitatory systems, and in Section 6 we will explore how stock similarity may be inferred from trading activity.
As another example, both the edges and vertices may be latent. In Section 7, we examine patterns of violence in Chicago, which can often be attributed to social structures in the form of gangs. We would expect that attacks from one gang onto another might induce cascades of violence, but the vertices (gang identity of both perpetrator and victim) are unobserved. As with the financial data, it should be possible to exploit dynamics to infer these social structures. In this case spatial information is available as well, which can help inform latent vertex identities.
In both of these examples, the noisy emissions have the form of events in time, or "spikes," and our intuition is that a spike at a vertex will induce activity at adjacent vertices. In this paper, we formalize this idea into a probabilistic model based on mutually-interacting point processes. Specifically, we combine the Hawkes process (Hawkes, 1971) with recently developed exchangeable random graph priors. This combination allows us to reason about latent networks in terms of the way that they regulate interaction in the Hawkes process. Inference in the resulting model can be done with Markov chain Monte Carlo, and an elegant data augmentation scheme results in efficient parallelism.
Preliminaries.
2.1. Poisson Processes. Point processes are fundamental statistical objects that yield random finite sets of events {s n } N n=1 ⊂ S, where S is a compact subset of R D , for example, space or time. The Poisson process is the canonical example. It is governed by a nonnegative "rate" or "intensity" function, λ(s) : S → R + . The number of events in a subset S ⊂ S follows a Poisson distribution with mean S λ(s)ds. Moreover, the number of events in disjoint subsets are independent.
We use the notation {s n } N n=1 ∼ PP(λ(s)) to indicate that a set of events {s n } N n=1 is drawn from a Poisson process with rate λ(s). The likelihood is given by
p({s n } N n=1 |λ(s)) = exp − S λ(s)ds N n=1 λ(s n ).(1)
In this work we will make use of a special property of Poisson processes, the Poisson superposition theorem, which states that {s n } ∼ PP(λ 1 (s) + . . . + λ K (s)) can be decomposed into K independent Poisson processes. Letting z n denote the origin of the n-th event, we perform the decomposition by independently sampling each z n from Pr(z n = k) ∝ λ k (s n ), for k ∈ {1 . . . K} (Daley & Vere-Jones, 1988).
Hawkes Processes.
Though Poisson processes have many nice properties, they cannot capture interactions between events. For this we turn to a more general model known as Hawkes processes. A Hawkes process consists of K point processes and gives rise to sets of marked events {s n , c n } N n=1 , where c n ∈ {1, . . . , K} specifies the process on which the n-th event occurred. For now, we assume the events are points in time, i.e., s n ∈ [0, T ]. Each of the K processes is a conditionally Poisson process with a rate λ k (t | {s n : s n < t}) that depends on the history of events up to time t.
Hawkes processes have additive interactions. Each process has a "background rate" λ 0,k (t), and each event s n on process k adds a nonnegative impulse response h k,k (t − s n ) to the intensity of other processes k . Causality and locality of influence are enforced by requiring h k,k (∆t) to be zero for ∆t / ∈ [0, ∆t max ]. By the superposition theorem for Poisson processes, these additive components can be considered independent processes, each giving rise to their own events. We augment our data with a latent random variable z n ∈ {0, . . . , n − 1} to indicate the cause of the n-th event (0 if the event is due to the background rate and 1 . . . n − 1 if it was caused by a preceding event).
Let C n,k denote the set of events on process k that were parented by event n. Formally, C n,k ≡ {s n : c n = k ∧ z n = n}. Let C 0,k be the set of events attributed to the background rate of process k. The augmented Hawkes likelihood is the product of likelihoods of each Poisson process:
p({(s n , c n , z n )} N n=1 | {λ 0,k (t)},{{h k,k (∆t)}}) = K k=1 p(C 0,k | λ 0,k (t)) × N n=1 K k=1 p(C n,k | h cn,k (t − s n )) ,(2)
where the densities in the product are given by Equation 1. Figure 1 illustrates a causal cascades of events for a simple network of three processes (I-III). The first event is caused by the background rate (z 1 = 0), and it induces impulse responses on processes II and III. Event 2 is spawned by the impulse on the third process (z 2 = 1), and feeds back onto processes I and II. In some cases a single parent event induces multiple children, e.g., event 4 spawns events 5a-c. In this simple example, processes excite one another, but do not excite themselves. Next we will introduce more sophisticated models for such interaction networks.
2.3. Random Graph Models. Graphs of K nodes correspond to K × K matrices. Unweighted graphs are binary adjacency matrices A where A k,k = 1 indicates a directed edge from node k to node k . Weighted directed graphs can be represented by a real matrix W whose entries indicate the weights of the edges. Random graph models reflect the probability of different network structures through distributions over these matrices.
Recently, many random graph models have been unified under an elegant theoretical framework due to Aldous and Hoover (Aldous, 1981;Hoover, 1979). See Lloyd et al. (2012) for an overview. Conceptually, the Aldous-Hoover representation characterizes the class of exchangeable random graphs, that is, graph models for which the joint probability is invariant under permutations of the node labels. Just as de Finetti's theorem equates exchangeable sequences (X n ) n∈N to independent draws from a random probability measure Θ, the Aldous-Hoover theorem relates random exchangeable graphs to the following generative model:
u 1 , u 2 , . . . ∼ i.i.d Uniform[0, 1], A k,k ∼ Bernoulli(Θ(u k , u k )),
for some random function Θ : [0, 1] 2 → [0, 1].
Empty graph models (A k,k ≡ 0) and complete models (A k,k ≡ 1) are trivial examples, but much more structure may be encoded. For example, consider a model in which nodes are endowed with a location in space, x k ∈ R D . This could be an abstract feature space or a real location like the center of a gang territory. The probability of connection between two notes decreases with distance between them as A k,k ∼ Bern(ρe −||x k −x k ||/τ ), where ρ is the overall sparsity and τ is the characteristic distance scale. This simple model can be converted to the Aldous-Hoover representation by transforming u k into x k via the inverse CDF.
Many models can be constructed in this manner. Stochastic block models, latent eigenmodels, and their nonparametric extensions all fall under this class (Lloyd et al., 2012). We will leverage the generality of the Aldous-Hoover formalism to build a flexible model and inference algorithm for Hawkes processes with structured interaction networks.
3. The Network Hawkes Model. In order to combine Hawkes processes and random network models, we decompose the Hawkes impulse response h k,k (∆t) as follows:
h k,k (∆t) = A k,k W k,k g θ k,k (∆t).(3)
Here, A ∈ {0, 1} K×K is a binary adjacency matrix and W ∈ R K×K + is a non-negative weight matrix. Together these specify the sparsity structure and strength of the interaction network, respectively. The non-negative function g θ k,k (∆t) captures the temporal aspect of the interaction. It is parameterized by θ k,k and satisfies two properties: a) it has bounded support for ∆t ∈ [0, ∆t max ], and b) it integrates to one. In other words, g is a probability density with compact support.
Decomposing h as in Equation 3 has many advantages. It allows us to express our separate beliefs about the sparsity structure of the interaction network and the strength of the interactions through a spike-and-slab prior on A and W (Mohamed et al., 2012). The empty graph model recovers independent background processes, and the complete graph recovers the standard Hawkes process. Making g a probability density endows W with units of "expected number of events" and allows us to compare the relative strength of interactions. The form suggests an intuitive generative model: for each impulse response draw m ∼ Poisson(W k,k ) number of induced events and draw the m child event times i.i.d. from g, enabling computationally tractable conjugate priors.
Intuitively, the background rates, λ 0,k (t), explain events that cannot be attributed to preceding events. In the simplest case the background rate is constant. However, there are often fluctuations in overall intensity that are shared among the processes, and not reflective of process-to-process interaction, as we will see in the daily variations in trading volume on the S&P100 and the seasonal trends in homicide. To capture these shared background fluctuations, we use a sparse Log Gaussian Cox process (Møller et al., 1998) to model the background rate:
λ 0,k (t) = µ k + α k exp{y(t)}, y(t) ∼ GP(0, K(t, t )).
The kernel K(t, t ) describes the covariance structure of the background rate that is shared by all processes. For example, a periodic kernel may capture seasonal or daily fluctuations. The offset µ k accounts for varying background intensities among processes, and the scaling factor α k governs how sensitive process k is to these background fluctuations (when α k = 0 we recover the constant background rate).
Finally, in some cases the process identities, c n , must also be inferred. With gang incidents in Chicago we may have only a location, x n ∈ R 2 . In this case, we may place a spatial Gaussian mixture model over the c n 's, as in Cho et al. (2013). Alternatively, we may be given the label of the community in which the incident occurred, but we suspect that interactions occur between clusters of communities. In this case we can use a simple clustering model or a nonparametric model like that of Blundell et al. (2012).
3.1. Inference with Gibbs Sampling. We present a Gibbs sampling procedure for inferring the model parameters, W , A, {{θ k,k }},{λ 0,k (t)}, and, if necessary, {c n }. In order to simplify our Gibbs updates, we will also sample a set of parent assignments for each event {z n }. Incorporating these parent variables enables conjugate prior distributions for W , θ k,k , and, in the case of constant background rates, λ 0,k .
Sampling weights W .. A gamma prior on the weights, W k,k ∼ Gamma(α 0 W , β 0 W ), results in the conditional distribution, W k,k | {s n , c n , z n } N n=1 , θ k,k ∼ Gamma(α k,k , β k,k ), α k,k = α 0 W + N n=1 N n =1 δ cn,k δ c n ,k δ z n , n β k,k = β 0 W + N n=1 δ cn,k .
This is a minor approximation valid for ∆t max T . Here and elsewhere, δ i,j is the Kronecker delta function. We use the inverse-scale parameterization of the gamma distribution, i.e.,
Gamma(x | α, β) = β α Γ(α) x α−1 exp{−β x}.
Sampling impulse response parameters θ k,k .. We let g k,k (∆t) be the logistic-normal density with parameters θ k,k = {µ, τ }:
g k,k (∆t | µ, τ ) = 1 Z exp −τ 2 σ −1 ∆t ∆t max − µ 2 σ −1 (x) = ln(x/(1 − x)) Z = ∆t(∆t max − ∆t) ∆t max τ 2π − 1 2 .
The normal-gamma prior µ, τ ∼ N G(µ, τ |µ 0 µ , κ 0 µ , α 0 τ , β 0 τ ) yields the standard conditional distribution (see Murphy, 2012) with the following sufficient statistics:
x n,n = ln(s n − s n ) − ln(t max − (s n − s n )), m = N n=1 N n =1 δ cn,k δ c n ,k δ z n ,n , x = 1 m N n=1 N n =1
δ cn,k δ c n ,k δ z n ,n x n,n .
Sampling background rates λ 0,k .. For background rates λ 0,k (t) ≡ λ 0,k , the prior λ 0,k ∼ Gamma(α 0 λ , β 0 λ ) is conjugate with the likelihood and yield the conditional distribution This conjugacy no longer holds for Gaussian process background rates, but conditioned upon the parent variables, we must simply fit a Gaussian process for those events for which z n = 0. We use elliptical slice sampling (Murray et al., 2010) for this purpose.
λ 0,k | {s n , c n , z n } N n=1 , ∼ Gamma(α λ , β λ ), α λ = α 0 λ + n δ cn,k δ zn,0 β λ = β 0 λ + T
Collapsed Gibbs sampling A and z n .. With Aldous-Hoover graph priors, the entries in the binary adjacency matrix A are conditionally independent given the parameters of the prior. The likelihood introduces dependencies between the rows of A, but each column can be sampled in parallel. Gibbs updates are complicated by strong dependencies between the graph and the parent variables, z n . Specifically, if z n = n, then we must have A cn,c n = 1. To improve the mixing of our sampling algorithm, first we update A | {s n , c n }, W , θ k,k by marginalizing the parent variables. The posterior is determined by the likelihood of the conditionally Poisson process λ k (t | {s n : s n < t}) (Equation 1) with and without interaction A k,k and the prior comes from the Aldous-Hoover graph model. Then we update z n | {s n , c n }, A, W , θ k,k by sampling from the discrete conditional distribution. Though there are N parent variables, they are conditionally independent and may be sampled in parallel. We have implemented our inference algorithm on GPUs to capitalize on this parallelism.
Sampling process identities c n .. As with the adjacency matrix, we use a collapsed Gibbs sampler to marginalize out the parent variables when sampling the process identities. Unfortunately, the c n 's are not conditionally independent and hence must be sampled sequentially. This limits the size of the datasets we can handle when the process identities are unknown, but our GPU implementation is still able to achieve upwards of 4 iterations (sampling all variables) per second on datasets with thousands of events.
Stability of Network Hawkes
Processes. Due to their recurrent nature, Hawkes processes must be constrained to ensure their positive feedback does not lead to infinite numbers of events. A stable system must satisfy 1 Daley & Vere-Jones, 1988). When we are conditioning on finite datasets we do not have to worry about this. We simply place weak priors on the network parameters, e.g., a beta prior on the sparsity ρ of an Erdős-Renyi graph, and a Jeffreys prior on the scale of the gamma weight distribution. For the generative model, however, we would like to set our hyperparameters such that the prior distribution places little mass on unstable networks. In order to do so, we use tools from random matrix theory.
λ max = max | eig(A W ) | < 1 (see
The celebrated circular law describes the asymptotic eigenvalue distribution for K × K random matrices with entries that are i.i.d. with zero mean and variance σ 2 . As K grows, the eigenvalues are uniformly distributed over a disk in the complex plane centered at the origin and with radius σ √ K. In our case, however, the mean of the entries, E[A k,k W k,k ] = µ, is not zero. Silverstein (1994) has shown that we can analyze noncentral random matrices by considering them to be perturbations about the mean.
Consider A W = V + U , where V = µKe K e T
K is a deterministic rank-one matrix with every entry equal to µ, e K ∈ R K is a column vector with all entries equal to K −1/2 , and U is a random matrix with i.i.d. zero-mean entries. Then, as K approaches infinity, the largest eigenvalue will come from V and will be distributed as λ max ∼ N (µK, σ 2 ), and the remaining eigenvalues will be uniformly distributed over the complex disc.
In the simple case of W k,k ∼ Gamma(α, β) and A k,k ∼ Bern(ρ), we have µ = ρα/β and σ = ρ((1 − ρ)α 2 + α)/β. For a given K, α and β, we can tune the sparsity parameter ρ to achieve stability with high probability. We simply set ρ such that the minimum of σ √ K and, say, µK + 3σ, equals one. Figures 2a and 2b show a variety of weight distributions and the maximum stable ρ. Increasing the network size, the mean, or the variance will require a concomitant increase in sparsity.
This approach relies on asymptotic eigenvalue distributions, and it is unclear how quickly the spectra of random matrices will converge to this distribution. To test this, we computed the empirical eigenvalue distribution for random matrices of various size, mean, and variance. We generated 10 4 random matrices for each weight distribution in Figure 2a with sizes K = 4, 64, and 1024, and ρ set to the theoretical maximum indicated by dots in Figure 2b. The theoretical and empirical distributions of the maximum eigenvalue are shown in Figures 2c and 2d. We find that for small mean and variance weights, for example Gamma(1, 5) in the Figure 2c, the empirical results closely match the theory. As the weights grow larger, as in Gamma(8, 12) in 2d, the empirical eigenvalue distributions have increased variance and lead to a greater than expected probability of unstable matrices for the range of network sizes tested here. We conclude that networks with strong weights should be counterbalanced by strong sparsity limits, or additional structure in the adjacency matrix that prohibits excitatory feedback loops.
Synthetic Results.
Our inference algorithm is first tested on synthetic data generated from the network Hawkes model. We perform two tests: a) a link prediction task where the process identities are given and the goal is to simply infer whether or not an interaction exists, and b) an event prediction task where we measure the probability of held-out event sequences. The network Hawkes model can be used for link prediction by considering the posterior probability of interactions P (A k,k | {s n , c n }). By thresholding at varying probabilities we compute a ROC curve. A standard Hawkes process assumes a complete set of interactions (A k,k ≡ 1), but we can similarly threshold its inferred weight matrix to perform link prediction.
Cross correlation provides a simple alternative measure of interaction. By summing the crosscorrelation over offsets ∆t ∈ [0, ∆t max ), we get a measure of directed interaction. A probabilistic alternative is offered by the generalized linear model for point processes (GLM), a popular model for spiking dynamics in computational neuroscience (Paninski, 2004). The GLM allows for constant background rates and both excitatory and inhibitory interactions. Impulse responses are modeled with linear basis functions. Area under the impulse response provides a measure of directed excitatory interaction that we use to compute a ROC curve. See the supplementary material for a detailed description of this model. Figure 3a, compared to a baseline of a Poisson process with constant rate. Improvement in predictive likelihood over baseline is normalized by the number of events in the test data to obtain units of "bits per spike." Again, the network Hawkes model outperforms the competitors in all but one sample network.
We sampled ten network Hawkes processes of 30 nodes each with Erdős-Renyi graph models, constant background rates, and the priors described in Section 3. The Hawkes processes were simulated for T = 1000 seconds. We used the models above to predict the presence or absence of interactions. The results of this experiment are shown in the ROC curves of Figure 3a. The network Hawkes model accurately identifies the sparse interactions, outperforming all other models.
With the Hawkes process and the GLM we can evaluate the log likelihood of held-out test data. On this task, the network Hawkes outperforms the competitors for 9 out 10 networks. On average, the network Hawkes model achieves 2.2 ± .1 bits/spike improvement in predictive log likelihood over a homogeneous Poisson process. Figure 3b shows that on average the standard Hawkes and the GLM provide only 60% and 72%, respectively, of this predictive power. See the supplementary material for further analysis.
6. Trades on the S&P 100. As an example of how Hawkes processes may discover interpretable latent structure in real-world data, we study the trades on the S&P 100 index collected at 1s intervals during the week of Sep. 28 through Oct. 2, 2009. Every time a stock price changes by ±0.1% of its current price an event is logged on the stock's process, yielding a total of K = 100 processes and N =182,037 events.
Trading volume varies substantially over the course of the day, with peaks at the opening and closing of the market. This daily variation is incorporated into the background rate via a Log Gaussian Cox Process (LGCP) with a periodic kernel (see supplementary material). We look for short-term interactions on top of this background rate with time scales of ∆t max = 60s. In Figure 4 we compare the predictive performance of independent LGCPs, a standard Hawkes process with LGCP background rates, and the network Hawkes model with LGCP background rates under two graph priors. The models are trained on four days of data and tested on the fifth. Though the network Hawkes is slightly outperformed by the standard Hawkes, the difference is small relative to the performance improvement from considering interactions, and the inferred network parameters provide interpretable insight into the market structure. In the latent distance model for A, each stock has a latent embedding x k ∈ R 2 such that nearby stocks are more likely to interact, as described in Section 2.3. Figure 5 shows a sample from the posterior distribution over embeddings in R 2 for ρ = 0.2 and τ = 1. We have plotted stocks in the six largest sectors, as listed on Bloomberg.com. Some sectors, notably energy and financials, tend to cluster together, indicating an increased probability of interaction between stocks in the same sector. Other sectors, such as consumer goods, are broadly distributed, suggesting that these stocks are less influenced by others in their sector. For the consumer industry, which is driven by slowly varying factors like inventory, this may not be surprising.
The Hinton diagram in the bottom panel of Figure 5 shows the top 4 eigenvectors of the interaction network. All eigenvalues are less than 1, indicating that the system is stable. The top row corresponds to first eigenvector (λ max = 0.74). Apple (AAPL), J.P. Morgan (JPM), and Exxon Mobil (XOM) have notably large entries in the eigenvector, suggesting that their activity will spawn cascades of self-excitation. The fourth eigenvector (λ 4 = 0.34) is dominated by Walgreens (WAG) and CVS (CVS), suggesting bursts of activity in these drug stores, perhaps due to encouraging quarterly reports during flu season (Associated Press, 2012).
7. Gangs of Chicago. In our final example, we study spatiotemporal patterns of gang-related homicide in Chicago. Sociologists have suggested that gang-related homicide is mediated by underlying social networks and occurs in mutually-exciting, retaliatory patterns (Papachristos, 2009). This is consistent with a spatiotemporal Hawkes process in which processes correspond to gang territories and homicides incite further homicides in rival territories.
We study gang-related homicides between 1980 and 1995 (Block et al., 2005). Homicides are labeled by the community in which they occurred. Over this time-frame there were N = 1637 gang-related homicides in the 77 communities of Chicago. We evaluate our model with an event-prediction task, training on 1980-1993 and testing on 1994-1995. We use a Log Gaussian Cox Process (LGCP) temporal background rate in all model variations. Our baseline is a single process with a uniform spatial rate for the city. We test two process identity models: a) the "community" model, which considers each community a separate process, and b) the "cluster" model, which groups communities into processes. The number of The community process identity model improves predictive performance by accounting for higher rates in South and West Chicago where gangs are deeply entrenched. Allowing for interactions between community areas, however, results in a decrease in predictive power due to overfitting (there is insufficient data to fit all 77 2 potential interactions). Interestingly, sparse graph priors do not help. They bias the model toward sparser but stronger interactions which are not supported by the test data. These results are shown in the "communities" group of Figure 6a. Clustering the communities improves predictive performance for all graph models, as seen in the "clusters" group. Moreover, the clustered models benefit from the inclusion of excitatory interactions, with the highest predictive log likelihoods coming from a four-cluster Erdős-Renyi graph model with interactions shown in Figure 6b. Distance-dependent graph priors do not improve predictive performance on this dataset, suggesting that either interactions do not occur over short distances, or that local rivalries are not substantial enough to be discovered in our dataset. More data is necessary to conclusively say which. Looking into the inferred clusters in Figure 6c and their rates in 6d, we can interpret the clusters as "safe suburbs" in gold, "buffer neighborhoods" in green, and "gang territories" in red and blue. Self-excitation in the blue cluster (Figure 6b) suggests that these regions are prone to bursts of activity, as one might expect during a turf-war. This interpretation is supported by reports of "a burst of street-gang violence in 1990 and 1991" in West Englewood (41.77 • N, −87.67 • W) (Block & Block, 1993). Figure 6d also shows a significant increase in the homicide rate between 1989 and 1995, consistent with reports of escalating gang warfare (Block & Block, 1993). In addition to this long-term trend, homicide rates show a pronounced seasonal effect, peaking in the summer and tapering in the winter. A LGCP with a quadratic kernel point-wise added to a periodic kernel captures both effects.
8. Related Work. Multivariate point processes are of great interest to the machine learning community as they are intuitive models for a variety of natural phenomena. We have leveraged previous work on Poisson processes with Gaussian process intensities in our background rate models (Cunningham et al., 2007). An expectation-maximization inference algorithm for Hawkes processes was put forth by Simma & Jordan (2010) and applied to very large social network datasets. We have adapted their latent variable formulation in our fully-Bayesian inference algorithm and introduced a framework for prior distributions over the latent network.
Others have considered special cases of the model we have proposed. Blundell et al. (2012) combine Hawkes processes and the Infinite Relational Model (a specific exchangeable graph model with an Aldous-Hoover representation) to cluster processes and discover interactions. Cho et al. (2013) applied Hawkes processes to gang incidents in Los Angeles. They developed a spatial Gaussian mixture model (GMM) for process identities, but did not explore structured network priors. We experimented with this process identity model but found that it suffers in predictive log likelihood tests (see supplementary material).
Recently, Iwata et al. (2013) developed a stochastic EM algorithm for Hawkes processes, leveraging similar conjugacy properties, but without network priors. Zhou et al. (2013) have developed a promising optimization-based approach to discovering low-rank networks in Hawkes processes, similar to some of the network models we explored.
Perhaps the most closely related work is that of Perry & Wolfe (2013). They provide a partial likelihood inference algorithm for Hawkes processes with a similar emphasis on structural patterns in the network of interactions. They provide an estimator capable of discovering homophily (the tendency for similar processes to interact) and other network effects. Our fully-Bayesian approach generalizes this method to capitalize on recent developments in random network models (Lloyd et al., 2012) and allows for nonparametric background rates.
Finally, generalized linear models (GLMs) are widely used in computational neuroscience (Paninski, 2004). GLMs allow for both excitatory and inhibitory interactions, but, as we have shown, when the data consists of purely excitatory interactions, Hawkes processes outperform GLMs in link-and event-prediction tests.
Conclusion.
We developed a framework for discovering latent network structure from spiking data. Our auxiliary variable formulation of the multivariate Hawkes process supported arbitrary Aldous-Hoover graph priors, Log Gaussian Cox Process background rates, and models of unobserved process identities. Our parallel MCMC algorithm allowed us to reason about uncertainty in the latent network in a fully-Bayesian manner, taking into account noisy observations and prior beliefs. We leveraged results from random matrix theory to analyze the conditions under which random network models will be stable, and our applications uncovered interpretable latent networks in a variety of synthetic and real-world problems. Generalizing beyond the Hawkes observation model is a promising avenue for future work.
APPENDIX A: INFERENCE DETAILS
A.1. Derivation of conjugate prior updates. By combining Equations 1 and 2 of the main text, we can write the joint likelihood, with the auxiliary parent variables, as,
p({s n , c n , z n } N n=1 , | {λ 0,k (t)} K k=1 , {h k,k (∆t)} k,k ) = K k=1 exp − T 0 λ 0,k (τ )dτ N n=1 λ 0,k (s n ) δ cn,k δ zn,0 × N n=1 K k =1 exp − T sn h cn,k (τ − s n )dτ N n =1
h cn,c n (s n − s n ) δ c n ,k δz n ,n .
The first line corresponds to the likelihood of the background processes; the second and third correspond to the likelihood of the induced processes triggered by each spike.
To derive the updates for weights, recall from Equation 3 of the main text that W k,k only appears in the impulse responses for which c n = k and c n = k . so we have,
p(W k,k | {s n , c n , z n } N n=1 , . . .) ∝ N n=1 exp − T sn h k,k (τ − s n )dτ N n =1 h k,k (s n − s n ) δ c n ,k δz n ,n δ cn,k × p(W k,k ) = N n=1 exp − T sn A k,k W k,k g k,k (τ − s n )dτ N n =1
A k,k W k,k g k,k (s n − s n ) δ c n ,k δz n ,n δ cn,k × p(W k,k ).
If A k,k = 1 and we ignore spikes after T − ∆t max , this is approximately proportional to
exp −W k,k N k W N k,k k,k p(W k,k ), where N k = N n=1 δ cn,k , and N k,k = N n=1 N n =1
δ cn,k δ c n ,k δ z n ,n .
When p(W k,k ) is a gamma distribution, the conditional distribution is also gamma. If A k,k = 0, the conditional distribution reduces to the prior, as expected.
Similar conjugate updates can be derived for constant background rates and the impulse response parameters, as stated in the main text.
A.2. Log Gaussian Cox Process background rates. In the Trades on the S&P100 and the Gangs of Chicago datasets, it was crucial to model the background fluctuations that were shared among all processes. However, if the background rate is allowed to vary at time scales shorter than ∆t max then it may obscure interactions between processes. To prevent this, we sample the Log Gaussian Cox Process (LGCP) at a sparse grid of M + 1 equally spaced points and linearly interpolate to evaluate the background rate at the exact time of each event. We have,
y = ŷ mT M M m=0
∼ GP(0, K(t, t )).
Then,
λ 0,k mT M M m=0 = µ k + α k exp ŷ mT M ,
and λ 0,k (s n ) is linearly interpolated between the rate at surrounding grid points. The equally spaced grid allows us to calculate the integral using the trapezoid quadrature rule. We use Elliptical Slice Sampling (Murray et al., 2010) to sample the conditional distribution of the vector y.
Kernel parameters are set empirically or with prior knowledge. For example, the period of the kernel is set to one day for the S&P100 dataset and one year for the Gangs of Chicago dataset since these are well-known trends. The scale and offset parameters have log Normal priors set such that the maximum and minimum homogeneous event counts in the training data are within two standard deviations of the expected value under the LGCP background rate. That is, the background rate should be able to explain all of the data without any observations if there is no evidence for interactions.
A.3. Priors on hyperparameters. When possible, we sample the parameters of the prior distributions. For example, in the Erdős-Renyi graph model we place a Beta(1, 1) prior on the sparsity ρ. For the latent distance model, we place a log normal prior on the characteristic length scale τ and sample it using Hamiltonian Monte Carlo.
For all of the results in this paper, we fixed the prior on the interaction kernel, g(∆t) to a weak Normal-Gamma distribution with parameters µ 0 µ = −1.0, κ 0 µ = 10, α 0 τ = 10, and β 0 τ = 1. Scale of gamma prior on weights.. For real data, we place an uninformative prior on the weight distribution. The gamma distribution is parameterized by a shape α 0 W and an inverse scale or rate β 0 W . The shape parameter α 0 W is chosen by hand (typically we use α 0 W = 2), but the inverse scale parameter β 0 W is sampled. We may not know a proper scale a priori, however we can use a scaleinvariant Jeffrey's prior to infer this parameter as well. Jeffrey's prior is proportional to the square root of the Fisher information, which for the gamma distribution is
Pr(β 0 W ) ∝ I(β 0 W ) = α 0 W β 0 W .
Hence the posterior is
Pr(β 0 W | {{W k,k }}) ∝ α 0 W β 0 W K k=1 K k =1 (β 0 W ) α 0 W Γ(α 0 W ) W α 0 W −1 k,k e −β 0 W W k,k ∝ (β 0 W ) K 2 α 0 W −1 exp −β 0 W K k=1 K k =1 W k,k .
This is a gamma distribution with parameters,
β 0 W ∼ Gamma(K 2 α 0 W , K k=1 K k =1
W k,k ).
APPENDIX B: SYNTHETIC TEST DETAILS
We generated T = 1000s of events for each synthetic network. The average number of spikes was 25,732 ± 9,425. Network 6, the only network for which the GLM outperformed the network Hawkes model in the event-prediction test, was an outlier with 44,973 events. For event prediction, we trained on the first 900 seconds and tested on the last 100 seconds of the data. We ran our Markov chain for 2500 iterations and computed the posterior probabilities of A and W using the last 500 samples.
A simple alternative to the Hawkes model is to look at cross-correlation between the event times. First, the event times are binned into an arrayŝ k of length M . Let (ŝ k ŝ k )[m] be the cross-correlation betweenŝ k andŝ k at discrete time lag m. Then,
W k,k = ∆tmaxM/T m=0 (ŝ k ŝ k )[m]
provides a simple measure of directed, excitatory interaction that can be thresholded to perform link prediction.
Additionally, we compare the network Hawkes process to the generalized linear model for point processes, a popular model in computational neuroscience (Paninski, 2004). Here, the event counts are modeled asŝ k,m ∼ Poisson(λ k,m ). The mean depends on external covariates and other events according to
λ k,m = exp α T k y m + K k =1 B b=1 β k,k ,b (g b * ŝ k )[m] ,
where y m is an external covariate at time m, {g b (∆m)} B b=1 are a set of basis functions that model impulse responses, and α and β are parameters to be inferred. Under this formulation the loglikelihood of the events is concave function of the parameters and is easily maximized. Unlike the Hawkes process, however, this model allows for inhibitory interactions.
For link prediction, b β k,k ,b provides a measure of directed excitatory interaction that can be used to compute an ROC curve. In our comparisons, we used y m ≡ 1 to allow for time-homogeneous background activity and set {g b (∆m)} to the top B = 6 principal components of a set of logistic normal impulse responses randomly sampled from the Hawkes prior.
We used an L1 penalty to promote sparsity in the parameters of the GLM, and chosen the penalty using cross validation on the last 100 seconds of the training data. Figure 3b of the main text shows the predictive log likelihoods for the Hawkes model with the correct Erdös-Renyi prior, the standard Hawkes model with a complete graph of interactions, and a GLM. On all but network 6, the network Hawkes model outperforms the competing models in terms of predictive log likelihood. Table 7 shows the average predictive performance across sample nextworks. The standard Hawkes and the GLM provide only 59.2% and 71.6%, respectively, of this predictive power.
APPENDIX C: TRADES ON THE S&P100 MODEL DETAILS
We study the trades on the S&P 100 index collected at 1s intervals during the week of Sep. 28 through Oct. 2, 2009. We group both positive and negative changes in price into the same process in order to measure overall activity. Another alternative would be to generate an "uptick" and a "downtick" process for each stock. We ignored trades outside regular trading hours because they tend to be outliers with widely varying prices. Since we are interested in short term interactions, we chose ∆t max = 60s. This also limits the number of potential event parents. If we were interested in interactions over longer durations, we would have to threshold the price changes at a higher level. We precluded self-excitation for this dataset since upticks are often followed by downticks and vice-versa. We are seeking to explain these brief price jumps using the activity of other stocks.
We run our Markov chain for 2000 iterations and compute predictive log likelihoods and the eigenvalues of the expected interaction matrix, E[A W ], using the last 400 iterations of the chain. The posterior sample illustrated in the main text is the last sample of the chain.
Trading volume varies substantially over the course of the day, with peaks at the opening and closing of the market. This daily variation is incorporated into the background rate via a Log Gaussian Cox Process with a periodic kernel. We set the period to one day. Figure 8 shows the posterior distribution over the background rate.
Though it is not discussed in the main text, we also considered stochastic block model (SBM) priors as well (Hoff, 2008), in hopes of recovering latent sector affiliations based on patterns of interaction between sectors. For example, stocks in the financial sector may have 90% probability of interacting with one another, and 30% probability of interacting with stocks in the energy sector. Rather than trying to interpret this from the embedding of a latent distance model, we can capture this belief explicitly with a stochastic block model prior on connectivity. We suppose there are J sectors, and the probability of belonging to a given sector is α ∈ [0, 1] J ∼ Dirichlet(α 0 ). The latent sector assignments are represented by the vector b ∈ [1, J] K , where b k ∼ Cat(α). The probability of a directed interaction is Pr(A k,k = 1) = B b k ,b k , where B is a J × J matrix of Bernoulli probabilities. We place a beta prior on the entries of B.
Our experiments with the SBM prior yield comparable predictive performance to the latent distance prior, as shown in Figure 9. The inferred clusters (not shown) are correlated with the clusters identified by Bloomberg.com, but more analysis is needed. It would also be interesting to study the difference in inferred interactions under the various graph models; this is left for future work.
Mon. Tues. Wed. Thur. LGCP 0.579 ± 0.006 Std. Hawkes 0.903 ± 0.003 Net. Hawkes (Erdős-Renyi) 0.893 ± 0.003 Net. Hawkes (Latent Distance) 0.879 ± 0.004 Net. Hawkes (SBM) 0.882 ± 0.004 Fig 9: Comparison of financial models on a event prediction task, relative to a homogeneous Poisson process baseline.
APPENDIX D: GANGS OF CHICAGO MODEL DETAILS
The first 12 years are used for training, 1993 is reserved for cross-validation, and the remaining two years are used to test the predictive power of the models. We also considered the crime dataset from www.data.cityofchicago.org, but this does not identify gang-related incidents.
We run our Markov chain for 700 iterations and use the last 200 iterations to compute predictive likelihoods and expectations. The posterior sample illustrated in the figure in main text is the last sample of the chain. Since this is a spatiotemporal dataset, our intensities are functions of both spatial location and time. For simplicity we factorize the intensity into λ k,x (x)λ k,t (t), where λ k,t (t) is a Gaussian process as described above, and λ k,x (x) is uniformly distributed over the spatial region associated with process k and is normalized such that it integrates to 1.
In the case of the latent distance model with the community process model, each community's location is fixed to its center of mass. With the cluster process model, we introduce a latent location for each cluster and use a Gaussian distribution for the prior probability that a community belongs to a cluster. This encourages spatially localized clusters. Figure 10 shows the cross validation results used to select the number of clusters, K, in the clustered process identity model and each of the four graph models. For the empty, complete, and Erdös-Renyi graph priors, we discover K = 15, 4, and 4 clusters respectively. The latent distance model, with its prior for spatially localized clusters, has its best performance for K = 5 clusters.
The spatial GMM process ID model from Cho et al. (2013) fails on this dataset because it assigns its spatial intensity over all of R 2 , whereas the clustering model concentrates the rate on only the communities in which the data resides. Figure 11 shows the results of this spatial process ID model on the prediction task. We did not test a latent distance model with the spatial GMM, but it would likely suffer in the same way as the empty, complete, and Erdős-Renyi graph priors. Comparison of predictive log likelihoods for Chicago homicides. This is the same as Figure 6a of the main text, but also includes the spatial GMM process identity model.
Fig 1 :
1Illustration of a Hawkes process. Events induce impulse responses on connected processes and spawn "child" events. See the main text for a complete description.
Fig 2 :
2Empirical and theoretical distribution of the maximum eigenvalue for Erdős-Renyi graphs with gamma weights. (a) Four gamma weight distributions. The colors correspond to the curves in the remaining panels. (b) Sparsity that theoretically yields 99% probability of stability as a function of p(W ) and K. (c) and (d) Theoretical (solid) and empirical (dots) distribution of the maximum eigenvalue. Color corresponds to the weight distribution in (a) and intensity indicates K and ρ shown in (b).
of models on a link prediction test averaged across ten randomly sampled synthetic networks of 30 nodes each. The network Hawkes model with the correct Erdős-Renyi graph prior outperforms a standard Hawkes model, GLM, and simple thresholding of the crosscorrelation matrix. (b) Comparison of predictive log likelihoods for the same set of networks as in
Fig 5 :
5Top: A sample from the posterior distribution over embeddings of stocks from the six largest sectors of the S&P100 under a latent distance graph model with two latent dimensions. Scale bar: the characteristic length scale of the latent distance model. The latent embedding tends to embed stocks such that they are nearby to, and hence more likely to interact with, others in their sector. Bottom: Hinton diagram of the top 4 eigenvectors. Size indicates magnitude of each stock's component in the eigenvector and colors denote sectors as in the top panel, with the addition of Materials (aqua), Utilities (orange), and Telecomm (gray). We show the eigenvectors corresponding to the four largest eigenvalues λ max = 0.74 (top row) to λ 4 = 0.34 (bottom row). clusters is chosen by cross-validation (see supplementary material). For each process identity model, we compare four graph models: a) independent LGCPs (empty), b) a standard Hawkes process with all possible interactions (complete), c) a network Hawkes model with a sparsity-inducing Erdős-Renyi graph prior, and d) a network Hawkes model with a latent distance model that prefers short-range interactions.
Fig 6 :
6Inferred interactions among clusters of community areas in the city of Chicago. (a) Predictive log likelihood for "communities" and "clusters" process identity models and four graph models. Panels (b-d) present results for the model with the highest predictive log likelihood: an Erdős-Renyi graph with K = 4 clusters. (b) The weighted interaction network in units of induced homicides over the training period(1980)(1981)(1982)(1983)(1984)(1985)(1986)(1987)(1988)(1989)(1990)(1991)(1992)(1993). (c) Inferred clustering of the 77 community areas. (d) The intensity for each cluster, broken down into the offset, the shared background rate, and the interactions (units of 10 −3 homicides per day per square kilometer).
Fig 8 :
8Posterior distribution over shared background rates for the S&P100. Shading indicates two standard deviations from the mean.
Fig 10 :
10Cross validation results for Chicago models with K clusters for each of the four graph models.
Fig 7: Relative improvement in predictive log likelihood over a homogeneous Poisson process baseline. Relative to the network Hawkes, the standard Hawkes and the GLM yield significantly less predictive power.Model
Relative prediction improvement
Network Hawkes
100%
Standard Hawkes
59.2±14.2%
GLM
71.6±9.2%
In this context λmax refers to an eigenvalue rather than a rate, and denotes the Hadamard product.
Acknowledgements. The authors wish to thank Leslie Valiant for many valuable discussions. SWL is supported by a National Defense Science and Engineering Graduate Fellowship.
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| {'fraction_non_alphanumeric': 0.045693674672297, 'fraction_numerical': 0.018808832585554675, 'mean_word_length': 4.4367706141189975, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 18, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Networks play a central role in modern data analysis, enabling us to reason about systems by studying the relationships between their parts. Most often in network analysis, the edges are given. However, in many systems it is difficult or impossible to measure the network directly. Examples of latent networks include economic interactions linking financial instruments and patterns of reciprocity in gang violence. In these cases, we are limited to noisy observations of events associated with each node. To enable analysis of these implicit networks, we develop a probabilistic model that combines mutuallyexciting point processes with random graph models. We show how the Poisson superposition principle enables an elegant auxiliary variable formulation and a fully-Bayesian, parallel inference algorithm. We evaluate this new model empirically on several datasets.', 'arxivid': '1402.0914', 'author': ['Scott W Linderman \nHarvard University\n\n', 'Ryan P Adams \nHarvard University\n\n', 'Scott W Linderman \nHarvard University\n\n', 'Ryan P Adams \nHarvard University\n\n'], 'authoraffiliation': ['Harvard University\n', 'Harvard University\n', 'Harvard University\n', 'Harvard University\n'], 'corpusid': 6418786, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 15350, 'n_tokens_neox': 13439, 'n_words': 9244, 'pdfsha': '78ed5e56ecbdc5062398137f16aecd3d45995147', 'pdfurls': ['https://arxiv.org/pdf/1402.0914v1.pdf'], 'title': ['DISCOVERING LATENT NETWORK STRUCTURE IN POINT PROCESS DATA', 'DISCOVERING LATENT NETWORK STRUCTURE IN POINT PROCESS DATA', 'DISCOVERING LATENT NETWORK STRUCTURE IN POINT PROCESS DATA', 'DISCOVERING LATENT NETWORK STRUCTURE IN POINT PROCESS DATA'], 'venue': []} |
arxiv |
Biosensing with plasmonic nanosensors
2008
Frédéric Peyskens [email protected]
Department of Physics, KULeuven
Photonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics
Ghent University
Sint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium
Ashim Dhakal
Department of Physics, KULeuven
Photonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics
Ghent University
Sint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium
Pol Van Dorpe
Nicolas Le Thomas
Department of Physics, KULeuven
Photonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics
Ghent University
Sint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium
Roel Baets
Department of Physics, KULeuven
Photonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics
Ghent University
Sint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium
Center for Nano-and BioPhotonics
Department of Physics
Photonics Research Group
INTEC-department
Ghent University-imec
Ghent University
Sint-Pietersnieuwstraat 41, Belgium, and imec, Kapeldreef 75, Celestijnenlaan 200D9000, 3001, 3001Ghent, Heverlee, KULeuven, LeuvenBelgium, Belgium
Biosensing with plasmonic nanosensors
Nat. Mater
712008* To whom correspondence should be addressed 1
Surface Enhanced Raman Spectroscopy (SERS) is a well-established technique for en-hancing Raman signals. [1][2][3][4][5][6][7][8][9][10][11][12] Recently photonic integrated circuits have been used, as an alternative to microscopy based excitation and collection, to probe SERS signals from external metallic nanoparticles. [13][14][15] However, in order to develop quantitative on-chip SERS sensors, integration of dedicated nanoplasmonic antennas and waveguides [16][17][18][19][20][21][22] is desirable.Here we bridge this gap by demonstrating for the first time the generation of SERS signals from integrated bowtie nanoantennas, excited and collected by a single mode waveguide, and rigorously quantify the enhancement process. The guided Raman power generated by a 4-Nitrothiophenol coated bowtie antenna shows an 8 × 10 6 enhancement compared to the free-space Raman scattering. An excellent correspondence is obtained between the theoretically predicted and observed absolute Raman power. This work paves the way towards fully integrated lab-on-a-chip systems where the single mode SERS-probe can be combined with other photonic, fluidic or biological functionalities.A schematic of the device under study is shown inFigure 1(a). The fundamental TE-mode of a silicon nitride (SiN) rib waveguide excites a periodic array of gold bowtie antennas coated with a 4-Nitrothiophenol (NTP) monolayer. The pump wavelength for all experiments is set to λ P = 785 nm and NTP Stokes light (at λ S ) is subsequently collected back into the same waveguide mode.Fabrication details can be found in the Methods section and a description of the measurement setup is outlined in the Supplementary Information S1. A scanning electron microscope image of the functionalized waveguide, with cross-sectional area of 220 × 700 nm 2 , is depicted inFigure 1(b). Raman spectra of an uncoated and coated waveguide functionalized with 40 antennas are shown inFigure 1(c). The spectral regions where an NTP Stokes peak is expected (1,080, 1,110, 1,340 and 1,575 cm −1 ) 23 are highlighted by the cyan shaded areas. Before coating no NTP peaks can be distinguished from the inherent SiN background. The peaks at 1,250 and 1,518 cm −1 (marked by the black dashed lines) are attributed to interference effects of the Au array which act on the scattered background light, and they are also observed on the extinction curves of the functionalized waveguides (see Supporting Information S2). Hence they do not represent specific 2 Raman lines. After coating, four additional peaks appear and coincide with the expected NTP Stokes peaks. This demonstrates that SERS signals from single monolayer coated antennas can be efficiently excited and collected by the same fundamental waveguide mode.Subsequently the dependence of the SERS signal on the position of the plasmon resonance was investigated to verify that it can be attributed to a resonance effect and not to coincidental surface roughness. To this end, waveguides functionalized with a fixed number of antennas but varying bowtie geometries were considered. The relevant bowtie parameters are its length L, gap ∆ and apex angle α(Figure 2(a)). By changing the length, the antenna resonance can be tuned (L 1 = 90 nm, L 2 = 115 nm and L 3 = 140 nm for fixed α = 60 • and ∆ = 40 nm). Extinction spectra are plotted inFigure 2(b) while the corresponding Raman spectra are depicted inFigure 2(c). The Raman spectrum of a reference waveguide without any Au functionalization is also shown. Even after coating the reference waveguide does not generate NTP peaks, so any Raman signal indeed originates from the antenna region and does not contain contributions from spontaneous Raman scattering along the waveguide. 24 The L 1 resonance is detuned from the relevant pump and Stokes region, resulting in a poor Raman spectrum. By increasing the length (L 2 and L 3 bowties) the resonance redshifts and lines up with the pump and Stokes wavelengths. For these bowties the NTP spectrum starts to emerge. The reported SERS spectra can hence be attributed to a plasmon resonance effect such that a stable and reproducible enhancement factor can be associated with them, in contrast to SERS events originating from random surface defects. The increased overlap with the plasmon resonance, and hence extinction, also results in a decreased background.Due to the metal induced loss, there will exist an optimum number N opt of patterned antennas such that the SERS signal is maximized. Such an optimum is investigated inFigure 3for a fixed bowtie geometry (α = 60 • , L = 100 nm and ∆ = 40 nm) but varying N: N = 10, 20, 30, 40, 70 and N = 0 which is a reference waveguide. Each waveguide is measured 10 times and the averaged Raman spectra are reported inFigure 3(a). The N = 70 signal is not shown since it could not be distinguished from the inherent offset signal of the detector. For a given fixed input power, corresponding to roughly 5 mW guided power, the reference waveguide generates a considerable 3 background signal in the 1,340 cm −1 region, where the strongest NTP peak is expected. Functionalizing the waveguide with increasing N reduces this unwanted background due to the attenuation caused by the nanoantennas. In addition the 1,340 cm −1 peak starts to emerge when N increases.The smaller peaks at 1,080, 1,110 and 1,575 cm −1 only appear when the background is sufficiently low. A zoom on the dominant 1,340 cm −1 peak (cyan dashed line) is shown inFigure 3(b). For clarity the background is locally subtracted. As expected, the signal reaches a maximum value for 10 ≤ N ≤ 20 and then decays again with increasing N. Apart from signal optimization it is however equally important to reduce the SiN background in order to resolve the smallest spectral features. Therefore an analytical model is developed to outline the interplay between signal enhancement and background reduction, and to derive the relevant figure of merit for on-chip SERS.Figure 4(a) shows a schematic longitudinal cross-section of the chip consisting of N antennas spaced with period Λ = 10 µm. Each array is centered on the waveguide with a distance L 1 ≈ 0.5 cm to the front and back facet of the chip. The NTP monolayer on each antenna will generate a forward propagating Stokes power P A (λ P , λ S ) for a given pump power P pump . This single antenna conversion efficiency P A (λ P , λ S ) is an antenna dependent factor incorporating the integrated field enhancement profile near the metal surface and the molecular density and Raman cross-section.The total transmission loss induced by one antenna at wavelength λ is given by 1 − e −1 λ , whereby e λ is the linear antenna extinction. Apart from the intrinsic waveguide losses α wg , the pump and Stokes light will hence also be attenuated by e P and e S respectively. In the Supplementary Information S3 it is then shown that the total Stokes power P tot generated by an array of N coated antennas is approximately given byThe quantity FOM(N, λ P , λ S ) contains all necessary parameters to assess the SERS signal strength for a given waveguide geometry and is hence considered to be the relevant figure of merit (FOM) in comparing the performance of integrated antenna arrays. The optimum antenna number N opt = 4 log {log (e S ) / log (e P )} / log (e S /e P ). For the particular bowtie antenna studied inFigure 3, the extinction spectrum e(λ ) and single antenna conversion efficiency P A (λ P , λ S ) are numerically evaluated using Lumerical FDTD Solutions (see Methods section). The predicted N opt for the 1,340 cm −1 peak is 10 antennas, using the simulated extinctions e P = 1.14 (E P = 0.58 dB) and e S = 1.08 (E S = 0.34 dB), while P A ≈ 2.35 × 10 −15 . For each 1W of pump power the antenna is therefore expected to generate 2.35 fW of guided Stokes power. The Raman enhancement factor is calcu-the local field enhancement (see Methods section). In the center of the gap at 5 nm from the tip of the antenna (marked by the black dot in Figure 2(a)) an EF R ≈ 1.42 × 10 4 is expected . Apart from the relevant NTP signal, the SiN itself generates a considerable background while the pump beam is propagating along the waveguide.This background signal P bg can be approximated byin which P B (λ P , λ S ) is a waveguide dependent factor incorporating the specific modal field profile and the SiN molecular density and cross-section (seeSupplementary Information S4).Our analytical model and the associated numerical calculations will now be benchmarked against the spectra fromFigure3 to verify whether the theoretically estimated power values correspond to the experimentally obtained absolute Raman powers. To this end, the NTP signal strength at 1,340 cm −1 , obtained from Figure 3(b), is analyzed as a function of the antenna number N and compared with the theoretical estimations. Furthermore, the background associated shot noise is also calculated. The results are depicted in Figure 4(b). While the ideal model assumes N identical antennas, the fabricated antennas will show differences among each other resulting in changes of e P , e S and P A from one antenna to the other. A generalized model incorporating potential differences in e P , e S and P A is described in the Supplementary Information S5. In order to estimate the uncertainty on these experimental parameters a randomized fit to the generalized model has been applied. A set of normally distributed numbers is generated for each of the three 5 parameters and then plugged into the generalized model to calculate the distribution of signal and shot noise counts, defining an area within which the probable signal (blue area) and noise (red area) counts are situated. The mean values of these distributions are extracted from an initial constrained fit to the ideal model (dotted lines), while its standard deviations are chosen such that the corresponding signal and shot noise distributions cover all experimental datapoints (red and blue dots). Based on the randomized fit it is possible to estimate the spread on the experimental parameters: E P ≈ 0.49 ± 0.11 dB, E S ≈ 0.35 ± 0.11 dB and P A = (2.60 ± 0.77) × 10 −15 (theoretically E P = 0.58 dB, E S = 0.34 dB and P A = 2.35 × 10 −15 were predicted). The theoretically predicted parameters are all within the error bars of the experimentally fitted data, which clearly establishes the validity of our model and its ability to provide quantitative predictions of the absolute Raman power coupled into a single mode waveguide. Given this excellent correspondence, we expect the fabricated structures to have a Raman enhancement factor EF R on the order of 1.42 × 10 4 near the two antenna gap tips. Decreasing the gap size should boost EF R and P A by another two or three orders of magnitude. From the fitting values the optimum antenna number is estimated to be 11 ± 3 (compared to 10 theoretically). The single antenna conversion efficiency P A shows that the fabricated antennas produce (2.60 ± 0.77) fW of guided Stokes power for each 1W of guided pump power. Compared to the free space Raman scattering P 0 of a single NTP molecule in a bulk air environment P A ≈ 3.94 × 10 6 P 0 . This includes the excitation and emission enhancement of all molecules in the monolayer as well as the coupling efficiency to the guided mode. Since only halfof the Stokes light is carried by the forward propagating mode, the total power coupled into the fundamental TE-mode is therefore ≈ 8 × 10 6 P 0 .Our observations also reveal that a minimum number of antennas N min is required to generate a detectable signal (marked by the white square inFigure 4(b)). If N < N min then the shot noise still dominates on the signal. It has to be noted however that the relevant signal is generated in a very small region (N − 1)Λ compared to the overall length 2L 1 + (N − 1)Λ ≈ 2L 1 , while the shot noise is mainly attributed to this non-useful length 2L 1 . Chip designs which allow a separation of the signal from the background are expected to have N min = 1 such that signals originating from one 6 single antenna can still be detected. As a result it would become possible to simultaneously probe large areas of analytes (> λ 2 ) and detecting all SERS events, originating from different locations, by monitoring just a single waveguide output in contrast to microscopy based systems where one has to serially scan all hotspot locations.The work presented here paves the way towards the efficient design of evanescently coupled nanoantennas for on-chip excitation and emission enhancement in the 700-1000 nm region. Due to the low fluorescence, negligible water absorption and the availability of high quality and low-cost sources and detectors this region is of particular interest for Raman sensing. 28 In combination with other on-chip spectral functionalities, such as arrayed waveguide gratings, 28 the presented platform is forecasted to allow multiplexed detection of extremely weak Raman signals on a highly dense integrated platform. We also envisage that integrated nanoantennas, similar to the ones reported here, could be used as transducer between quantum dot emitters and the fundamental waveguide mode, potentially enabling applications in on-chip quantum communication and quantum computation. 29,30 Methods Fabrication details The fabrication consists of a 2-step e-beam lithography process. In the first step the nanoplasmonic antennas are patterned on top of a slab Si/SiO 2 /SiN wafer using a positive PMMA e-beam resist. After PMMA exposure, the samples are developed in a 1:1 MIBK:IPA solution after which a 2 nm Ti adhesion layer and 30 nm Au layer are deposited in a commercial Pfeiffer Spider sputter system. The samples are then immersed in acetone for lift-off. In the second step the waveguides are defined. After metal lift-off a negative ma-N 2403 resist is spun, exposed and developed in ma-D 525. An e-spacer is also spun on top of the ma-N 2403 to avoid charging effects. The developed samples are then etched with an ICP plasma (C 4 F 8 /SF 6 mixture) in a commercial Oxford Plasmalab system. After resist strip and cleaning, the samples are immersed overnight in a 1 mM NTP:EtOH 7solution and subsequently rinsed with pure ethanol to remove the residual NTP. A self-assembled monolayer of NTP is assumed to form on the Au surface through a Au-S bond. 23Numerical SimulationsNumerical simulations were performed with Lumerical FDTD Solutions. We used a refractive index of n rib = 1.9 for the SiN rib (with width w rib = 700 nm and height h rib = 220 nm), n uclad = 1.45 for the SiO 2 undercladding and n tclad = 1 for the top cladding (air). The Si substrate was not taken into account since the real oxide cladding is thick enough to avoid substantial power leakage to the Si such that the numerical results faithfully represent the actual experimental conditions. A thin native oxide layer (t nox = 2 nm) between the SiN and the Ti has also been incorporated. 19The metal stack thicknesses are fixed to t Ti = 2 nm and t Au = 30 nm and a built-in refractive index model for Au (Johnson and Christy 25 ) and Ti (CRC 26 ) is used. An additional surface layer with thickness t NT P = 1 nm and index n NT P = 3 is used to model the NTP monolayer. The antenna region (including the Ti adhesion layer and the NTP monolayer) is meshed with a uniform mesh of 0.5 nm in the plane of the antenna (yz-plane) and 2 nm in the x-direction. A mesh refinement to 1 nm is applied in regions where the thicknesses in the x-direction are ≤ 2 nm. The estimated surface area of an NTP molecule is 0.18 nm 2 , so the surface density is then ρ s = 5.56 × 10 18 molecules/m 2 . 23 The Raman cross section is σ ≈ 0.358 × 10 −30 cm 2 /sr, which was obtained by applying the λ −4 S scaling to the original data of the 1,340 cm −1 line. 27 Single antenna extinction spectra E(λ ) (in dB) are calculated through E(λ ) = T re f (λ ) − T ant (λ ) in which T re f (λ ) is the power transmission (in dB) through the reference waveguide and T ant (λ ) the power transmission (in dB) of a waveguide functionalized with one antenna. Linear extinction spectra e(λ ) ∆ = e λ are then given by e(λ ) = 10 E(λ )/10 . A field and index profile monitor are used to extract the local field |E(r, λ )| and index n(r) around the antenna. The single antenna conversion efficiency P A (λ P , λ S ) = ρ s σ 2t NT P V m n g (λ P )n g (λ S )λ 2 S |E(r, λ P )| 2 |E(r, λ S )| 2 dr n(r) 2 |E m (r, λ P )| 2 dr n(r) 2 |E m (r, λ S )| 2 dr 8 is calculated by integrating the local fields over the effective monolayer volume V m in which the index satisfies n(r) | r∈V m = n NT P (seeSupplementary Information S3). The group index of the waveguide mode is n g (λ ). The denominator is calculated using the modal fields E m (r, λ ) of a non-functionalized reference waveguide and the local field enhancement is given by the ratio of the local and modal electric fields: β (r, λ ) = |E(r,λ )| |E m (r,λ )| . At a certain position, the Raman enhancement factor EF R is calculated as EF R = β (λ P ) 2 β (λ S ) 2 . Numerically calculated values are mentioned in the main text.9
Surface Enhanced Raman Spectroscopy (SERS) is a well-established technique for en-
hancing Raman signals. [1][2][3][4][5][6][7][8][9][10][11][12] Recently photonic integrated circuits have been used, as an alternative to microscopy based excitation and collection, to probe SERS signals from external metallic nanoparticles. [13][14][15] However, in order to develop quantitative on-chip SERS sensors, integration of dedicated nanoplasmonic antennas and waveguides [16][17][18][19][20][21][22] is desirable.
Here we bridge this gap by demonstrating for the first time the generation of SERS signals from integrated bowtie nanoantennas, excited and collected by a single mode waveguide, and rigorously quantify the enhancement process. The guided Raman power generated by a 4-Nitrothiophenol coated bowtie antenna shows an 8 × 10 6 enhancement compared to the free-space Raman scattering. An excellent correspondence is obtained between the theoretically predicted and observed absolute Raman power. This work paves the way towards fully integrated lab-on-a-chip systems where the single mode SERS-probe can be combined with other photonic, fluidic or biological functionalities.
A schematic of the device under study is shown in Figure 1 Fabrication details can be found in the Methods section and a description of the measurement setup is outlined in the Supplementary Information S1. A scanning electron microscope image of the functionalized waveguide, with cross-sectional area of 220 × 700 nm 2 , is depicted in Figure 1(b). Raman spectra of an uncoated and coated waveguide functionalized with 40 antennas are shown in Figure 1(c). The spectral regions where an NTP Stokes peak is expected (1,080, 1,110, 1,340 and 1,575 cm −1 ) 23 are highlighted by the cyan shaded areas. Before coating no NTP peaks can be distinguished from the inherent SiN background. The peaks at 1,250 and 1,518 cm −1 (marked by the black dashed lines) are attributed to interference effects of the Au array which act on the scattered background light, and they are also observed on the extinction curves of the functionalized waveguides (see Supporting Information S2). Hence they do not represent specific Raman lines. After coating, four additional peaks appear and coincide with the expected NTP Stokes peaks. This demonstrates that SERS signals from single monolayer coated antennas can be efficiently excited and collected by the same fundamental waveguide mode.
Subsequently the dependence of the SERS signal on the position of the plasmon resonance was investigated to verify that it can be attributed to a resonance effect and not to coincidental surface roughness. To this end, waveguides functionalized with a fixed number of antennas but varying bowtie geometries were considered. The relevant bowtie parameters are its length L, gap ∆ and apex angle α (Figure 2(a)). By changing the length, the antenna resonance can be tuned (L 1 = 90 nm, L 2 = 115 nm and L 3 = 140 nm for fixed α = 60 • and ∆ = 40 nm). Extinction spectra are plotted in Figure 2(b) while the corresponding Raman spectra are depicted in Figure 2(c). The Raman spectrum of a reference waveguide without any Au functionalization is also shown. Even after coating the reference waveguide does not generate NTP peaks, so any Raman signal indeed originates from the antenna region and does not contain contributions from spontaneous Raman scattering along the waveguide. 24 The L 1 resonance is detuned from the relevant pump and Stokes region, resulting in a poor Raman spectrum. By increasing the length (L 2 and L 3 bowties) the resonance redshifts and lines up with the pump and Stokes wavelengths. For these bowties the NTP spectrum starts to emerge. The reported SERS spectra can hence be attributed to a plasmon resonance effect such that a stable and reproducible enhancement factor can be associated with them, in contrast to SERS events originating from random surface defects. The increased overlap with the plasmon resonance, and hence extinction, also results in a decreased background.
Due to the metal induced loss, there will exist an optimum number N opt of patterned antennas such that the SERS signal is maximized. Such an optimum is investigated in Figure 3 for a fixed bowtie geometry (α = 60 • , L = 100 nm and ∆ = 40 nm) but varying N: N = 10, 20, 30, 40, 70 and N = 0 which is a reference waveguide. Each waveguide is measured 10 times and the averaged Raman spectra are reported in Figure 3(a). The N = 70 signal is not shown since it could not be distinguished from the inherent offset signal of the detector. For a given fixed input power, corresponding to roughly 5 mW guided power, the reference waveguide generates a considerable background signal in the 1,340 cm −1 region, where the strongest NTP peak is expected. Functionalizing the waveguide with increasing N reduces this unwanted background due to the attenuation caused by the nanoantennas. In addition the 1,340 cm −1 peak starts to emerge when N increases.
The smaller peaks at 1,080, 1,110 and 1,575 cm −1 only appear when the background is sufficiently low. A zoom on the dominant 1,340 cm −1 peak (cyan dashed line) is shown in Figure 3(b). For clarity the background is locally subtracted. As expected, the signal reaches a maximum value for 10 ≤ N ≤ 20 and then decays again with increasing N. Apart from signal optimization it is however equally important to reduce the SiN background in order to resolve the smallest spectral features. Therefore an analytical model is developed to outline the interplay between signal enhancement and background reduction, and to derive the relevant figure of merit for on-chip SERS.
Figure 4(a) shows a schematic longitudinal cross-section of the chip consisting of N antennas spaced with period Λ = 10 µm. Each array is centered on the waveguide with a distance L 1 ≈ 0.5 cm to the front and back facet of the chip. The NTP monolayer on each antenna will generate a forward propagating Stokes power P A (λ P , λ S ) for a given pump power P pump . This single antenna conversion efficiency P A (λ P , λ S ) is an antenna dependent factor incorporating the integrated field enhancement profile near the metal surface and the molecular density and Raman cross-section.
The total transmission loss induced by one antenna at wavelength λ is given by 1 − e −1 λ , whereby e λ is the linear antenna extinction. Apart from the intrinsic waveguide losses α wg , the pump and Stokes light will hence also be attenuated by e P and e S respectively. In the Supplementary Information S3 it is then shown that the total Stokes power P tot generated by an array of N coated antennas is approximately given by
P tot P pump ≈ P A (λ P , λ S )e −2α wg L 1 e (1−N) S 1 − e S e P N 1 − e S e P = FOM(N, λ P , λ S )e −2α wg L 1 .
The quantity FOM(N, λ P , λ S ) contains all necessary parameters to assess the SERS signal strength for a given waveguide geometry and is hence considered to be the relevant figure of merit (FOM) in comparing the performance of integrated antenna arrays. The optimum antenna number N opt = log {log (e S ) / log (e P )} / log (e S /e P ). For the particular bowtie antenna studied in Figure 3, the extinction spectrum e(λ ) and single antenna conversion efficiency P A (λ P , λ S ) are numerically evaluated using Lumerical FDTD Solutions (see Methods section). The predicted N opt for the 1,340 cm −1 peak is 10 antennas, using the simulated extinctions e P = 1.14 (E P = 0.58 dB) and e S = 1.08 (E S = 0.34 dB), while P A ≈ 2.35 × 10 −15 . For each 1W of pump power the antenna is therefore expected to generate 2.35 fW of guided Stokes power. The Raman enhancement factor is calcu- This background signal P bg can be approximated by
lated through EF R = β (λ P ) 2 β (λ S ) 2 in which β (λ ) isP bg P pump ≈ P B (λ P , λ S )e −2α wg L 1 e −N P + e −N S
in which P B (λ P , λ S ) is a waveguide dependent factor incorporating the specific modal field profile and the SiN molecular density and cross-section (see Supplementary Information S4).
Our analytical model and the associated numerical calculations will now be benchmarked against the spectra from Figure Our observations also reveal that a minimum number of antennas N min is required to generate a detectable signal (marked by the white square in Figure 4(b)). If N < N min then the shot noise still dominates on the signal. It has to be noted however that the relevant signal is generated in a very small region (N − 1)Λ compared to the overall length 2L 1 + (N − 1)Λ ≈ 2L 1 , while the shot noise is mainly attributed to this non-useful length 2L 1 . Chip designs which allow a separation of the signal from the background are expected to have N min = 1 such that signals originating from one single antenna can still be detected. As a result it would become possible to simultaneously probe large areas of analytes (> λ 2 ) and detecting all SERS events, originating from different locations, by monitoring just a single waveguide output in contrast to microscopy based systems where one has to serially scan all hotspot locations.
The work presented here paves the way towards the efficient design of evanescently coupled nanoantennas for on-chip excitation and emission enhancement in the 700-1000 nm region. Due to the low fluorescence, negligible water absorption and the availability of high quality and low-cost sources and detectors this region is of particular interest for Raman sensing. 28 In combination with other on-chip spectral functionalities, such as arrayed waveguide gratings, 28 the presented platform is forecasted to allow multiplexed detection of extremely weak Raman signals on a highly dense integrated platform. We also envisage that integrated nanoantennas, similar to the ones reported here, could be used as transducer between quantum dot emitters and the fundamental waveguide
Numerical Simulations
Numerical simulations were performed with Lumerical FDTD Solutions. We used a refractive index of n rib = 1.9 for the SiN rib (with width w rib = 700 nm and height h rib = 220 nm), n uclad = 1.45 for the SiO 2 undercladding and n tclad = 1 for the top cladding (air). The Si substrate was not taken into account since the real oxide cladding is thick enough to avoid substantial power leakage to the Si such that the numerical results faithfully represent the actual experimental conditions. A thin native oxide layer (t nox = 2 nm) between the SiN and the Ti has also been incorporated. 19
The metal stack thicknesses are fixed to t Ti = 2 nm and t Au = 30 nm and a built-in refractive index model for Au (Johnson and Christy 25 ) and Ti (CRC 26 ) is used. An additional surface layer with thickness t NT P = 1 nm and index n NT P = 3 is used to model the NTP monolayer. The antenna region (including the Ti adhesion layer and the NTP monolayer) is meshed with a uniform mesh of 0.5 nm in the plane of the antenna (yz-plane) and 2 nm in the x-direction. A mesh refinement to 1 nm is applied in regions where the thicknesses in the x-direction are ≤ 2 nm. The estimated surface area of an NTP molecule is 0.18 nm 2 , so the surface density is then ρ s = 5.56 × 10 18 molecules/m 2 . 23 The Raman cross section is σ ≈ 0.358 × 10 −30 cm 2 /sr, which was obtained by applying the λ −4 S scaling to the original data of the 1,340 cm −1 line. 27 Single antenna extinction
spectra E(λ ) (in dB) are calculated through E(λ ) = T re f (λ ) − T ant (λ ) in which T re f (λ ) isP A (λ P , λ S ) = ρ s σ 2t NT P V m n g (λ P )n g (λ S )λ 2 S |E(r, λ P )| 2 |E(r, λ S )| 2 dr n(r) 2 |E m (r, λ P )| 2 dr n(r) 2 |E m (r, λ S )| 2 dr
is calculated by integrating the local fields over the effective monolayer volume V m in which the index satisfies n(r) | r∈V m = n NT P (see Supplementary Information S3). The group index of the waveguide mode is n g (λ ). The denominator is calculated using the modal fields E m (r, λ ) of a non-functionalized reference waveguide and the local field enhancement is given by the ratio of the local and modal electric fields:
β (r, λ ) = |E(r,λ )| |E m (r,λ )| .
At a certain position, the Raman enhancement factor EF R is calculated as EF R = β (λ P ) 2 β (λ S ) 2 . Numerically calculated values are mentioned in the main text. Longitudinal cross-section of the chip. The pump beam, with power P pump at wavelength λ P , excites a Stokes signal λ S of which the total power at the output facet is P tot . The plasmonic array consists of N antennas with period Λ and is separated L 1 from both the input and output facet of the chip. Each of the antennas generates P A guided Stokes power for a given pump power. Apart from the waveguide losses α wg , the pump and Stokes light is attenuated by the pump e P and Stokes e S extinction respectively. (b) Signal (blue circles) and shot noise (red circles) at the 1,340 cm −1 peak. The dotted lines represent a constrained fit to the ideal model while the shaded areas represent a randomized fit to estimate the uncertainty on the experimental parameters. The white square denotes the minimum number of antennas N min required to generate a detectable signal.
1 Figure 4 Supporting Information S1: SERS measurement setup SERS spectra are measured with the setup depicted in Figure S1. A tunable Ti:saph laser is set to a pump wavelength of λ P = 785 nm (red) after which the polarized beam passes through a half-wave plate (λ /2) in order to rotate the polarization to a TE-polarized beam. A beamsplitter BS1 then splits the beam into two parts (solid and dashed red line). The solid path is used to generate the forward propagating Raman beam and passes through a laser line filter (LLF) at λ P for side-band suppression before it is coupled into the chip by an aspheric lens (ASPH). The output beam is then collected with an objective (OBJ) and passes through a polarizer P (set to TE) before it is filtered by a dichroic mirror which reflects the pump beam and transmits all Stokes wavelengths (green).
The Stokes light is collected into a fiber using a parabolic mirror collimator (PMC) after which the fiber is split by a fiber splitter (FS) of which 1% goes to a power meter (PM) and 99% to a commercial spectrometer from ANDOR (Shamrock 303i spectrometer and iDus 416 cooled CCD camera). Mirror M2 blocks the second (dashed) path during the measurement but can be removed for alignment purposes. The camera (CAM) and the 1% fiber tap are used during alignment and to measure the transmitted power P T . In order to align the sample we initially set the wavelength of the Ti:saph to λ T = 800 nm (such that it can be transmitted through DM1 and collected in the power meter) and maximize the transmitted power P T in the forward path. After optimizing the transmission the laser is tuned back to λ P = 785 nm.
S3: Derivation of the analytical on-chip SERS model
The power P wg (r 0 , λ ) coupled into the forward propagating waveguide mode E m (r, λ ) as a result of a radiating dipole at position r 0 is given by
P wg (r 0 , λ ) P 0 = 3 8π n g (λ ) n m λ n 2 ε o ε(r 0 ) |e d · E m (r 0 , λ )| 2 A wg ε o ε(r) |E m (r, λ )| 2 dr | z=z 0 (1)
where P 0 = ω 4 |d 0 | 2 12πε 0 c 3 is the power radiated in free space at a wavelength λ = 2πc ω , ε(r) the relative permittivity at position r, n g (λ ) the group index of the waveguide mode, n m the refractive index of the medium in which the dipole is placed and e d the unit vector along the dipole moment vector d 0 . 31 The integral in the denominator is calculated over a waveguide cross-section A wg in the xyplane and evaluated at the dipole position z = z 0 (coordinate axes are defined in Figure S4(a) and S4(b)).
Functionalizing the waveguide with an antenna however introduces a perturbation to the modal field E m (r, λ ) of the waveguide. In order to investigate whether formula (1) is still valid when the waveguide is functionalized with a metallic antenna, two sets of simulations were performed . In the first set ( Figure S4 Figure S4(a)). In the second set ( Figure S4(b)) the fundamental TE-mode is launched into the waveguide functionalized with the same antenna in order to calculate the fields at positions r 0 where the dipole sources in the first simulation set were located. In this way it is possible to calculate the right-hand side of equation (1). The integral in the denominator is approximated in all calculations by evaluating it on a reference waveguide without antenna, so using the true modal fields and not the perturbed fields. This approximation allows a sufficient accurate evaluation of the right-hand side of equation (1) as is confirmed in Figure S4(c) where the simulation results are depicted. The red curve represents an incoherent superposition of the power coupled into the TE-mode as a result of the complete set of dipole emitters. The dashed blue curve is the predicted power that couples into the TE-mode, and is calculated as an incoherent superposition of the predicted coupled powers for each dipole. It is clear that the predicted power coupled into the TE-mode (simulation set 2) matches very well with the explicit calculation using dipole emitters (simulation set 1).
|d 0 | 2 = α 2 m |E m d (r 0 , λ )| 2 n g (λ )P pump cε o ε(r) |E m (r, λ )| 2 dr | z=z input
in which α m is the molecular polarizability and E m d (r 0 , λ ) = e d · E m (r 0 , λ ) the field strength at the dipole position. The integral is evaluated at the input z input = 0 of the waveguide. 24 The quantity
η ( r 0 , λ P , λ S ) = n g (λ P )n g (λ S )λ 2 S n m |E m d (r 0 , λ P )| 2 |E m d (r 0 , λ S )| 2 ε(r) |E m (r, λ P )| 2 dr ε(r) |E m (r, λ S )| 2 dr .(2)
solely depends on the modal fields and can be evaluated on a reference waveguide. Both integrals in the denominator are evaluated along the same cross-sectional area of the reference waveguide (in all simulations n m = 1). The power P n wg coupled into the fundamental TE-mode due to a collection of incoherently radiating dipoles around antenna n (n = 1 . . . N) is then approximately given by
P n wg P pump ≈ ρσ (λ S ) 2 e −α(λ P )(L 1 +(n−1)Λ) e(λ P ) 1−n V m η ( r 0 , λ P , λ S )β (r 0 , λ P ) 2 β (r 0 , λ S ) 2 dr 0 (3)
in which ρ is the molecular density, σ (λ S ) = π 2 α 2 m /(ε 2 0 λ 4 S ) the Raman cross section, 32 α(λ P ) = α P the waveguide loss at the pump wavelength, e(λ P ) = e P the linear antenna extinction at the pump wavelength, V m the volume in which the molecules are situated and β (r 0 , λ P/S ) = β P/S (r 0 ) the field enhancement factor, at the dipole position r 0 , at the pump (P) and Stokes (S) wavelength respectively:
β (r 0 , λ ) = E ant d (r 0 , λ ) |E m d (r 0 , λ )| whereby E ant d (r 0 , λ )
is the local field around the antenna surface. For notational simplicity we rewrite formula (3) as P n wg P pump ≈ P A (λ P , λ S )e −α P (L 1 +(n−1)Λ) e 1−n P in which Since N antennas contribute incoherently to the signal, the total amount of Stokes light at the output of the waveguide is given by
P A (λ P , λ S ) = ρσ (λ S ) 2 V m η ( r 0 , λ P , λ S )β (r 0 , λ P ) 2 β (r 0 , λ S ) 2 drP tot wg,out P pump ≈ P A (λ P , λ S ) e −α P (L 1 −Λ)−α S (L 1 +NΛ) e P e −N S N ∑ n=1 e S e P e (α S −α P )Λ n .
For the considered waveguide platform it is reasonable to approximate α P ≈ α S = α wg such that the forward propagating Raman power is given by For the specific case of a monolayer, one has a molecular surface density ρ s (number of molecules per m 2 ) rather than a volume density ρ (number of molecules per m 3 ). In this limiting case the single antenna conversion efficiency is theoretically defined by
P A (λ P , λ S ) = ρ s σ (λ S ) 2 A m η ( r 0 , λ P , λ S )β (r 0 , λ P ) 2 β (r 0 , λ S ) 2 dr 0
where the integration is now performed over the surface area A m covered by the monolayer. Numerically one however always needs to model the presence of such a monolayer by introducing a finite thickness t m . As a result an effective monolayer volume V m ≈ t m × A m is defined. For any numerical evaluation the fields are hence integrated over this volume V m such that
V m η ( r 0 , λ P , λ S )β (r 0 , λ P ) 2 β (r 0 , λ S ) 2 dr 0 ≈ t m × A m η ( r 0 , λ P , λ S )β (r 0 , λ P ) 2 β (r 0 , λ S ) 2 dr 0
The single antenna conversion efficiency is then given by
P A (λ P , λ S ) ≈ ρ s σ (λ S ) 2t m V m η ( r 0 , λ P , λ S )β (r 0 , λ P ) 2 β (r 0 , λ S ) 2 dr 0
From a numerical point of view, an effective volume density ρ e f f = ρ s /t m is introduced which is then multiplied with the integral over the effective monolayer volume V m .
In order to correlate the Stokes power at the output facet of the chip and the actual detected signal counts C S , normalized with the integration time T , we need to take into account the characteristics of all optics in our setup. The Stokes power reaching the detector surface is given by
P S = γ out × T out × P tot = γ out × T out × FOMe −α wg L P pump = γ out × T out × FOMe −α wg L γ in P in = FOM γ in γ out P in e −α wg L × T out
in which γ in and γ out are the coupling efficiencies in and out of the chip respectively, P in is the power just before the input facet of the chip and T out is the transmission through all optics between the output of the chip and the detector surface. The quantity γ in γ out P in e −α wg L can be written as γ in γ out P in e −α wg L = P T /T PM in which P T is the transmitted power as measured by the power meter (see Supplementary Information S1) and T PM is the optical transmission between the output of the chip and the input of the powermeter. This transmission is related to T out by T out ≈ 100 × T PM × T spec in which T spec is the transmission between the input slit of the spectrometer and the detector surface. The factor 100 stems from the fact that the fiber splitter only transmits 1% of the power to the power meter. So eventually P S can be written as P S = FOM × P T × 100T spec . By using the spectrometer sensitivity χ (defined as the number of electrons per count) and quantum efficiency QE(λ S ) (number of electrons per number of incident photons) P S can be related to the measured signal counts C S per unit integration time T by
P S = C S T hc λ S χ QE(λ S ) .
Finally one gets
C S T = FOM × 100T spec λ S hc QE(λ S ) χ P T = FOM ×C * (λ S ).
The transmitted power 100P T is about 1mW for the given measurement setup. If we consider the 1340 cm −1 line then QE(λ S ) χ ≈ 1 counts/photon and T spec ≈ 0.44 (using data supplied by the manufacturer) such that C * ≈ 1.94 × 10 15 counts/sec. C * (λ S ) can now be used as a conversion factor when the experimental signal counts are fitted to the analytical model (and hence the figure of merit). In this way one can rigorously quantify the parameters e P , e S and P A and compare them with the theoretical predictions.
S4: Derivation of the background signal
Similar reasonings as in the Supplementary Information S3 can be applied to derive the total background signal generated along the waveguide. This background mostly stems from the SiN core and is hence generated when the pump light propagates along the total length L of the waveguide.
For a functionalized waveguide the attenuation of the pump and Stokes light due to the antenna array also needs to be incorporated. The dipoles which give rise to the background are mainly situated in the core, so for a given core cross-sectional area A core equation (2) has to be integrated over the complete waveguide core. This quantity is defined as
η core (λ P , λ S ) = η c = A core η ( r 0 , λ P , λ S )dr 0 .
For a given molecular core density ρ c and core scattering cross section σ c , the total background P bg propagating in the forward direction is then calculated through After some calculation and using α P ≈ α S = α wg and L ≈ L 1 /2 (2L 1 >> (N − 1)Λ), the total forward propagating background is given by
P bg P pump = ρ c σ c η c 2 e −α wg L (e −N P + e −N S ) L 2 + Λ e 1−N S e P 1 − e S e P N−1 1 − e S e P ≈ ρ c σ c η c 2 e −α wg L (e −N P + e −N S ) L 2 = P B (λ P , λ S )e −α wg L e −N P + e −N S ≈ P B (λ P , λ S )e −2α wg L 1 e −N P + e −N S .
S5: Randomized fit to a generalized model
The model derived in the Supplementary Information S3 and S4 assumes identical antennas. Due to fabrication errors there will always be differences among each of the antennas in the array.
These differences have an impact on both the extinction e P and e S as well as on the single antenna conversion efficiency P A (λ P , λ S ). While the experimental data described in this Letter can be fitted well to the ideal model, the fit is not perfect. In this section a randomized fit model that takes into account the potential deviations among different antennas is outlined. The power generated by antenna n is given by:
P n wg P pump ≈ P A (λ P , λ S )e −α P (L 1 +(n−1)Λ) e 1−n P .
Since L 1 >> NΛ this can be simplified to P n wg P pump ≈ P A (λ P , λ S )e −α P L 1 e 1−n P .
The formula however assumes a constant pump extinction from the previous n − 1 antennas. Furthermore the factor P A (λ P , λ S ) is also assumed constant for each antenna. If we allow that each of the antennas has a different extinction e m P and antenna dependent factor P m A (λ P , λ S ) (m = 1 . . . N), then the power generated by antenna n is given by P n wg P pump e −α P L 1 ≈ P n A (λ P , λ S )
n−1 ∏ m=1 (e m P ) −1 .
Applying a similar reasoning to the Stokes light that has to propagate along N − n other antennas (and a length L 1 ), the Stokes power (generated by antenna n) reaching the output is given by P n wg,out P pump e −α P 2L 1 ≈ P n A (λ P , λ S )
In this case P B is a constant since it only depends on the waveguide parameters. The shot noise associated to the background is then simply proportional to P bg . The number of counts/sec can be obtained by multiplying equations (4) and (5) with C * (λ S ) (see Supplementary Information S3).
Based on the conversion factor C * (λ S ) and a fit to the experimental data, one can quantitatively determine a value for the parameters in the model. For a fixed λ P and λ S , P A (λ P , λ S )C * (λ S ) will be denoted by P * A .
In order to make a randomized fit of the signal and shot noise at the same time, N random numbers (based on a normal distribution) are independently generated for the pump and Stokes extinction and for the single antenna conversion efficiency (so three random numbers are generated for each antenna in the array). Using these random numbers, formulas (4) and (5) are evaluated.
This process is repeated 1000 times in order to generate a statistically relevant distribution of the possible signal and shot noise counts. Ultimately the mean value µ and standard deviation σ from the obtained signal and shot noise distributions are extracted for each number of antennas N. The 3σ -intervals for both distributions ([µ − 3σ , µ + 3σ ]) define an area that marks the possible signal and noise counts for a given uncertainty on the antenna parameters (according to formulas (4) and
(5)). These areas are plotted in Figure 4(b) of the main text.
The mean value of each of the three normal distributions (for e P , e S and P A ) is determined by a fit of the experimental data to the ideal model. One could obtain a perfect fit to either the signal or shot noise data by fitting only one of the two equations ((4) or (5)). The fitting parameters would however not generate a perfect fit to the other non-fitted equation since both equations depend on e P and e S . Therefore a constrained fit, in which one minimizes the fitting error to both equations simultaneously, is performed such that both the signal and shot noise data are represented well using the fitted values for e P and e S . The values e P , e S and P * A (mind that P * A and not P A is used because we fit the counts/sec) obtained from the constrained fit are then chosen to be the mean values (µ e P , µ e S and µ P * A ) of their respective normal distributions. The standard deviation σ x is chosen such that the 3σ x intervals represent realistic deviations from the mean value (x denotes one the three parameters). For the extinction this means e.g. that µ e P/S − 3σ e P/S ≥ 1 (since 1 is the lower boundary for the linear extinction). Since P * A is always positive, µ P * A − 3σ P * A > 0 should also be satisfied. For the experimental data shown in Figure 3 of the main text, µ e P = 1.1182, µ e S = 1.0845 and µ P * A = 5.0394 has been derived from the constrained fit. The sigma value for the extinction is chosen to be σ = 0.028 such that E P ≈ 0.49 ± 0.11 dB and E S ≈ 0.35 ± 0.11 dB. For P * A we similarly get P * A ≈ 5.04 ± 1.5 such that the single antenna conversion efficiency for the 1340 cm −1 line is P A = (2.60 ± 0.77) × 10 −15 . All of these values satisfy the constraints mentioned above and represent realistic deviations of the three fitting parameters.
(a). The fundamental TE-mode of a silicon nitride (SiN) rib waveguide excites a periodic array of gold bowtie antennas coated with a 4-Nitrothiophenol (NTP) monolayer. The pump wavelength for all experiments is set to λ P = 785 nm and NTP Stokes light (at λ S ) is subsequently collected back into the same waveguide mode.
the local field enhancement (see Methods section). In the center of the gap at 5 nm from the tip of the antenna (marked by the black dot in Figure 2(a)) an EF R ≈ 1.42 × 10 4 is expected . Apart from the relevant NTP signal, the SiN itself generates a considerable background while the pump beam is propagating along the waveguide.
3 to verify whether the theoretically estimated power values correspond to the experimentally obtained absolute Raman powers. To this end, the NTP signal strength at 1,340 cm −1 , obtained from Figure 3(b), is analyzed as a function of the antenna number N and compared with the theoretical estimations. Furthermore, the background associated shot noise is also calculated. The results are depicted inFigure 4(b). While the ideal model assumes N identical antennas, the fabricated antennas will show differences among each other resulting in changes of e P , e S and P A from one antenna to the other. A generalized model incorporating potential differences in e P , e S and P A is described in the Supplementary Information S5. In order to estimate the uncertainty on these experimental parameters a randomized fit to the generalized model has been applied. A set of normally distributed numbers is generated for each of the three parameters and then plugged into the generalized model to calculate the distribution of signal and shot noise counts, defining an area within which the probable signal (blue area) and noise (red area) counts are situated. The mean values of these distributions are extracted from an initial constrained fit to the ideal model (dotted lines), while its standard deviations are chosen such that the corresponding signal and shot noise distributions cover all experimental datapoints (red and blue dots). Based on the randomized fit it is possible to estimate the spread on the experimental parameters: E P ≈ 0.49 ± 0.11 dB, E S ≈ 0.35 ± 0.11 dB and P A = (2.60 ± 0.77) × 10 −15 (theoretically E P = 0.58 dB, E S = 0.34 dB and P A = 2.35 × 10 −15 were predicted). The theoretically predicted parameters are all within the error bars of the experimentally fitted data, which clearly establishes the validity of our model and its ability to provide quantitative predictions of the absolute Raman power coupled into a single mode waveguide. Given this excellent correspondence, we expect the fabricated structures to have a Raman enhancement factor EF R on the order of 1.42 × 10 4 near the two antenna gap tips. Decreasing the gap size should boost EF R and P A by another two or three orders of magnitude. From the fitting values the optimum antenna number is estimated to be 11 ± 3 (compared to 10 theoretically). The single antenna conversion efficiency P A shows that the fabricated antennas produce (2.60 ± 0.77) fW of guided Stokes power for each 1W of guided pump power. Compared to the free space Raman scattering P 0 of a single NTP molecule in a bulk air environment P A ≈ 3.94 × 10 6 P 0 . This includes the excitation and emission enhancement of all molecules in the monolayer as well as the coupling efficiency to the guided mode. Since only half of the Stokes light is carried by the forward propagating mode, the total power coupled into the fundamental TE-mode is therefore ≈ 8 × 10 6 P 0 .
mode, potentially enabling applications in on-chip quantum communication and quantum computation. 29,30MethodsFabrication detailsThe fabrication consists of a 2-step e-beam lithography process. In the first step the nanoplasmonic antennas are patterned on top of a slab Si/SiO 2 /SiN wafer using a positive PMMA e-beam resist.After PMMA exposure, the samples are developed in a 1:1 MIBK:IPA solution after which a 2 nm Ti adhesion layer and 30 nm Au layer are deposited in a commercial Pfeiffer Spider sputter system. The samples are then immersed in acetone for lift-off. In the second step the waveguides are defined. After metal lift-off a negative ma-N 2403 resist is spun, exposed and developed in ma-D 525. An e-spacer is also spun on top of the ma-N 2403 to avoid charging effects. The developed samples are then etched with an ICP plasma (C 4 F 8 /SF 6 mixture) in a commercial Oxford Plasmalab system. After resist strip and cleaning, the samples are immersed overnight in a 1 mM NTP:EtOH solution and subsequently rinsed with pure ethanol to remove the residual NTP. A self-assembled monolayer of NTP is assumed to form on the Au surface through a Au-S bond.23
the power transmission (in dB) through the reference waveguide and T ant (λ ) the power transmission (in dB) of a waveguide functionalized with one antenna. Linear extinction spectra e(λ ) ∆ = e λ are then given by e(λ ) = 10 E(λ )/10 . A field and index profile monitor are used to extract the local field |E(r, λ )| and index n(r) around the antenna. The single antenna conversion efficiency
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Figure 1 : 1 Figure 1
111Evanescent excitation and collection of SERS spectra. (a) Schematic of the chip consisting of single mode SiN waveguides (blue) on an SiO 2 undercladding (gray), functionalized with an array of gold bowtie antennas (yellow). All antennas are coated with an NTP monolayer (purple dots), evanescently excited by the fundamental TE-mode (red). The NTP Stokes signal (green) is collected by the same mode. (b) Scanning electron microscope image of a functionalized waveguide. The white arrows indicate antenna positions. The inset shows a zoom of a typical antenna. (c) Raman spectra of a waveguide functionalized with 40 antennas. The cyan shaded areas mark the NTP Stokes peaks while the black dashed lines represent peaks attributed to interference effects of the plasmonic array. Before coating, the waveguide already generates a Raman background in itself (top). After coating, NTP peaks emerge (bottom).
Figure 2 : 1 Figure 2
212Signal dependence on the plasmon resonance.(a) Bowtie antenna geometrical paramters: length L, gap ∆ and apex angle α. (b) Single antenna extinction spectra for 3 different bowtie antennas with fixed apex angle α = 60 • and gap ∆ = 40 nm but varying length: L 1 = 90 nm (orange), L 2 = 115 nm (green), L 3 = 140 nm (blue). The red and cyan shaded lines correspond to the pump and Stokes wavelengths respectively. (c) Corresponding Raman spectra of the waveguides functionalized with these 3 bowtie anntenas (orange,green,blue) and Raman spectrum of the reference waveguide (red).
Figure 3 :
3Signal dependence on the number of antennas N . (a) Raman spectra of a reference waveguide (REF) and waveguides functionalized with N = 10, 20, 30, 40 antennas. (b) Zoom on the 1,340 cm −1 signal peak. The background is locally subtracted to obtain the pure NTP signal.
Figure 3
3Figure 3
Figure 4 :
4On-chip SERS model and fit to the experimental data. (a)
Figure S1 :Figure S2 :
S1S2Measurement setup. Ti:saph: tunable Ti:saphire laser emitting the pump beam at λ P = 785 nm, CAM: camera, PM: power meter, SR 303i and iDus 416: spectrometer and cooled CCD detector from ANDOR, BS1: beamsplitter, λ /2: half-wave plate, LLF: laser line filter for 785 nm, P: polarizer, M1/M3/M4: fixed mirrors, M2: removable mirror/beam block, OBJ: objective (50X, NA=0.9), ASPH: aspheric lens (NA=0.5), S: sample stage, DM1: dichroic mirror (reflection R and transmission T shown), PMC: parabolic mirror collimator (EFL=15 mm, NA=0.2), FS: fiber splitter.S2: Extinction measurementSingle antenna extinction spectra, resulting from the plasmon resonance, are measured with the setup depicted inFigure S2. Light from an NKT EXR-4 supercontinuum source (SC) is filtered through a near-IR filter (NIRF) to filter out the relevant wavelength region. Subsequently it is coupled in a fiber with a fiber coupling unit (FC). This fiber is plugged into a fiberbench consisting of 3 parts (fixed to the same bench): an achromatic fiber collimator (C) which converts the fiberized light to a free-space beam, a free-space broadband polarizer (P) which polarizes the unpolarized light into a TE-beam and an aspheric lens (ASPH) used to focus the free-space beam on the input facet of the chip. This fiberbench (marked by the light-blue area) is mounted on a piezo-controlled stage (XYZ) in order to precisely couple the supercontinuum light into the chip. At the output facet a lensed fiber (LF) is used to capture the transmitted light. This lensed fiber is also connected to a piezo-controller for accurate positioning. Finally the light is coupled to an optical spectrum analyzer (OSA) and the spectra are read out by Python controlled software (PC). Resonance measurement. SC: supercontinuum source, NIRF: near-IR filter, FC: fiber coupling unit, C: achromatic fiber collimator, P: free-space broadband polarizer, ASPH: aspheric lens (NA=0.68), LF: lensed fiber, XYZ: piezo controller stage, S: sample stage, OSA: Optical Spectrum Analyzer, PC: OSA control using Python based measurement framework.The single antenna extinction curves E(λ ) (in dB) can then be calculated through E(λ ) = (T re f − T Nant )/N in which T re f is the power transmission (in dB) through the reference waveguide and T Nant the power transmission (in dB) of a waveguide functionalized with N antennas.16 InFigure S3the single antenna extinction curve of a waveguide functionalized with N = 40 bowtie antennas (α ≈ 60 • , H ≈ 100 nm and ∆ ≈ 40 nm) is shown. The extinction exhibits periodic fringes on the broad envelope which are attributed to interference effects of the plasmonic array. On one hand, the array forms a multiple Fabry-Perot resonator of which the expected free spectral range FSR of 19.9 nm around 1,250 cm −1 (using the fabricated array period of Λ = 10µm) matches well with the experimentally obtained value (≈ 20.8 nm). On the other hand, the waveguide mode interferes with the radiative decay of the plasmon mode. This will affect the specific lineshape and strength of the fringes. The spectral positions at which we see sudden changes in the Raman background (see main text) coincide with the fringes observed on the extinction curves. Hence these features are not attributed to specific Raman lines.
Figure S3 :
S3Single antenna extinction spectrum. Single antenna extinction spectrum of the waveguide functionalized with N = 40 bowtie antennas (α ≈ 60 • , H ≈ 100 nm and ∆ ≈ 40 nm).
(a)) the coupling of dipole radiation into the fundamental TE-mode is investigated. This allows an explicit evaluation of the left-hand side of equation (1). Dipole sources with fixed dipole moment vector d 0 were placed at different positions r 0 around the antenna (shown as black dots in
Figure S4 :
S4Dipole radiation. (a) Simulation set 1: collection of incoherent dipoles emitting into the fundamental TE-mode (red). (b) Simulation set 2: Calculation of the electric fields at the dipole positions of simulation set 1 using a fundamental TE-mode excitation. Based on these fields one can extract the predicted dipole emission (dashed blue). (c) Simulation results: incoherent superposition of the TE-coupled power due to all dipoles (red) and predicted TE-coupled power using the electric field values (dashed blue). Since the right-hand side of equation (1) faithfully represents the power coupled into the fundamental TE-mode, it is used to derive an analytical model predicting the total Stokes power coupled into the waveguide as a result of an array of N antennas on top of the waveguide (see Figure 4(a) in the main text). First of all we write the dipole strength |d 0 | 2 as a function of the guided pump power P pump in the waveguide
=
antenna dependent factor which incorporates the specific field enhancement profile near the antenna surface. Formula (3) takes into account the waveguide loss and the loss induced by the antennas in front of the n th antenna as a result of which the actual excitation power decays. The signal (at Stokes wavelength λ S ) still has to pass N − n antennas and propagate along a distance L 1 + (N − n)Λ such that the power P n wg,out reaching the output of the waveguide ≈ P A (λ P , λ S ) e −α P (L 1 +(n−1)Λ) P A (λ P , λ S ) e −α P (L 1 −Λ)−α S (L 1 +NΛ)
≈≈
P A (λ P , λ S ) e −α wg (2L 1 +(N−1)Λ) P A (λ P , λ S ) e −2α wg L 1 e FOM(N, λ P , λ S )e −2α wg L 1 , where L = 2L 1 + (N − 1)Λ ≈ 2L 1 has been used. This result applies to Stokes light co-propagating with the pump beam. For a fixed antenna geometry and λ P and λ S , the optimum number of antennas N opt that should be patterned on a waveguide to maximize the SERS signal is given by
AcknowledgementThe authors acknowledge Josine Loo (imec) for performing the e-beam lithography, Liesbet Van Landschoot for making the SEM images, Ananth Subramanian for useful discussions and Stephane Clemmen for providing feedback on the manuscript. This research was funded by the ERC grant InSpectra. F.P acknowledges support from the Bijzonder Onderzoeksfonds (BOF) fellowship of Ghent University.Author ContributionsCompeting financial interestsThe authors declare no competing financial interests.
Quantum plasmonics. M S Tame, Nature Phys. 9Tame, M.S., et al. Quantum plasmonics. Nature Phys. 9, 329-340 (2013).
Broadband enhancement of light emission in silicon slot waveguides. Y Chul Jun, R M Briggs, H A Atwater, M L Brongersma, Opt. Express. 179Chul Jun, Y., Briggs, R.M., Atwater, H.A. & Brongersma, M.L. Broadband enhancement of light emission in silicon slot waveguides. Opt. Express 17 (9), 7479-7490 (2009).
The Raman Effect: A Unified Treatment Of The Theory Of Raman Scattering By Molecules pp. D A Long, John Wiley & Sons LtdChichester, EnglandLong, D.A. The Raman Effect: A Unified Treatment Of The Theory Of Raman Scattering By Molecules pp 96-97 (John Wiley & Sons Ltd, Chichester, England, 2002).
| {'fraction_non_alphanumeric': 0.04900244484017989, 'fraction_numerical': 0.028236394917147082, 'mean_word_length': 4.055157155935307, 'pattern_counts': {'":': 0, '<': 2, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 1, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 32, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Surface Enhanced Raman Spectroscopy (SERS) is a well-established technique for en-hancing Raman signals. [1][2][3][4][5][6][7][8][9][10][11][12] Recently photonic integrated circuits have been used, as an alternative to microscopy based excitation and collection, to probe SERS signals from external metallic nanoparticles. [13][14][15] However, in order to develop quantitative on-chip SERS sensors, integration of dedicated nanoplasmonic antennas and waveguides [16][17][18][19][20][21][22] is desirable.Here we bridge this gap by demonstrating for the first time the generation of SERS signals from integrated bowtie nanoantennas, excited and collected by a single mode waveguide, and rigorously quantify the enhancement process. The guided Raman power generated by a 4-Nitrothiophenol coated bowtie antenna shows an 8 × 10 6 enhancement compared to the free-space Raman scattering. An excellent correspondence is obtained between the theoretically predicted and observed absolute Raman power. This work paves the way towards fully integrated lab-on-a-chip systems where the single mode SERS-probe can be combined with other photonic, fluidic or biological functionalities.A schematic of the device under study is shown inFigure 1(a). The fundamental TE-mode of a silicon nitride (SiN) rib waveguide excites a periodic array of gold bowtie antennas coated with a 4-Nitrothiophenol (NTP) monolayer. The pump wavelength for all experiments is set to λ P = 785 nm and NTP Stokes light (at λ S ) is subsequently collected back into the same waveguide mode.Fabrication details can be found in the Methods section and a description of the measurement setup is outlined in the Supplementary Information S1. A scanning electron microscope image of the functionalized waveguide, with cross-sectional area of 220 × 700 nm 2 , is depicted inFigure 1(b). Raman spectra of an uncoated and coated waveguide functionalized with 40 antennas are shown inFigure 1(c). The spectral regions where an NTP Stokes peak is expected (1,080, 1,110, 1,340 and 1,575 cm −1 ) 23 are highlighted by the cyan shaded areas. Before coating no NTP peaks can be distinguished from the inherent SiN background. The peaks at 1,250 and 1,518 cm −1 (marked by the black dashed lines) are attributed to interference effects of the Au array which act on the scattered background light, and they are also observed on the extinction curves of the functionalized waveguides (see Supporting Information S2). Hence they do not represent specific 2 Raman lines. After coating, four additional peaks appear and coincide with the expected NTP Stokes peaks. This demonstrates that SERS signals from single monolayer coated antennas can be efficiently excited and collected by the same fundamental waveguide mode.Subsequently the dependence of the SERS signal on the position of the plasmon resonance was investigated to verify that it can be attributed to a resonance effect and not to coincidental surface roughness. To this end, waveguides functionalized with a fixed number of antennas but varying bowtie geometries were considered. The relevant bowtie parameters are its length L, gap ∆ and apex angle α(Figure 2(a)). By changing the length, the antenna resonance can be tuned (L 1 = 90 nm, L 2 = 115 nm and L 3 = 140 nm for fixed α = 60 • and ∆ = 40 nm). Extinction spectra are plotted inFigure 2(b) while the corresponding Raman spectra are depicted inFigure 2(c). The Raman spectrum of a reference waveguide without any Au functionalization is also shown. Even after coating the reference waveguide does not generate NTP peaks, so any Raman signal indeed originates from the antenna region and does not contain contributions from spontaneous Raman scattering along the waveguide. 24 The L 1 resonance is detuned from the relevant pump and Stokes region, resulting in a poor Raman spectrum. By increasing the length (L 2 and L 3 bowties) the resonance redshifts and lines up with the pump and Stokes wavelengths. For these bowties the NTP spectrum starts to emerge. The reported SERS spectra can hence be attributed to a plasmon resonance effect such that a stable and reproducible enhancement factor can be associated with them, in contrast to SERS events originating from random surface defects. The increased overlap with the plasmon resonance, and hence extinction, also results in a decreased background.Due to the metal induced loss, there will exist an optimum number N opt of patterned antennas such that the SERS signal is maximized. Such an optimum is investigated inFigure 3for a fixed bowtie geometry (α = 60 • , L = 100 nm and ∆ = 40 nm) but varying N: N = 10, 20, 30, 40, 70 and N = 0 which is a reference waveguide. Each waveguide is measured 10 times and the averaged Raman spectra are reported inFigure 3(a). The N = 70 signal is not shown since it could not be distinguished from the inherent offset signal of the detector. For a given fixed input power, corresponding to roughly 5 mW guided power, the reference waveguide generates a considerable 3 background signal in the 1,340 cm −1 region, where the strongest NTP peak is expected. Functionalizing the waveguide with increasing N reduces this unwanted background due to the attenuation caused by the nanoantennas. In addition the 1,340 cm −1 peak starts to emerge when N increases.The smaller peaks at 1,080, 1,110 and 1,575 cm −1 only appear when the background is sufficiently low. A zoom on the dominant 1,340 cm −1 peak (cyan dashed line) is shown inFigure 3(b). For clarity the background is locally subtracted. As expected, the signal reaches a maximum value for 10 ≤ N ≤ 20 and then decays again with increasing N. Apart from signal optimization it is however equally important to reduce the SiN background in order to resolve the smallest spectral features. Therefore an analytical model is developed to outline the interplay between signal enhancement and background reduction, and to derive the relevant figure of merit for on-chip SERS.Figure 4(a) shows a schematic longitudinal cross-section of the chip consisting of N antennas spaced with period Λ = 10 µm. Each array is centered on the waveguide with a distance L 1 ≈ 0.5 cm to the front and back facet of the chip. The NTP monolayer on each antenna will generate a forward propagating Stokes power P A (λ P , λ S ) for a given pump power P pump . This single antenna conversion efficiency P A (λ P , λ S ) is an antenna dependent factor incorporating the integrated field enhancement profile near the metal surface and the molecular density and Raman cross-section.The total transmission loss induced by one antenna at wavelength λ is given by 1 − e −1 λ , whereby e λ is the linear antenna extinction. Apart from the intrinsic waveguide losses α wg , the pump and Stokes light will hence also be attenuated by e P and e S respectively. In the Supplementary Information S3 it is then shown that the total Stokes power P tot generated by an array of N coated antennas is approximately given byThe quantity FOM(N, λ P , λ S ) contains all necessary parameters to assess the SERS signal strength for a given waveguide geometry and is hence considered to be the relevant figure of merit (FOM) in comparing the performance of integrated antenna arrays. The optimum antenna number N opt = 4 log {log (e S ) / log (e P )} / log (e S /e P ). For the particular bowtie antenna studied inFigure 3, the extinction spectrum e(λ ) and single antenna conversion efficiency P A (λ P , λ S ) are numerically evaluated using Lumerical FDTD Solutions (see Methods section). The predicted N opt for the 1,340 cm −1 peak is 10 antennas, using the simulated extinctions e P = 1.14 (E P = 0.58 dB) and e S = 1.08 (E S = 0.34 dB), while P A ≈ 2.35 × 10 −15 . For each 1W of pump power the antenna is therefore expected to generate 2.35 fW of guided Stokes power. The Raman enhancement factor is calcu-the local field enhancement (see Methods section). In the center of the gap at 5 nm from the tip of the antenna (marked by the black dot in Figure 2(a)) an EF R ≈ 1.42 × 10 4 is expected . Apart from the relevant NTP signal, the SiN itself generates a considerable background while the pump beam is propagating along the waveguide.This background signal P bg can be approximated byin which P B (λ P , λ S ) is a waveguide dependent factor incorporating the specific modal field profile and the SiN molecular density and cross-section (seeSupplementary Information S4).Our analytical model and the associated numerical calculations will now be benchmarked against the spectra fromFigure3 to verify whether the theoretically estimated power values correspond to the experimentally obtained absolute Raman powers. To this end, the NTP signal strength at 1,340 cm −1 , obtained from Figure 3(b), is analyzed as a function of the antenna number N and compared with the theoretical estimations. Furthermore, the background associated shot noise is also calculated. The results are depicted in Figure 4(b). While the ideal model assumes N identical antennas, the fabricated antennas will show differences among each other resulting in changes of e P , e S and P A from one antenna to the other. A generalized model incorporating potential differences in e P , e S and P A is described in the Supplementary Information S5. In order to estimate the uncertainty on these experimental parameters a randomized fit to the generalized model has been applied. A set of normally distributed numbers is generated for each of the three 5 parameters and then plugged into the generalized model to calculate the distribution of signal and shot noise counts, defining an area within which the probable signal (blue area) and noise (red area) counts are situated. The mean values of these distributions are extracted from an initial constrained fit to the ideal model (dotted lines), while its standard deviations are chosen such that the corresponding signal and shot noise distributions cover all experimental datapoints (red and blue dots). Based on the randomized fit it is possible to estimate the spread on the experimental parameters: E P ≈ 0.49 ± 0.11 dB, E S ≈ 0.35 ± 0.11 dB and P A = (2.60 ± 0.77) × 10 −15 (theoretically E P = 0.58 dB, E S = 0.34 dB and P A = 2.35 × 10 −15 were predicted). The theoretically predicted parameters are all within the error bars of the experimentally fitted data, which clearly establishes the validity of our model and its ability to provide quantitative predictions of the absolute Raman power coupled into a single mode waveguide. Given this excellent correspondence, we expect the fabricated structures to have a Raman enhancement factor EF R on the order of 1.42 × 10 4 near the two antenna gap tips. Decreasing the gap size should boost EF R and P A by another two or three orders of magnitude. From the fitting values the optimum antenna number is estimated to be 11 ± 3 (compared to 10 theoretically). The single antenna conversion efficiency P A shows that the fabricated antennas produce (2.60 ± 0.77) fW of guided Stokes power for each 1W of guided pump power. Compared to the free space Raman scattering P 0 of a single NTP molecule in a bulk air environment P A ≈ 3.94 × 10 6 P 0 . This includes the excitation and emission enhancement of all molecules in the monolayer as well as the coupling efficiency to the guided mode. Since only halfof the Stokes light is carried by the forward propagating mode, the total power coupled into the fundamental TE-mode is therefore ≈ 8 × 10 6 P 0 .Our observations also reveal that a minimum number of antennas N min is required to generate a detectable signal (marked by the white square inFigure 4(b)). If N < N min then the shot noise still dominates on the signal. It has to be noted however that the relevant signal is generated in a very small region (N − 1)Λ compared to the overall length 2L 1 + (N − 1)Λ ≈ 2L 1 , while the shot noise is mainly attributed to this non-useful length 2L 1 . Chip designs which allow a separation of the signal from the background are expected to have N min = 1 such that signals originating from one 6 single antenna can still be detected. As a result it would become possible to simultaneously probe large areas of analytes (> λ 2 ) and detecting all SERS events, originating from different locations, by monitoring just a single waveguide output in contrast to microscopy based systems where one has to serially scan all hotspot locations.The work presented here paves the way towards the efficient design of evanescently coupled nanoantennas for on-chip excitation and emission enhancement in the 700-1000 nm region. Due to the low fluorescence, negligible water absorption and the availability of high quality and low-cost sources and detectors this region is of particular interest for Raman sensing. 28 In combination with other on-chip spectral functionalities, such as arrayed waveguide gratings, 28 the presented platform is forecasted to allow multiplexed detection of extremely weak Raman signals on a highly dense integrated platform. We also envisage that integrated nanoantennas, similar to the ones reported here, could be used as transducer between quantum dot emitters and the fundamental waveguide mode, potentially enabling applications in on-chip quantum communication and quantum computation. 29,30 Methods Fabrication details The fabrication consists of a 2-step e-beam lithography process. In the first step the nanoplasmonic antennas are patterned on top of a slab Si/SiO 2 /SiN wafer using a positive PMMA e-beam resist. After PMMA exposure, the samples are developed in a 1:1 MIBK:IPA solution after which a 2 nm Ti adhesion layer and 30 nm Au layer are deposited in a commercial Pfeiffer Spider sputter system. The samples are then immersed in acetone for lift-off. In the second step the waveguides are defined. After metal lift-off a negative ma-N 2403 resist is spun, exposed and developed in ma-D 525. An e-spacer is also spun on top of the ma-N 2403 to avoid charging effects. The developed samples are then etched with an ICP plasma (C 4 F 8 /SF 6 mixture) in a commercial Oxford Plasmalab system. After resist strip and cleaning, the samples are immersed overnight in a 1 mM NTP:EtOH 7solution and subsequently rinsed with pure ethanol to remove the residual NTP. A self-assembled monolayer of NTP is assumed to form on the Au surface through a Au-S bond. 23Numerical SimulationsNumerical simulations were performed with Lumerical FDTD Solutions. We used a refractive index of n rib = 1.9 for the SiN rib (with width w rib = 700 nm and height h rib = 220 nm), n uclad = 1.45 for the SiO 2 undercladding and n tclad = 1 for the top cladding (air). The Si substrate was not taken into account since the real oxide cladding is thick enough to avoid substantial power leakage to the Si such that the numerical results faithfully represent the actual experimental conditions. A thin native oxide layer (t nox = 2 nm) between the SiN and the Ti has also been incorporated. 19The metal stack thicknesses are fixed to t Ti = 2 nm and t Au = 30 nm and a built-in refractive index model for Au (Johnson and Christy 25 ) and Ti (CRC 26 ) is used. An additional surface layer with thickness t NT P = 1 nm and index n NT P = 3 is used to model the NTP monolayer. The antenna region (including the Ti adhesion layer and the NTP monolayer) is meshed with a uniform mesh of 0.5 nm in the plane of the antenna (yz-plane) and 2 nm in the x-direction. A mesh refinement to 1 nm is applied in regions where the thicknesses in the x-direction are ≤ 2 nm. The estimated surface area of an NTP molecule is 0.18 nm 2 , so the surface density is then ρ s = 5.56 × 10 18 molecules/m 2 . 23 The Raman cross section is σ ≈ 0.358 × 10 −30 cm 2 /sr, which was obtained by applying the λ −4 S scaling to the original data of the 1,340 cm −1 line. 27 Single antenna extinction spectra E(λ ) (in dB) are calculated through E(λ ) = T re f (λ ) − T ant (λ ) in which T re f (λ ) is the power transmission (in dB) through the reference waveguide and T ant (λ ) the power transmission (in dB) of a waveguide functionalized with one antenna. Linear extinction spectra e(λ ) ∆ = e λ are then given by e(λ ) = 10 E(λ )/10 . A field and index profile monitor are used to extract the local field |E(r, λ )| and index n(r) around the antenna. The single antenna conversion efficiency P A (λ P , λ S ) = ρ s σ 2t NT P V m n g (λ P )n g (λ S )λ 2 S |E(r, λ P )| 2 |E(r, λ S )| 2 dr n(r) 2 |E m (r, λ P )| 2 dr n(r) 2 |E m (r, λ S )| 2 dr 8 is calculated by integrating the local fields over the effective monolayer volume V m in which the index satisfies n(r) | r∈V m = n NT P (seeSupplementary Information S3). The group index of the waveguide mode is n g (λ ). The denominator is calculated using the modal fields E m (r, λ ) of a non-functionalized reference waveguide and the local field enhancement is given by the ratio of the local and modal electric fields: β (r, λ ) = |E(r,λ )| |E m (r,λ )| . At a certain position, the Raman enhancement factor EF R is calculated as EF R = β (λ P ) 2 β (λ S ) 2 . Numerically calculated values are mentioned in the main text.9', 'arxivid': '1508.02189', 'author': ['Frédéric Peyskens [email protected] \nDepartment of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium\n', 'Ashim Dhakal \nDepartment of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium\n', 'Pol Van Dorpe ', 'Nicolas Le Thomas \nDepartment of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium\n', 'Roel Baets \nDepartment of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium\n', '\nCenter for Nano-and BioPhotonics\nDepartment of Physics\nPhotonics Research Group\nINTEC-department\nGhent University-imec\nGhent University\nSint-Pietersnieuwstraat 41, Belgium, and imec, Kapeldreef 75, Celestijnenlaan 200D9000, 3001, 3001Ghent, Heverlee, KULeuven, LeuvenBelgium, Belgium\n'], 'authoraffiliation': ['Department of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium', 'Department of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium', 'Department of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium', 'Department of Physics, KULeuven\nPhotonics Research Group, INTEC-department, Ghent University-imec; Center for Nano-and BioPhotonics\nGhent University\nSint-Pietersnieuwstraat 41, Belgium ‡ imec, Kapeldreef 759000, 3001, 200D, 3001Ghent, Heverlee, LeuvenCelestijnenlaanBelgium;, Belgium', 'Center for Nano-and BioPhotonics\nDepartment of Physics\nPhotonics Research Group\nINTEC-department\nGhent University-imec\nGhent University\nSint-Pietersnieuwstraat 41, Belgium, and imec, Kapeldreef 75, Celestijnenlaan 200D9000, 3001, 3001Ghent, Heverlee, KULeuven, LeuvenBelgium, Belgium'], 'corpusid': 55685424, 'doi': '10.1021/acsphotonics.5b00487', 'github_urls': [], 'n_tokens_mistral': 19938, 'n_tokens_neox': 16665, 'n_words': 11551, 'pdfsha': '4ccfe45c5c4cdf82a1c821f6c711d8e5ab5326a1', 'pdfurls': ['https://arxiv.org/pdf/1508.02189v1.pdf'], 'title': ['Biosensing with plasmonic nanosensors', 'Biosensing with plasmonic nanosensors'], 'venue': ['Nat. Mater']} |
arxiv |
Nanoscale visualization of the thermally-driven evolution of antiferromagnetic domains in FeTe thin films
Shrinkhala Sharma
Department of Physics
Boston College
140 Commonwealth Ave, Chestnut Hill02467MA
Hong Li
Department of Physics
Boston College
140 Commonwealth Ave, Chestnut Hill02467MA
Zheng Ren
Department of Physics
Boston College
140 Commonwealth Ave, Chestnut Hill02467MA
Wilber Alfaro Castro
Department of Physics
Boston College
140 Commonwealth Ave, Chestnut Hill02467MA
Ilija Zeljkovic
Department of Physics
Boston College
140 Commonwealth Ave, Chestnut Hill02467MA
Nanoscale visualization of the thermally-driven evolution of antiferromagnetic domains in FeTe thin films
Antiferromagnetic order, being a ground state of a number of exotic quantum materials, is of immense interest both from the fundamental physics perspective and for driving potential technological applications. For a complete understanding of antiferromagnetism in materials, nanoscale visualization of antiferromagnetic domains, domain walls and their robustness to external perturbations is highly desirable. Here, we synthesize antiferromagnetic FeTe thin films using molecular beam epitaxy. We visualize local antiferromagnetic ordering and domain formation using spin-polarized scanning tunneling microscopy. From the atomically-resolved scanning tunneling microscopy topographs, we calculate local structural distortions to find a high correlation with the distribution of the antiferromagnetic order. This is consistent with the monoclinic structure in the antiferromagnetic state. Interestingly, we observe a substantial domain wall change by small temperature variations, unexpected for the low temperature changes used compared to the much higher antiferromagnetic ordering temperature of FeTe. This is in contrast to electronic nematic domains in the cousin FeSe multilayer films, where we find no electronic or structural change within the same temperature range. Our experiments provide the first atomic-scale imaging of perturbation-driven magnetic domain evolution simultaneous with the ensuing structural response of the system. The results reveal surprising thermally-driven modulations of antiferromagnetic domains in FeTe thin films well below the Neel temperature.
Introduction
Antiferromagnetic (AF) ordering is an important presence in the phase diagrams of various correlated electron systems. For example, in cuprate high-temperature superconductors, the ground state of the undoped parent system is an AF Mott insulator 1,2 , which becomes superconducting as the AF Mott insulating state is weakened by chemical doping. In Fe-based high-temperature superconductors, AF order has been widely observed in the parent systems 3,4 , which may have a close relationship with the subsequent emergence of electronic nematicity and superconductivity 3,5 . More recently, it has been discovered that many kagome metals also host AF ordering, which coexists with flat bands, Dirac fermions and charge density waves [6][7][8][9] .
While magnetic domains are nearly ubiquitous in ferromagnets due to uncompensated stray magnetic fields and the resulting interactions, domains in antiferromagnets are generally less frequent. Nevertheless, antiferromagnets still exhibit a tendency to form domains [10][11][12][13] . A domain state denotes an ordered state associated with the possible order parameter orientation or phase; a domain is then defined as a translationally-invariant region in a solid that can take one of the possible domain states. Probing the AF domain structure and domain wall formation is of immense importance for the understanding of fundamental phenomena and for driving potential technological applications. AF spintronics, for example, is proposed to utilize AF moments as carriers for information, and strongly relies on the robustness of the AF structure to external perturbations [14][15][16] . AF domain walls may also have different properties compared to the interior of the domains 17,18 .
Resolving magnetic domain structures at the nanoscale has been challenging but it can bring exciting new insights 12,19,20 . Probes that are crucial in unveiling magnetic ordering in solids, such as neutron scattering, nuclear magnetic resonance, muon spin rotation and resonant inelastic xray scattering, average the signal from the entire beam-spot and therefore lack the capability to visualize domains at the atomic scale. Complementary to these, scanning probe microscopy can be a very useful tool to visualize domains down to the nanoscale. Here we focus on using spinpolarized scanning tunneling microscopy (SP-STM), which measures the local spin-polarized density of states, to image magnetic ordering at the nanoscale 21 . In particular, we investigate the AF domain formation in FeTe thin films synthesized by molecular beam epitaxy (MBE). We identify the double stripe antiferromagnetic order in FeTe, as well as the domain structures consisting of nanometer-scale domains. By extracting local structural distortion from the atomically-resolved STM topographs, we find a high correlation between the antisymmetric strain and the AF domains, which is consistent with the structural distortion accompanying the AF phase in bulk FeTe. Surprisingly, we discover substantial fluctuations of magnetic domains after thermal cycling up to relatively low temperatures compared to the Neel temperature of FeTe.
Results
While the superconducting FeSe does not exhibit any magnetic ordering, double-stripe antiferromagnetic order emerges when Se is completely substituted with Te, accompanied by stronger electron-electron correlation [3][4][5] . As such, FeTe is often viewed as a parent compound of the superconducting chalcogenide Fe(Se,Te). FeTe has the PbO-type crystal structure (Fig. 1a), and it goes through a bulk antiferromagnetic phase transition at T N ≈70 K, accompanied by a tetragonal-monoclinic structural transition 22 . In contrast to the (π, π) in-plane stripe antiferromagnetic order in Fe pnictides, the in-plane antiferromagnetic order in FeTe appears to be a (π, 0) order characterized by the double stripe structure 3-5,22,23 .
We synthesize FeTe thin films on Nb-doped SrTiO 3 (001) substrates by using MBE (Methods). The quality of our films is confirmed by the sharp streaks in the reflection high energy electron diffraction (RHEED) pattern of FeTe (Fig. 1b). X-ray diffraction measurement exhibits only the (00l) (l=1,2,3,4) reflections, which further demonstrates the layered structure of our films that grow along the crystalline c-axis (Fig. 1d). Based on the zero-field cooled resistance measurements as a function of temperature, we determine the antiferromagnetic ordering temperature to be T N ≈ 62 K (Fig. 1c). This is comparable but slightly lower than that of bulk single crystals of FeTe 22 . The surface morphology of the FeTe films is displayed in the STM topograph acquired using a conventional (spin-averaged) tungsten tip (Fig. 1e,f). As expected from the RHEED measurement, we observe an atomically flat surface with terraces. A step height of about 0.7 nm is measured, which is consistent with the expected unit cell height of bulk single crystals (inset of Fig. 1e). Top-most Te atoms are clearly visible in the atomically resolved topograph (Fig. 1f).
To gain insight into the magnetic structure of the FeTe films, we perform SP-STM measurements using spin-polarized tips (Methods). The same technique has been applied to a variety of different antiferromagnets, including bulk single crystals of FeTe 23-25 , iridates 10,11 , and chiral 20 and kagome magnets 6,9 . We focus on the area as shown in Fig. 2a,b. The SP-STM topograph T(r, B = 4 T), acquired when a 4 T magnetic field is applied parallel to the c-axis, resolves the top-most Te atoms arranged on a square lattice (Fig. 2a). In addition to the Te atoms, a stripe-like superstructure is discernable propagating along the Te-Te x-axis, which has a wavelength of 2a Te (Fig. 2a,d). These stripes are not seen in STM topographs obtained by a spin-averaged STM tip (Fig. 1f) and are consistent with the orientation and periodicity of the double stripe antiferromagnetic ordering observed in FeTe bulk crystals and in previous SP-STM work [23][24][25] . To further confirm the AF origin of these features, we reverse the direction of the magnetic field. This flips the polarization direction of the STM tip, without significantly affecting the magnetic ordering in the FeTe film, since the Zeeman energy induced by the 4 T field is much smaller than the energy scale of the exchange interaction. We image the identical region of the sample to find similar stripe features in the SP-STM topograph T(r, B = -4 T), but with one noticeable difference -the bright stripes now shift by a Te along the x-axis (Fig. 2b,e). We obtain a spin-resolved magnetic contrast M(r) map by subtracting the STM topographs in Fig. 2a and Fig. 2b, which emphasizes the AF contrast and now more clearly shows the 2a Te stripe modulation related to the AF order (Fig. 2c,f). The topographs in Fig. 2a,b were drift-corrected and aligned with atomicregistry using Lawler-Fujita drift-correction algorithm 26 prior to subtraction. This conclusively demonstrates that the stripe-like features in SP-STM data are intimately related to the same double stripe antiferromagnetic ordering observed in FeTe bulk crystal.
We turn to a larger field of view to investigate the existence of AF domains (Fig. 3). We again acquire the SP-STM topographs over an identical field-of-view at different magnetic fields, T(r, B=4 T) and T(r, B=-4 T), and subtract the two to extract the AF signal (Fig. 3a). The M(r) map shows the AF ordering stripes oriented in either near-horizontal, or near-vertical direction. The two types of orientational AF domains are enclosed by solid white lines, across which the orientation of the AF ordering stripes rotates by 90° (Fig. 3a). Within each orientational domain, we also find several smaller sub-domains with the same wave vector, but offset by π phase with respect to one another (denoted by dashed white lines in Fig. 3a, Supplementary Figure 7).
In FeTe bulk single crystals, it has been established that the antiferromagnetic phase transition is accompanied by a tetragonal-to-monoclinic structural transition. The in-plane lattice constants along the two lattice directions are different by about 1% in the AF phase 27,28 . To measure if the AF order in our FeTe film is also accompanied by a similar structural change, we use the Lawler-Fujita drift-correction algorithm to calculate the antisymmetric strain map U(r)=u xx (r)-u yy (r) (u ii (r)= du i (r)/dr i , i=x,y, where u i (r) is the displacement field), which quantifies local structural anisotropy between different crystalline directions 26,[29][30][31][32][33] (Fig. 3b, Supplementary Note 1). By visually comparing the spin-resolved magnetic contrast M(r) map and the U(r) map (Fig. 3a,b), it is evident that the shape of the antiferromagnetic domains is correlated with the distribution of the structural anisotropy. To quantitatively establish the cross-correlation between the strength of AF ordering and the local structural anisotropy, we Fourier-filter the M(r) map by choosing the q AF peak along the x-axis (Fig. 3c) or along the y-axis (Fig. 3d) in the FT. We then calculate the radially-averaged cross-correlation coefficient of the Fourier-filtered M(r) map and the U(r) map, and find it to be 0.5 and -0.37 (Fig. 3c,d insets). Consistent with the expectation from the bulk FeTe, the lattice constant along which q AF develops is larger by about 1 % 22 .
Magnetic domains we visualized in Figure 3 are formed spontaneously when the sample is cooled across the ordering temperature. Thermal cycling through the ordering temperature can reform the domain structure 34,35 . However, it is not expected that the antiferromagnetic domains would significantly change at a temperature much lower than the Neel temperature. To investigate the robustness of the AF domain structure, we warm up the sample to a higher temperature, cool it back down to the base temperature and compare the AF orders before and after the thermal cycle. In Figure 4, we show one such process, where we warmed up the sample to ~10 K and cooled it back down to ~4.5 K (additional data is shown in Supplementary Figure 4). Focusing on the same field-of-view, we repeat the SP-STM experiment sequence to generate spin-resolved magnetic contrast maps before (M(r)) and after (M'(r)) thermal cycling (Fig. 4c,f). Surprisingly, by comparing M(r) and M'(r) maps, we observe a notable change of the orientational AF domain structure. In particular, a substantial part of the AF domain where the stripes were initially aligned along the x-axis in the M(r) map rotated by ~90° to merge with a part of the AF domain along the other stripe orientation (along the y-axis) in the M'(r) map. We also observe a change in the anti-phase sub-domains within each orientational domain (Figure 4c,f, Supplementary Figure 7). The considerable change in the AF domain structure with thermal cycling in evident in subsequent thermal cycles over the same area of the sample (Supplementary Figure 4). In particular, we cycled the system to 21 K and to 52 K (two times in a row) to find a different resulting domain structure at low temperature each time (Supplementary Figure 4).
Discussion
We can rule out the possibility that the AF domain modulation is driven by structural degrees of freedom because 10 K is extremely low compared to the synthesis temperature of FeTe films (about 570 K). Furthermore, in a similar heterostructure FeSe/SrTiO 3 (001), we do not observe any change in the structure of electronic nematic domains, which also locally carry a 1% lattice anisotropy, by warming up to 30 K (Supplementary Figure 6). Future experiments could explore if monolayer FeTe grown on SrTiO 3 also shows the surprising T N enhancement similar to that reported in ultrathin FeTe films grown on Bi 2 Te 3 36 . It is interesting to note that the irregularlyshaped AF domain walls observed in thin films are markedly different than the larger scale FeTe domains with straight-line domain walls in FeTe bulk single crystals [23][24][25]37,38 . This is similar to the electronic nematic domain morphology in non-magnetic FeSe, where the domains in thin films are similarly irregularly shaped 31,33,39,40 and smaller compared to bulk FeSe 41,42 . The susceptibility of AF domain structure to change with small temperature variations may suggest the domain walls are not strongly pinned to defects or impurities in this system. As our FeTe films also exhibit strain at the order of a few percent due to the lattice mismatch with the substrate (Methods), this points to an important role of the substrate in partitioning the domains to smaller length scales compared to bulk single crystals.
Methods
MBE growth.
SrTiO 3 (100) substrates (5 mm x 5 mm x 0.5 mm) were cleaned in acetone and 2propanol in an ultrasonic bath and then introduced into our MBE system (Fermion Instruments) with a base pressure of ~ 5 x 10 -10 Torr. Nb-doped (0.05 wt%) SrTiO 3 was used as the substrate for the growth of FeTe films subsequently studied by STM, and undoped SrTiO 3 was used as the substrate for FeTe films characterized by transport measurements in Fig. 1c. The substrate was first slowly heated to the growth temperature at ~300℃, which was continuously monitored by a pyrometer (emissivity=0.7). Thereafter, Fe (99%) and Te (99.9999%) were co-evaporated from individual Knudsen cells after the flux rates were calibrated using a quartz crystal microbalance (QCM). A flux ratio of Fe:Te=1:15 was roughly achieved as the temperatures of Fe and Te were set at 1100℃ and 240℃, respectively. Films studied in this work were 24 nm thick, determined from Fe flux calibration by QCM. XRD measurements of our films in Fig. 1d reveal the average caxis lattice constant reduction by 3.8%, and in turn suggest, on average, in-plane tensile strain. For STM measurements, thin films were quickly transferred using a vacuum suitcase chamber held at 5 x 10 -11 Torr, and were never exposed to air. STM measurements. STM data was acquired using a customized Unisoku USM1300 STM at the base temperature of about 4.5 K. Spectroscopic measurements were acquired using a standard lock-in technique at 915 Hz and bias excitations as detailed in the figure captions. STM tips were home-made chemically etched tungsten tips, annealed in UHV to bright orange color before STM imaging. Spin polarization of STM tips was obtained in-situ on FeTe films, by fast scanning and bias pulsing, which likely leads to the tip picking up a few Fe atoms at its apex. To obtain spinresolved magnetic contrast M(r) maps, the two topographs obtained at different magnetic fields were drift-corrected and aligned using Lawler-Fujita drift-correction algorithm 26 prior to subtraction.
(a,b). The M(r) map reveals the underlying AF order more clearly. The thick white solid lines in (c) outline the rotational AF domain walls, across which the wave vector rotates by about 90 degrees inplane. Thinner white dashed lines denote smaller anti-phase sub-domains within each rotational domain, with the same wave vector but offset by π phase with respect to one another. (d-f) Equivalent panels to those in (a-c), over the same area of the sample, but after the sample was warmed up to 10 K and cooled back down. All STM topographs are acquired at 4.5 K on the same area of the sample surface. The arrows in (a,d) point to the same set of defects in both images to demonstrate the identical positions of the scan frames. The domain walls clearly shift after a thermal cycle, demonstrating their sensitivity to thermal fluctuations. STM setup conditions: (ab, d-e) I set = 300 pA, V sample = 100 mV.
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| {'fraction_non_alphanumeric': 0.05514220151588902, 'fraction_numerical': 0.037340935957165707, 'mean_word_length': 4.46202050892518, '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': 'Antiferromagnetic order, being a ground state of a number of exotic quantum materials, is of immense interest both from the fundamental physics perspective and for driving potential technological applications. For a complete understanding of antiferromagnetism in materials, nanoscale visualization of antiferromagnetic domains, domain walls and their robustness to external perturbations is highly desirable. Here, we synthesize antiferromagnetic FeTe thin films using molecular beam epitaxy. We visualize local antiferromagnetic ordering and domain formation using spin-polarized scanning tunneling microscopy. From the atomically-resolved scanning tunneling microscopy topographs, we calculate local structural distortions to find a high correlation with the distribution of the antiferromagnetic order. This is consistent with the monoclinic structure in the antiferromagnetic state. Interestingly, we observe a substantial domain wall change by small temperature variations, unexpected for the low temperature changes used compared to the much higher antiferromagnetic ordering temperature of FeTe. This is in contrast to electronic nematic domains in the cousin FeSe multilayer films, where we find no electronic or structural change within the same temperature range. Our experiments provide the first atomic-scale imaging of perturbation-driven magnetic domain evolution simultaneous with the ensuing structural response of the system. The results reveal surprising thermally-driven modulations of antiferromagnetic domains in FeTe thin films well below the Neel temperature.', 'arxivid': '2305.18197', 'author': ['Shrinkhala Sharma \nDepartment of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA\n', 'Hong Li \nDepartment of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA\n', 'Zheng Ren \nDepartment of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA\n', 'Wilber Alfaro Castro \nDepartment of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA\n', 'Ilija Zeljkovic \nDepartment of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA\n'], 'authoraffiliation': ['Department of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA', 'Department of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA', 'Department of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA', 'Department of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA', 'Department of Physics\nBoston College\n140 Commonwealth Ave, Chestnut Hill02467MA'], 'corpusid': 258960297, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8819, 'n_tokens_neox': 7297, 'n_words': 4336, 'pdfsha': 'b99fc181c01be47e38c4b7b4c5d96983f6efef7e', 'pdfurls': ['https://export.arxiv.org/pdf/2305.18197v1.pdf'], 'title': ['Nanoscale visualization of the thermally-driven evolution of antiferromagnetic domains in FeTe thin films', 'Nanoscale visualization of the thermally-driven evolution of antiferromagnetic domains in FeTe thin films'], 'venue': []} |
arxiv |
GENERALIZED BLOCH SPACES, INTEGRAL MEANS OF HYPERBOLIC HARMONIC MAPPINGS IN THE UNIT BALL
18 Nov 2017
Jiaolong Chen
GENERALIZED BLOCH SPACES, INTEGRAL MEANS OF HYPERBOLIC HARMONIC MAPPINGS IN THE UNIT BALL
18 Nov 2017arXiv:1612.06542v3 [math.CV]
In this paper, we investigate the properties of hyperbolic harmonic mappings in the unit ball B n in R n (n ≥ 2). Firstly, we establish necessary and sufficient conditions for a hyperbolic harmonic mapping to be in the Bloch space B(B n ) and the generalized Bloch space L ∞,ω B 0 α,a (B n ), respectively. Secondly, we discuss the relationship between the integral means of hyperbolic harmonic mappings and that of their gradients. The obtained results are the generalizations of Hardy and Littlewood's related ones in the setting of hyperbolic harmonic mappings. Finally, we characterize the weak uniform boundedness property of hyperbolic harmonic mappings in terms of the quasihyperbolic metric.2000 Mathematics Subject Classification. Primary: 31B05; Secondary: 31C05.
Introduction and main results
For n ≥ 2, let B n (x 0 , r) = {x ∈ R n : |x − x 0 | < r}, S n−1 (x 0 , r) = ∂B n (x 0 , r) and B
n (x 0 , r) = B n (x 0 , r) ∪ S n−1 (x 0 , r). In particular, we write B n = B n (0, 1), S n−1 = S n−1 (0, 1) and B n = B n ∪ S n−1 .
The purpose of this paper is to consider the hyperbolic harmonic mappings whose definition is as follows.
Definition 1.1. A mapping u = (u 1 , · · · , u n ) ∈ C 2 (B n , R n ) is said to be hyperbolic harmonic if ∆ h u = (∆ h u 1 , · · · , ∆ h u n ) = 0, that is, for each j ∈ {1, · · · , n}, u j satisfies the hyperbolic Laplace equation
∆ h u j = 0, where (1.1) ∆ h u j (x) = (1 − |x| 2 ) 2 ∆u j (x) + 2(n − 2)(1 − |x| 2 ) n i=1 x i ∂u j ∂x i (x).
We refer to [4,15,19,28,35,36,37] for basic properties of this class of mappings. For convenience, in the following of this paper, we always use the notation ∆ h u = 0 to mean that u = (u 1 , · · · , u n ) is hyperbolic harmonic in B n . Obviously, for n = 2, hyperbolic harmonic mappings coincide with harmonic mappings. See [9,11] and the references therein for the basic properties of harmonic mappings. Here and hereafter, dσ always denotes the normalized surface measure on S n−1 so that σ(S n−1 ) = 1. The classical Hardy space H p (D) consisting of related analytic functions is a subspace of H p g (D), where D denotes the unit disk in the complex plane C (In this paper, we always identify R 2 with C and B 2 with D, respectively).
In order to introduce the definition of the generalized Bloch space, we need the following notion.
A continuous increasing function ω : [0, ∞) → [0, ∞) with ω(0) = 0 is called a majorant if ω(t)/t is non-increasing for t > 0 (cf. [12,13,24,25]).
Given a subset Ω of R n , a function f : Ω → R n is said to belong to the Lipschitz space L ω (Ω) if there is a positive constant µ 01 such that for all x, y ∈ Ω, |f (x) − f (y)| ≤ µ 01 ω(|x − y|).
First, we define the Bloch space of B n , denoted by B(B n ), as the space of functions
f in C 1 (B n , R n ) such that ||f || B < ∞, where ||f || B = sup x∈B n ||Df (x)||(1 − |x| 2 )
and ||Df (x)|| denotes the matrix norm of the usual Jacobian matrix Df (x) of f at x (See §2.1 below for the precise definition of ||Df (x)||). This is only a semi-norm. Obviously, ||f || B = 0 if and only if f is constant. The Bloch space B(B n ) becomes a Banach space with the following norm:
||f || B = |f (0)| + ||f || B .
For an analytic function f in D, obviously, ||Df (z)|| = |f ′ (z)|. Therefore, the classical Bloch space B(D) consisting of the analytic functions f satisfying
||f || B = |f (0)| + sup x∈D |f ′ (z)|(1 − |z| 2 ) < ∞
is a subspace of B(B n ) (cf. [2,18,40]).
Next, we introduce the notion: Generalized Bloch spaces. Definition 1.3. For p ∈ (0, ∞], α > 0, β ∈ R and a majorant ω, we use L p,ω B β α,a (B n ) to denote the generalized Bloch space, which consists of all functions
f ∈ C 1 (B n , R n ) with ||f || Lp,ωB β α,a (B n ) < ∞, where ||f || Lp,ωB β α,a (B n ) = |f (0)| + sup x∈B n M p (|x|, ||Df ||)ω φ α,β,a (x) , φ α,β,a (x) = (1 − |x|) α log a 1 − |x| β and a is a constant satisfying (1) a > 1, if β ≤ 0; and (2) a ≥ e β α , if β > 0.
The space L p,ω B β α,e (D) was discussed in [5] and the space L p,id B β α,e β/α (D) of analytic functions was introduced in [34]. Note that, when β = 0, the space L p,ω B 0 α,a (B n ) has nothing to do with the parameter a.
Obviously, L ∞,id B 0 1,a (B n ) = B(B n ), where id denotes the identity mapping. Further, we have the following.
(1) The special case L ∞,ω B 0 α,a (D) is called the ω-α-Bloch space (cf. [14,39] and the related references therein).
(2) The special case L ∞,ω B β 1,a (D) is called the logarithmic ω-Bloch space (cf. [33,34] and the related references therein).
(3) The special case L ∞,id B 0 α,a (D) is called the generalized α-Bloch space (cf. [20,21,22] and the related references therein). (4) The special case L ∞,id B β 1,a (D) is called the generalized logarithmic Bloch space (cf. [27,39] and the related references therein). In [12], Dyakonov discussed the relationship between the Lipschitz space L ω (D) and the bounded mean oscillation on analytic functions in D ([12, Theorem 1]). In [5] and [7], the authors extended [12,Theorem 1] As the first aim of this paper, we consider the similar results of the above type for hyperbolic harmonic mappings. The following is our first result in this line.
Theorem 1.1. Suppose α ∈ [1, 2)
, ω is a majorant and ∆ h u = 0. Then u ∈ L ∞,ω B 0 α,a (B n ) if and only if there is a positive constant µ 1 such that for all r ∈ (0, 1 − |x|),
1 |B n (x, r)| B n (x,r) |u(y) − u(x)|dν(y) ≤ µ 1 ω(r α ) r,
where |B n (x, r)| means the Lebesgue volume of the ball B n (x, r) and dν denotes the normalized Lebesgue volume measure in B n . To state our next result, let us recall the following notion. The hyperbolic distance between two points x and y in B n is defined by
ρ(x, y) = inf γ∈Γxy(B n ) γ 2 1 − |z| 2 ds(z),
where ds is the length element on γ and Γ xy (B n ) stands for the collection of all rectifiable curves in B n joining x and y (cf. [38]). See §2.2 below for more properties of ρ.
In [40], Zhu characterized the holomorphic Bloch space in C n in terms of the Bergman metric ([40, Theorem 3.6 and Corollary 3.7]). Now, we establish the following necessary and sufficient condition for hyperbolic harmonic mappings to be in B(B n ) in terms of the hyperbolic metric. |u(x) − u(y)| ≤ µ 2 ρ(x, y).
1.2.
Integral means. First, we recall the following well known result on analytic functions due to Hardy and Littlewood.
(1) M p (r, f ′ ) = O 1 (1−r) α as r → 1; (2) M p (r, f ) = O 1 (1−r) α−1 as r → 1.
Obviously, the above result of Hardy and Littlewood provides a close relationship between the integral means of analytic functions and that of their derivatives.
As the second aim of this paper, we consider Theorem A in the setting of hyperbolic harmonic mappings. Our first result is the following analog of the implication from (1) to (2) in Theorem A for hyperbolic harmonic mappings.
Theorem 1.3. Suppose ∆ h u = 0 and u ∈ L p,ω B β α,a (B n ). Then for r ∈ [0, 1) and p ∈ [1, ∞], M p (r, u) ≤ |u(0)| + (log a) β ||u|| Lp,ωB β α,a (B n ) ω (log a) β 1 0 r φ α,β,a (rt) dt.
Remark 1.2. By taking n = 2, ω(t) = t, α > 1 and β = 0, we obtain that Theorem 1.3 is a generalization of the implication from (1) to (2) in Theorem A even in the case of harmonic mappings when p ∈ [1, ∞].
We also consider the converse of Theorem 1.3, and we get the following analog of the implication from (2) to (1) in Theorem A for hyperbolic harmonic mappings.
Theorem 1.4. Suppose ∆ h u = 0 and M p (r, u) = O 1 (1 − r) α as r → 1,
where p ∈ (0, ∞) and α > 1 p . Then for q ∈ (0, ∞], u ∈ L q,id B 0
α+1+ n−1 p ,a (B n ). 1.3. Weak uniform boundedness property.
Let Ω be a proper domain of R n . For x ∈ Ω, we use d Ω (x) to denote the Euclidean distance from x to the boundary ∂Ω of Ω. For x, y ∈ Ω, let [23,38]).
r Ω (x, y) = |x − y| min{d Ω (x), d Ω (y)} and k Ω (x, y) = inf γ∈Γxy(Ω) γ 1 d Ω (z) ds(z) (cf.
We say that f : Ω → f (Ω) ⊂ R n satisfies the weak uniform boundedness property in Ω (with respect to r Ω ) if there is a constant µ 02 > 0 such that for all x, y ∈ Ω,
(1.3) r Ω (x, y) ≤ 1 2 implies r f (Ω) f (x), f (y) ≤ µ 02 . Remark 1.3.
The above definition of the weak uniform boundedness property is equivalent to the following one: f : Ω → f (Ω) is said to satisfy the weak uniform boundedness property in Ω (with respect to r Ω ) if for any µ 03 ∈ (0, 1), there is a constant µ 04 > 0 such that for all x, y ∈ Ω,
r Ω (x, y) ≤ µ 03 implies r f (Ω) f (x), f (y) ≤ µ 04 .
In [23, Theorem 2.8], Mateljević and Vuorinen proved that a harmonic mapping f satisfies the weak uniform boundedness property in G ⊂ R n if and only if there exists a constant µ 05 such that for all x, y ∈ G,
k f (G) (f (x), f (y)) ≤ µ 05 k G (x, y).
Recently, Chen and Rasila generalized this result to the case of the solutions to Yukawa equation in B n ([6, Theorem 2]). As the last aim of this paper, we consider the weak uniform boundedness property of hyperbolic harmonic mappings. Our result reads as follows.
x, y ∈ B n , k u(B n ) u(x), u(y) ≤ µ 3 k B n (x, y).
This paper is organized as follows. In Section 2, some necessary terminology and notation will be introduced. In Section 3, we shall prove Theorem 1.1, and in Section 4, we shall show Theorem 1.2. The proofs of Theorems 1.3 and 1.4 will be presented in Section 5. Section 6 will be devoted to the proof of Theorem 1.5.
Preliminaries
In this section, we recall some necessary terminology and notation.
Matrix notations. For a natural number n, let
A = a ij n×n ∈ R n×n . For A ∈ R n×n , denote by A the matrix norm A = sup{|Ax| : x ∈ R n , |x| = 1}.
For a domain Ω ⊂ R n , let f = (f 1 , . . . , f n ) : Ω → R n be a function that has all partial derivatives at x = (x 1 , . . . , x n ) in Ω. Then Df (x) denotes the usual Jacobian matrix
Df = ∂f 1 (x) ∂x 1 · · · ∂f 1 (x) ∂xn . . . . . . . . . ∂fn(x) ∂x 1 · · · ∂fn(x) ∂xn = ∇f 1 (x) · · · ∇f n (x)
T at x, where T is the transpose and the gradients ∇f j (x) are understood as column vectors (cf. [4]). For two column vectors x, y ∈ R n , we use x, y to denote the inner product of x and y. For j ∈ {1, · · · , n}, it follows from
||Df (x)|| 2 = sup ξ∈S n−1 n j=1 ∇f j (x), ξ 2 that (2.1) ||Df (x)|| 2 ≤ n j=1 |∇f j (x)| 2 = n i=1 ∂ ∂x i f (x) 2 and ||Df (x)|| ≥ |∇f j (x)|.
2.2. Hyperbolic metric. For any w ∈ B n , let
ϕ w (x) = |x − w| 2 w − (1 − |w| 2 )(x − w) [x, w] 2 in B n , where [x, w] = |x|w − x |x| . Then ϕ w is a Möbius transformation from B n onto B n with ϕ w (w) = 0, ϕ w (0) = w and ϕ w ϕ w (x) = x.
We denote by M(B n ) the set of all Möbius transformations in B n . It is well known that if ϕ ∈ M(B n ), then there exist w ∈ B n and an orthogonal transformation A such that [37]). For more information on Möbius transformations in B n , see e.g. [1,3,38]. In terms of ϕ w , the hyperbolic metric ρ in B n can be given by
ϕ(x) = Aϕ w (x) (cf. [35, Theorem 2.1] or(2.2) ρ(x, w) = log 1 + |ϕ w (x)| 1 − |ϕ w (x)| for x, w ∈ B n .(2.3) |ϕ w (x)| = |ϕ x (w)| = |x − w| [x, w] .
Therefore, ρ(x, w) = ρ(w, x). In particular,
ρ(0, x) = log 1 + |x| 1 − |x| and ρ(x, y) = ρ ϕ(x), ϕ(y) for all ϕ ∈ M(B n ) (cf. [38, Chapter 1]).
For any w ∈ B n and 0 < r < 1, we define the pseudo-hyperbolic ball with center w and radius r as
(2.4) E(w, r) = {x ∈ B n : |ϕ w (x)| < r}.
Clearly, E(w, r) = ϕ w B n (0, r) (cf. [36]). It is well known that E(w, r) is also a Euclidean ball with centre c w and radius r w which are as follows:
(2.5) c w = 1 − r 2 1 − |w| 2 r 2 w and r w = 1 − |w| 2 1 − |w| 2 r 2 r. By [29, Lemma 2.1], we obtain that for any δ ∈ (0, 1) and y ∈ E(x, δ),
1 − |y| 2 ≤ |x|y − x |x| (1 + |y|) ≤ 2(1 + δ) 1 − δ (1 − |x| 2 ).
By (2.3) and (2.4), we see that y ∈ E(x, δ) if and only if x ∈ E(y, δ). Therefore,
(2.6) 1 − |x| 2 ≤ 2(1 + δ) 1 − δ (1 − |y| 2 ).
2.3. Hyperbolic harmonic mappings. For all ϕ ∈ M(B n ) and f ∈ C 2 (B n , R n ), we have the following Möbius invariance property (cf. [36, Section 2]):
(2.7) ∆ h (f • ϕ)(x) = ∆ h f ϕ(x) .
Obviously, (1.1) implies that
∆ h f (x) = ∆ h f ϕ x (0) = ∆ h (f • ϕ x )(0) = ∆(f • ϕ x )(0).
It is well known that if ψ ∈ C(S n−1 , R n ), then the Dirichlet problem
∆ h f = 0 in B n , f = ψ on S n−1
has a unique solution in C(B n ) and can be represented by
f (x) = P h [ψ](x) = S n−1 P h (x, ξ)ψ(ξ) dσ(ξ) (cf. [8] or [35]), where P h (x, ξ) = 1 − |x| 2 |x − ξ| 2 n−1 .
For f ∈ C 1 (B n , R), the gradient ∇ h f with respect to the hyperbolic metric is given by [26,28,35]). In particular,
∇ h f (x) = (1 − |x| 2 )∇f (x) (cf.(2.8) |∇ h f (x)| = (1 − |x| 2 )|∇f (x)|.
Furthermore, for all ϕ ∈ M(B n ), by [35,Theorem 3.2] or [36, Equation (2.14)], we have
(2.9) |∇ h (f • ϕ)(x)| = ∇ h f ϕ(x) .
From [36,Lemma 3.3] or [26, Theorem 2.1], we can easily obtain the following useful result.
Lemma B. Suppose p ∈ (0, ∞) and δ ∈ (0, 1 2 ). Then there exists a constant µ 06 such that for any f ∈ C 2 (B n , R n ) with ∆ h f = 0,
|∇ h f (x)| p ≤ µ 06 δ −n E(x,δ) |f (y)| p dτ (y)
in B n , where µ 06 = µ 06 (p, δ) (which means that the constant µ 06 depends only on the given parameters p and δ) and dτ denotes the Möbius invariant measure in B n , which is given by
dτ (x) = dν(x) (1 − |x| 2 ) n .
Generalized Bloch spaces and bounded mean oscillation
The purpose of this section is to prove Theorem 1.1. Before the proof, we need the following lemma. where µ 07 is a constant depending only on µ 06 (1, 1 9 ) and µ 06 is the constant from Lemma B.
Proof. For x ∈ B n , by (2.1), we know that
(3.1) Du(x) ≤ n j=1 |∇u j (x)| 2 1 2 .
Hence, to prove this lemma, it suffices to estimate the quantity |∇u j (x)|. Since for a fixed x ∈ B n , ∆ h u(w) − u(x) = 0 in B n , by taking p = 1 and δ = 1 9 , we see from Lemma B and (2.8) that there is a constant µ 08 such that for any w ∈ B n and each j ∈ {1, · · · , n},
(1 − |w| 2 )|∇u j (w)| ≤ µ 08 E(w, 1 9 ) |u j (y) − u j (x)|dτ (y) (3.2)
≤ µ 08 E(w, 1 9 ) |u(y) − u(x)|dτ (y). Therefore, (3.1) guarantees the following:
(1 − |w| 2 )||Du(w)|| ≤ (1 − |w| 2 ) n j=1 |∇u j (w)| 2 1 2 (3.3) ≤ √ nµ 08 E(w, 1 9 ) |u(y) − u(x)|dτ (y) = √ nµ 08 E(w, 1 9 ) |u(y) − u(x)| (1 − |y| 2 ) n dν(y).
Moreover, by (2.5), we know that
(3.4) E(x, 1 9 ) = B n 80 81 − |x| 2 x, 9(1 − |x| 2 ) 81 − |x| 2 ⊂ B n x, 1 4 (1 − |x|) .
Further, for any y ∈ E(x, 1 9 ), (2.6) implies
(3.5) 1 1 − |y| 2 ≤ 5 2(1 − |x|)
.
By letting w = x, it follows from (3.3)∼(3.5) that (1 − |x| 2 )||Du(x)|| ≤ 5 n √ nµ 08 2 n (1 − |x|) n B n (x, 1 4 (1−|x|))
|u(y) − u(x)|dν(y).
By taking µ 07 = µ 08 , we know that the lemma is proved.
Proof of Theorem 1.1. First, we show the "if" part in the theorem. For any x ∈ B n , we see that
1 2 n (1 − |x|) n+1 = 1 8 n (1 − |x|) 1 4 (1 − |x|) n = |B n | 8 n (1 − |x|) B n x, 1 4 (1 − |x|)
. Then by Lemma 3.1, we know that there is a constant µ 07 such that for any x ∈ B n , ||Du(x)|| ≤ 5 n √ nµ 07 |B n | 8 n (1 − |x|) B n x, 1 4 (1 − |x|) B n (x, 1 4 (1−|x|)) |u(y) − u(x)| dν(y).
Then the assumption in the theorem implies
||Du(x)|| ≤ 5 n √ nµ 1 µ 07 2 3n+2 ω (1−|x|) α 4 α |B n |.
Moreover, it follows from α ∈ [1, 2) and [8, Lemma 2.2(2)] that
ω 4 α · (1 − |x|) α 4 α ≤ 4 α ω (1 − |x|) α 4 α ,
and thus, for all x ∈ B n , ||Du(x)|| ≤ 5 n √ nµ 1 µ 07 2 3n+2−2α ω (1 − |x|) α |B n |, which means that u ∈ L ∞,ω B 0 α,a (B n ).
Next, we prove the "only if" part. Let x ∈ B n . For any r ∈ (0, 1 − |x|) and y ∈ B n (x, r), obviously, we have (3.6) |y − x| < r < 1 − |x| and t|x − y| < 1 − |x|,
where t ∈ [0, 1]. For the proof, we need an upper bound on the quantity |u(y)−u(x)|. For this, we let γ [x,y] denote the segment between x and y with the parametrization γ(t) = (1 − t)x + ty, where t ∈ [0, 1]. By the well-known gradient theorem (see, e.g.
[30, Theorem 6.24]), we have that for each j ∈ {1, . . . , n},
γ [x,y] ∇u j (γ), dγ = 1 0 ∇u j γ(t) , γ ′ (t) dt = 1 0 d dt u j • γ(t) dt = u j (y) − u j (x).
Note that
Du γ(t) × γ ′ (t) = ∇u 1 γ(t) , γ ′ (t) . . . ∇u n γ(t) , γ ′ (t) ,
where A × B denotes the product of two matrices A and B. Hence
u(y) − u(x) = 1 0 Du γ(t) × γ ′ (t) dt,
and therefore
(3.7) |u(y) − u(x)| = 1 0 Du γ(t) × γ ′ (t) dt ≤ γ [x,y] Du(γ) · |dγ|.
Moreover, it follows from the assumption u ∈ L ∞,ω B 0 α,a (B n ) that for x ∈ B n ,
||Du(x)|| ≤ 1 ω (1 − |x|) α ||u|| L∞,ωB 0 α,a (B n ) .
We infer from the easy fact |γ ′ (t)| = |x − y| that
|u(y) − u(x)| ≤ ||u|| L∞,ωB 0 α,a (B n ) 1 0 |x − y| ω (1 − |(1 − t)x + ty|) α dt ≤ ||u|| L∞,ωB 0 α,a (B n ) 1 0 |x − y| ω (1 − |x| − t|x − y|) α dt = ||u|| L∞,ωB 0 α,a (B n ) |x−y| 0 1 ω (1 − |x| − t) α dt,
since the inequalities (3.6) and the assumption α ∈ [1, 2) guarantee that
ω (1 − |(1 − t)x + ty|) α ≥ ω (1 − |x| − t|y − x|) α .
This is our needed upper bound on |u(y) − u(x)|. Now, we are ready to finish the proof. Let x − y = η ∈ B n . A similar argument as in the proof of [6, Theorem 1] implies that
1 |B n (x, r)| B n (x,r) |u(y) − u(x)|dν(y) ≤ ||u|| L∞,ωB 0 α,a (B n ) |B n (x, r)| B n (x,r) |x−y| 0 1 ω (1 − |x| − t) α dt dν(y) = ||u|| L∞,ωB 0 α,a (B n ) |B n (0, r)| B n (0,r) |η| 0 1 ω (1 − |x| − t) α dt dν(η) = n||u|| L∞,ωB 0 α,a (B n ) r n r 0 ρ n−1 ρ 0 1 ω (1 − |x| − t) α dt dρ ≤ n||u|| L∞,ωB 0 α,a (B n ) r n r 0 r t ρ n−1 dρ 1 ω (r − t) α dt = ||u|| L∞,ωB 0 α,a (B n ) r n r 0 (r − t)(r n−1 + r n−2 t + · · · + t n−1 ) ω (r − t) α dt ≤ n||u|| L∞,ωB 0 α,a (B n ) r r 0 (r − t) ω (r − t) α dt ≤ n||u|| L∞,ωB 0 α,a (B n ) r r 0 (r − t) α ω (r − t) α (r − t) 1−α dt ≤ nr α−1 ||u|| L∞,ωB 0 α,a (B n ) ω(r α ) r 0 (r − t) 1−α dt ≤ n||u|| L∞,ωB 0 α,a (B n ) (2 − α) r ω(r α ) ,
which is what we need.
Bloch space and hyperbolic metric
The aim of this section is to prove Theorem 1.2. We start this section with a lemma. Then the assumption
||u|| B = sup x∈B n (1 − |x| 2 )||Du(x)|| < ∞ implies that |u(z) − u(0)| ≤ ||u|| B 1 0 |z| 1 − |tz| 2 dt = ||u|| B 2 log 1 + |z| 1 − |z| ,
as needed.
Proof of Theorem 1.2. First, we show the "if" part in the theorem. Since
||u|| B = |u(0)| + sup w∈B n (1 − |w| 2 )||Du(w)|| ,
we know that, to prove this part, we need to estimate the quantity (1−|w| 2 )||Du(w)||.
Since for a fixed x ∈ B n ,
∆ h u(w) − u(x) = 0
in B n , by taking p = 1 and δ = 1 9 , we know from Lemma B, (3.1) and (3.2) that there is a constant µ 08 such that for any w ∈ B n and each j ∈ {1, · · · , n},
(1 − |w| 2 ) 2 ||Du(w)|| 2 ≤ (1 − |w| 2 ) 2 n j=1 |∇u j (w)| 2 ≤ µ 2 08 n j=1 E(w, 1 9 ) |u j (y) − u j (x)|dτ (y) 2 .
Further, Hölder inequality leads to Then the arbitrariness of x in B n ensures the following:
(1 − |w| 2 ) 2 ||Du(w)|| 2 ≤ µ 2 08 E(w, 1 9 ) dτ (y) · E(w, 1 9 ) n j=1 |u j (y) − u j (x)| 2 dτ (y) = µ 2 08 τ E(w,1 9 )E(w, 1 9 ) |u(y) − u(x)| 2 dτ (y) ≤ µ 2 08 τ E(w,1 9 )||u|| B = |u(0)| + sup x∈B n
(1 − |x| 2 )||Du(x)|| ≤ |u(0)| + µ 2 µ 08 9 n 80 n log 10 8 < ∞.
Next, we prove the "only if" part. We shall show this part by applying Lemma 4.1 to the mapping u • ϕ y , where y ∈ B n . Hence we have to verify that u • ϕ y satisfies the conditions in Lemma 4.1. Obviously, the hyperbolic harmonicity of u • ϕ y easily follows from (2.7). It remains to check that u • ϕ y satisfies the second assumption in Lemma 4.1, which is stated in the following claim. To reach this goal, we first obtain from (2.8) and (2.9) that for each j ∈ {1, · · · , n},
sup x∈B n (1 − |x| 2 ) ∇(u j • ϕ y )(x) = sup x∈B n ∇ h (u j • ϕ y )(x) (4.1) = sup x∈B n ∇ h u j ϕ y (x) = sup w∈B n |∇ h u j (w)| = sup w∈B n (1 − |w| 2 ) ∇u j (w) < ∞,
where in the last inequality, the assumption u ∈ B(B n ) and (2.1) are exploited. Then
||u • ϕ y || B 2 = sup x∈B n (1 − |x| 2 ) 2 ||D(u • ϕ y )(x)|| 2 ≤ sup x∈B n (1 − |x| 2 ) 2 n j=1 |∇(u j • ϕ y )(x)| 2 (by (2.1)) ≤ n j=1 sup x∈B n (1 − |x| 2 ) 2 |∇(u j • ϕ y )(x)| 2 = n j=1 sup w∈B n (1 − |w| 2 ) 2 ∇u j (w) 2 .
(by (4.1)) Again, by (2.1) and the assumption u ∈ B(B n ), we have
||u • ϕ y || B 2 ≤ n j=1 sup w∈B n (1 − |w| 2 ) 2 ||Du(w)|| 2 = n ||u|| B 2 < ∞. (4.2)
Hence the claim is proved. Now, we have known that u • ϕ y satisfies the conditions in Lemma 4.1, and so we are ready to finish the proof of the theorem by applying Lemma 4.1. For x, y ∈ B n , let x = ϕ y (z). Obviously, z = ϕ y (x) ∈ B n . Then it follows that
|u(x) − u(y)| = |u • ϕ y (z) − u • ϕ y (0)| ≤ ||u • ϕ y || B 2 log 1 + |z| 1 − |z| (by Lemma 4.1) ≤ √ n||u|| B 2 log 1 + |ϕ y (x)| 1 − |ϕ y (x)| (by (4.2)) ≤ √ n||u|| B 2 ρ(x, y), (by (2.2))
as required.
Generalized Bloch spaces and integral means
The aim of this section is to prove Theorems 1.3 and 1.4. Before the proofs, we need some preparation. ω (log a) β φ α,β,a (r) .
Proof. By direct calculations, we see that
φ ′ α,β,a (r) = (1 − r) α−1 log a 1 − r β−1 β − α log a 1 − r .
It follows from the assumptions in the lemma that φ ′ α,β,a (r) ≤ 0. Hence, φ α,β,a (r) is non-increasing in (0, 1). Then the assumption " ω(t) t being non-increasing in (0, 1)" implies that φ α,β,a (r) ω(φ α,β,a (r)) is also non-increasing in (0,1). Therefore,
M p (r, ||Du||) ≤ ||u|| Lp,ωB β α,a (B n ) ω φ α,β,a (r) ≤ ||u|| Lp,ωB β α,a (B n ) φ α,β,a (0) ω φ α,β,a (0) φ α,β,a (r) = (log a) β ||u|| Lp,ωB β α,a (B n ) ω (log a) β φ α,β,a (r) ,
which is what we want.
Proof of Theorem 1.3. We divide the proof into two cases according to the values of the parameter p.
Case 5.1. p ∈ [1, ∞).
Let y = |y|ξ and γ(t) = ty, where ξ ∈ S n−1 and t ∈ [0, 1]. Since For any y ∈ B n , it follows from the assumption u ∈ L p,ω B β α,a (B n ) and Lemma 5.1 that We conclude from (5.2) and (5.3) that the proof of this theorem is complete.
M ∞ (|y|, u) ≤ |u(0)| +
Proof of Theorem 1.4. We prove this theorem by considering two possibilities according to the values of the parameter q.
Case 5.3. q = ∞.
To prove the theorem in this case, we need to estimate the operator norm: ||Du(x)||. We start with some preparation. First, we shall estimate the quantity |∇u j (x)| p in terms of the integral B n (0, 1+4|x| 4+|x| ) |u j (y)| p dν(y). For this, we take δ = 1 4 in Lemma B. Then by (2.8), we know that there is a constant µ 09 such that for any x ∈ B n and each j ∈ {1, · · · , n},
(5.4) (1 − |x| 2 ) p |∇u j (x)| p ≤ µ 09 E(x, 1 4 ) (1 − |y| 2 ) −n |u j (y)| p dν(y),
where µ 09 = µ 09 µ 06 (p, 1 4 ) . Moreover, by (2.5), we obtain that
E(x, 1 4 ) ⊂ B n 0, 1 + 4|x| 4 + |x| .
Then we infer from (2.6) and (5.4) that
|∇u j (x)| p ≤ 10 n µ 09 3 n (1 − |x|) n+p E(x, 1 4 ) |u j (y)| p dν(y) (5.5) ≤ 10 n µ 09 3 n (1 − |x|) n+p B n (0, 1+4|x| 4+|x| ) |u j (y)| p dν(y).
Second, we shall estimate the integral B n (0, 1+4|x| 4+|x| ) |u j (y)| p dν(y). Since the assumption M p (r, u j ) ≤ µ 10 (1 − r) α as r → 1 − . It follows from the assumption αp > 1 that for each j ∈ {1, · · · , n}, Now, we are ready to get the estimate on ||Du(x)||. We deduce from (2.1), together with the combination of (5.5) and (5.7), that ||Du(x)|| ≤ (1 − r) α+1+ n−1 p as r → 1 − . Now, we conclude from (5.8) and (5.9) that for q ∈ (0, ∞], u ∈ L q,id B 0 α+1+ n−1 p ,a (B n ), and so the proof of this theorem is finished.
Weak uniform boundedness and quasihyperbolic metric
In this section, we shall prove Theorem 1.5.
Proof of Theorem 1.5. The sufficiency in the theorem easily follows from the proof of [23,Theorem 2.8] or the proof of [6,Theorem 2]. So we only need to prove the necessity. Obviously, for every x ∈ B n and y ∈ B n x, d B n (x)
4 , d B n (y) ≥ d B n (x) − 1 4 d B n (x) = 3 4 d B n (x). Hence |x − y| ≤ d B n (x) 4 ≤ min d B n (x) 2 , d B n (y) 2 ,
and so r B n (x, y) ≤ 1 2 .
Then the assumption "u satisfying the weak uniform boundedness property" implies that |u(y) − u(x)| ≤ µ 02 d u(B n ) u(x) ,
where µ 02 is the constant from (1.3). From Lemma 3.1, we deduce that (6.1) ||Du(x)|| ≤ √ nµ 02 µ 07 |B n | 5 n · d u(B n ) u(x) 8 n · d B n (x) .
Since |du(z)| ≤ ||Du(z)|| · |dz|, by taking µ 3 = 5 n 8 n √ nµ 02 µ 07 |B n |, it follows from (6.1) that for all x, y ∈ B n , k u(B n ) (u(x), u(y)) ≤ inf
γ∈Γxy(B n ) γ 1 d u(B n ) u(z) |du(z)| ≤ inf γ∈Γxy(B n ) γ ||Du(z)|| d u(B n ) u(z) |dz| ≤ inf γ∈Γxy(B n ) γ µ 3 d B n (z) |dz| = µ 3 k B n (x, y),
and so Theorem 1.5 is proved.
.
For p ∈ (0, ∞], the generalized Hardy space H p g (B n ) consists of all those functions f : B n → R n such that each f i is measurable, M p (r, f ) exists for all r ∈ (0, 1) and||f || p < ∞, where f = (f 1 , · · · , f n ), ||f || p rξ)| , if p = ∞.
Theorem 1 . 2 .
12Suppose ∆ h u = 0. Then u ∈ B(B n ) if and only if there exists a positive constant µ 2 such that for all x, y ∈ B n , (1.2)
Theorem A. ([16, Theorems 2 and 3] and [17, Theorem 46] or [10, Theorem 5.5]) Suppose p ∈ (0, ∞], α ∈ (1, ∞) and f is an analytic function in D. Then the following statements are equivalent.
Theorem 1. 5 .
5Suppose ∆ h u = 0. Then u satisfies the weak uniform boundedness property in B n if and only if there exists a positive constant µ 3 such that for all
Lemma 3. 1 .
1Suppose ∆ h u = 0. Then there is a constant µ 07 such that for any x ∈ B n , ||Du(x)|| ≤ 5 n √ nµ 07 2 n (1 − |x|) n+1 B n (x, 1 4 (1−|x|)) |u(y) − u(x)| dν(y),
Lemma 4. 1 .
1Suppose ∆ h u = 0 and u ∈ B(B n ). Then for any z ∈ B n , For any z ∈ B n , we let γ [0,z] denote the segment between 0 and z with the parametrization γ(t) = tz, where t ∈ [0, 1]. A similar argument as in (3.7) leads to |u(z) − u(0)| ≤ γ [0,z] Du(γ) · |dγ| = |z| 1 0 Du(tz) dt.
n−1 (1 − t 2 ) −n dt ≤ 9 n 80 n .By letting w = x, we see that(1 − |x| 2 ) 2 ||Du(x)|| 2
Claim 4 . 1 .
41For any y ∈ B n , u • ϕ y ∈ B(B n ).Since u • ϕ y ∈ B(B n ) if and only if ||u • ϕ y || B = sup x∈B n {||D(u • ϕ y )(x)||(1 − |x| 2 )} < ∞,clearly, to check this claim, it needs to estimate the quantity ||D(u • ϕ y )(x)||(1 − |x| 2 ).
Lemma 5 . 1 .
51Suppose u ∈ L p,ω B β α,a (B n ). Then for p ∈ (0, ∞] and r ∈ [0, 1), M p (r, ||Du||) ≤ (log a) β ||u|| Lp,ωB β α,a (B n )
.
Further, Minkowski's integral inequality (cf. [32, A. 1]) gives M p (|y|, u) ≤ |u(that for all y ∈ B n , (5.2) M p (|y|, u) ≤ |u(0)| + (log a) β ||u|| Lp,ωB β α,a (B n ) Case 5.2. p = ∞.
=
|u(0)| + (log a) β ||u|| Lp,ωB β α,a (B n ) ω (log a) β 1 0 |y| φ α,β,a (ρ|y|)dρ.
r) α as r → 1 − , together with the obvious fact |u(x)| ≥ |u j (x)| in B n , impliesM p (r, u j ) ≤ M p (r, u),we see that there exists a constant µ 10 such that(5.6)
− ρ) αp dρ (by (5.6)) ≤ n5 αp−1 µ p 10
to the case of complex-valued harmonic mappings ([5, Theorem 4] and [7, Theorem 3]). Recently, Chen and Rasila generalized [5, Theorem 4], [7, Theorem 3] and [12, Theorem 1] to the setting of the solutions to the non-homogenous Yukawa PDE ∆f = λf , where λ : B n → R is a nonnegative continuous function with sup x∈B n {λ(x)} < ∞ ([6, Theorem 1]).
αp−1 (αp − 1)(1 − |x|) αp−1 . (since αp > 1)
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arxiv |
On the missing log in upper tail estimates
24 May 2019 29 February, 2016; revised January 11, 2019
Lutz Warnke
On the missing log in upper tail estimates
24 May 2019 29 February, 2016; revised January 11, 2019
In the late 1990s, Kim and Vu pioneered an inductive method for showing concentration of certain random variables X. Shortly afterwards, Janson and Ruciński developed an alternative inductive approach, which often gives comparable results for the upper tail P(X ≥ (1 + ε)EX). In some cases, both methods yield upper tail estimates which are best possible up to a logarithmic factor in the exponent, but closing this narrow gap has remained a technical challenge. In this paper we present a BK-inequality based combinatorial sparsification idea that can recover this missing logarithmic term in the upper tail.As an illustration, we consider random subsets of the integers {1, . . . , n}, and prove sharp upper tail estimates for various objects of interest in additive combinatorics. Examples include the number of arithmetic progressions, Schur triples, additive quadruples, and (r, s)-sums.
Introduction
Concentration inequalities are of great importance in discrete mathematics, theoretical computer science, and related fields. They intuitively quantify random fluctuations of a given random variable X, by bounding the probability that X differs substantially from its expected value µ = EX. In combinatorial applications, X often counts certain objects (e.g., the number of subgraphs or arithmetic progressions), in which case the random variable X can usually be written as a low-degree polynomial of many independent random variables. In this context concentration inequalities with exponentially small estimates are vital (e.g., to make union bound arguments amenable), and here Kim and Vu [20,31,33] achieved a breakthrough in the late 1990s. Their powerful concentration inequalities have since then, e.g., been successfully applied to many combinatorial problems, been included in standard textbooks, and earned Vu the George Pólya Prize in 2008.
In probabilistic combinatorics, the exponential rate of decay of the lower tail P(X ≤ µ − t) and upper tail P(X ≥ µ + t) have received considerable attention, since they are of great importance in applications (of course, this is also an interesting problem in concentration of measure). The behaviour of the lower tail is nowadays well-understood due to the celebrated Janson-and Suen-inequalities [11,22,18,17,13]. By contrast, the behaviour of the 'infamous' upper tail has remained a well-known technical challenge (see also [14,12]). Here the inductive method of Kim and Vu [20,33] from around 1998 often yields inequalities of the form
P(X ≥ (1 + ε)µ) ≤ exp −c(ε)µ 1/q ,(1)
where q ≥ 1 is some constant. In 2000, Janson and Ruciński [15] developed an alternative inductive approach, which often gives comparable results for the upper tail, i.e., which recovers (1) up to the usually irrelevant numerical value of the parameter c. Studying the sharpness of the tail inequality (1) is an important problem according to Vu (see Section 4.8 in [33]). In fact, one main aim of the paper [15] was 'to stimulate more research into these methods' since 'neither of [them] seems yet to be fully developed'. In other words, Janson and Ruciński were asking for further improvements of the aforementioned fundamental proof techniques (the papers [15,33] already contained several tweaking options for decreasing q).
In this paper we address this technical challenge in cases where the inductive methods of Kim-Vu and Janson-Ruciński are nearly sharp. The crux is that, for several interesting classes of examples (naturally arising, e.g., in additive combinatorics), the upper tail inequality (1) is best possible up to a logarithmic factor in the exponent. Closing such narrow gaps has recently become an active area of research in combinatorial probability (see, e.g, [14,12,16,6,7,36]). The goal of this paper is to present a new idea that can add such missing logarithmic terms to the upper tail. From a conceptual perspective, this paper thus makes a new effect amenable to the rich toolbox of the Kim-Vu and Janson-Ruciński methods (we believe that our techniques will be useful elsewhere). For example, under certain somewhat natural technical assumptions, our methods allow us to improve the classical upper tail inequality (1) to estimates of the form P(X ≥ (1 + ε)µ) ≤ exp −c(ε) min µ, µ 1/q s with s ∈ log n, log(1/p) ,
where the reader may wish to tentatively think of the parameters n = ω(1) and p = o(1) as those in the binomial random graph G n,p (here some extra assumptions are necessary, since there are examples where (1) is sharp, see Sections 1.1 and 6.1). This seemingly small improvement of (1) is conceptually important, since in several interesting applications the resulting inequality is best possible up to the value of c. Indeed, as we shall see, sharp examples with P(X ≥ (1+ε)µ) = exp −Θ(min µ, µ 1/q log(1/p)) for ε = Θ(1) naturally arise when X counts various objects of great interest in additive combinatorics, such as the number of arithmetic progressions (of given length) or additive quadruples in random subsets of the integers [n] = {1, . . . , n}.
In the remainder of this introduction we illustrate our methods with some applications, outline our highlevel proof strategy, and discuss the structure of this paper. Noteworthily, our proof techniques do not solely rely on induction, but a blend of combinatorial and probabilistic arguments.
Flavour of the results
We now illustrate the main flavour of our upper tail results with some concrete examples. Many important counting problems can be rephrased as the number of edges induced by the random induced subhypergraph H p = H[V p (H)] (see, e.g., [14,23,16,36,38]), where V p (H) denotes the binomial random subset where each vertex v ∈ V (H) is included independently with probability p. Our methods yield the following upper tail inequality for H p , which extends one of the main results from [36] for the special case q = 2, and sharpens one of the principle results of Janson and Ruciński [16] by a logarithmic factor in the exponent.
Theorem 1 (Counting edges of random induced subhypergraphs). Let 1 ≤ q < k and γ, D > 0. Assume that H is a k-uniform hypergraph with v(H) ≤ N vertices and e(H) ≥ γN q edges. Suppose that ∆ q (H) ≤ D, where ∆ q (H) denotes the maximum number of edges of H that contain q given vertices. Let X = e(H p ) and µ = EX. Then for any ε > 0 there is c = c(ε, k, γ, D) > 0 such that for all p ∈ (0, 1] we have P(X ≥ (1 + ε)µ) ≤ exp −c min µ, µ 1/q log(e/p) .
(
This upper tail inequality is conceptually best possible in several ways. First, the restriction to q < k is necessary (see Section 6.1 for a counterexample when q = k), Second, in several important applications (3) is sharp (yields the correct exponential rate of decay), i.e., there is a matching lower bound of form P(X ≥ (1 + ε)µ) ≥ ½ {1≤(1+ε)µ≤e(H)} exp −C(ε) min µ, µ 1/q log(e/p) ,
where the restriction 1 ≤ (1 + ε)µ ≤ e(H) is natural. 1 In particular, letting the edges of the hypergraph H with vertex-set V (H) = [n] encode classical objects from additive combinatorics and Ramsey Theory, sharp examples of type (3)-(4) include the number of k-term arithmetic progressions, Schur triples x + y = 2z, additive quadruples x 1 + x 2 = y 1 + y 2 , and (r, s)-sums x 1 + · · · + x r = y 1 + · · · + y s in the binomial random subset [n] p = V p (H) of the integers; see Section 1.1.1 and 6.1 for more details/concrete examples.
The two expressions in the exponent of the upper tail (3)-(4) correspond to different phenomena. 2 Namely, in some range we expect that X = e(H p ) is approximately Poisson, in which case P(X ≥ 2µ) decays roughly like exp(−cµ). Similarly, the exp(−cµ 1/q log(1/p)) = p cµ 1/q term intuitively corresponds to 'clustered' behaviour (see also [36,28,12]), where few vertices U ⊆ V p (H) induce many edges in H p = H[V p (H)]: e.g., in each of the above-mentioned examples there always is such a set with |U | = cµ 1/q and e(H[U ]) ≥ 2µ, which readily implies P(X ≥ 2µ) ≥ P(U ⊆ V p (H)) = p cµ 1/q . Note that classical tail inequalities of form (1) fail to handle these phenomena properly (lacking Poisson behaviour and the extra log(1/p) term).
Upper tail examples from additive combinatorics and Ramsey theory
In the following exemplary upper tail bounds (5)- (8) we tacitly allow the implicit constants to depend on ε.
Example 2. Arithmetic progressions (APs) are central objects in additive combinatorics. Given k ≥ 3, let X = X n,k,p denote the number of arithmetic progressions of length k in the binomial random subset [n] p of the integers (to clarify: we count k-subsets {x 1 , . . . , x k } ⊆ [n] p forming APs); note that µ = EX = Θ(n 2 p k ). Then, for any ε > 0 and p = p(n) ∈ (0, 1] satisfying 1 ≤ (1 + ε)µ ≤ X n,k,1 , we have P(X ≥ (1 + ε)µ) = exp −Θ min µ, µ 1/2 log(1/p) .
Example 3. Schur triples {x, y, z} ⊆ [n] with x+y = z (where x = y) are classical objects in Number theory and Ramsey theory (see, e.g., [10] and [9,25]). Let X = X n,p denote the number of Schur triples in [n] p ; note that µ = EX = Θ(n 2 p 3 ). Then, for any ε > 0 and p = p(n) ∈ (0, 1] satisfying 1 ≤ (1 + ε)µ ≤ X n,1 , we have P(X ≥ (1 + ε)µ) = exp −Θ min µ, µ 1/2 log(1/p) .
The same tail bound also holds for ℓ-sums (studied, e.g., in [1]), where the 3-element subsets satisfy x+y = ℓz.
Example 4. Additive quadruples are 4-subsets {x 1 , x 2 , y 1 , y 2 } ⊆ [n] satisfying x 1 +x 2 = y 1 +y 2 . The number of these quadruples is also called additive energy, which is an important quantity in additive combinatorics (see, e.g., [2,5]). Let X = X n,p denote the number of additive quadruples in [n] p ; note that µ = EX = Θ(n 3 p 4 ). Then, for any ε > 0 and p = p(n) ∈ (0, 1] satisfying 1 ≤ (1 + ε)µ ≤ X n,1 , we have
P(X ≥ (1 + ε)µ) = exp −Θ min µ, µ 1/3 log(1/p) .(7)
Example 5. (r, s)-sums are (r+s)-subsets {x 1 , . . . , x r , y 1 , . . . , y 2 } ⊆ [n] satisfying x 1 +· · ·+x r = y 1 +· · ·+y s . In the special case r = s the number of these sets is called 2r-fold additive energy, which is useful in the context of Roth's theorem (see, e.g., [5]). Given r, s ≥ 1 satisfying r + s ≥ 3, let X = X n,r,s,p denote the number of (r, s)-sums in [n] p ; note that µ = EX = Θ(n r+s−1 p r+s ). Then, for any ε > 0 and p = p(n) ∈ (0, 1] satisfying 1 ≤ (1 + ε)µ ≤ X n,r,s,1 , we have
P(X ≥ (1 + ε)µ) = exp −Θ min µ, µ 1/(r+s−1) log(1/p) .(8)
Similar tail bounds also hold for integer solutions of linear homogeneous systems, see Section 6.1 for the details.
Subgraph counts in random graphs: sub-Gaussian type upper tail bounds
As a side-product, our proof techniques also yield new results with a slightly different flavour. To illustrate this with subgraph counts in the binomial random graph G n,p , let X = X H denote the number of copies of H in G n,p . Set µ = EX. Here sub-Gaussian type upper tail estimates 3 of the form
P(X ≥ µ + t) ≤ C exp(−ct 2 / Var X)(9)
have been extensively studied [24,31,15,26,19,37,38] during the last decades, usually with emphasis on small deviations of form √ Var X ≤ t = o(µ), say (differing from the large deviations regime t = Θ(µ) considered in the classical upper tail problem for subgraph counts). In particular, for so-called 'strictly balanced' graphs H three different approaches [31,15,26] have been developed during the years 2000-2012, which each establish a form of inequality (9) for t ≤ µ = O(log n). Our methods allow us to break this logarithmic barrier slightly, answering a question of Janson and Ruciński [13]; see Section 6.2.1 for more details.
Theorem 6 (Subgraph counts: sub-Gaussian type upper tail bounds). For any strictly balanced graph H there are n 0 , c, C, ξ > 0 such that inequality (9) holds whenever n ≥ n 0 and 0 < t ≤ µ ≤ (log n) 1+ξ .
Glimpse of the proof strategy
In contrast to most of the previous work, in this paper we take a more combinatorial perspective to concentration of measure (and avoid induction via a more iterative point of view). Our high-level proof strategy proceeds roughly as follows. In the deterministic part of the argument, we define several 'good' events E i = E i (H, ε), and show that the following implication holds:
all E i hold =⇒ X < (1 + ε)EX.(10)
In the probabilistic part of the argument, we show that for some suitable parameter Ψ we have
P(some E i fails) ≤ exp(−Ψ).(11)
Combining both parts then readily yields an exponential upper tail estimate of the form
P(X ≥ (1 + ε)EX) ≤ P(some E i fails) ≤ exp(−Ψ).
In this paper we illustrate the above approach by implementing (10)-(11) in a general Kim-Vu/Janson-Ruciński type setup. To communicate our ideas more clearly, our below informal discussion again uses the simpler random induced subhypergraph setup (a more detailed sketch is given in
Sections 3.1.2-3.1.3).
For the deterministic part (10), we shall crucially exploit a good event E Q,ε of the following form: all subhypergraphs with 'small' maximum degree have 'not too many' edges, i.e., that e(J ) < (1 + ε/2)EX holds for all J ⊆ H p with ∆ 1 (J ) ≤ Q, say. Our sparsification idea proceeds roughly as follows. First, using combinatorial arguments (and further good events) we find a nested sequence of subhypergraphs
H p = J q ⊇ J q−1 ⊇ · · · ⊇ J 2 ⊇ J 1 ,(12)
which gradually decreases the maximum degree down to ∆ 1 (J 1 ) ≤ Q. The crux is that E Q,ε then implies e(J 1 ) < (1 + ε/2)EX. In the second step we exploit various good events (and properties of the constructed sequence) to show that we obtained J 1 by removing relatively few edges from H p , such that
X = e(H p ) = e(J 1 ) + 1≤j<q e(J j+1 \ J j ) < (1 + ε/2)EX + (ε/2)EX = (1 + ε)EX.(13)
In fact, the combinatorial arguments leading to (12)-(13) develop a 'maximal matching' based sparsification idea from [36], which is key for handling some vertices of H p with exceptionally high degrees, say. The probabilistic part (11) works hand in hand with the above deterministic arguments. Similar to E Q,ε , we shall throughout work with 'relative estimates', i.e., which are valid for all subhypergraphs of H p satisfying some extra properties (e.g., that ∆ j (J ) ≤ R j holds for all J ⊆ H p with ∆ j+1 (J ) ≤ R j+1 ). These estimates are crucial for bringing combinatorial arguments of type (12)-(13) into play (instead of relying solely on inductive reasoning), and they hinge on a concentration inequality from [36]. Perhaps surprisingly, this inequality allows us to estimate P(¬E Q,ε ) and similar 'relative' events without taking a union bound over all subhypergraphs. For the matching based sparsification idea briefly mentioned above, we exploit the fact that the relevant 'matchings' guarantee the 'disjoint occurrence' of suitably defined events. This observation allows us to estimate the probability of certain 'bad' events via BK-inequality based moment arguments.
Finally, in our probabilistic estimates the logarithmic terms in (2)-(3) arise in a fairly delicate way (which comes as no surprise, since there are examples where (1) is sharp). We now illustrate the underlying technical idea for binomial random variables X ∼ Bin(n, p) with µ = np, where for x ≥ e(e/p) α µ we have
P(X ≥ x) ≤ n x p x ≤ eµ x x ≤ p e αx = exp −αx log e/p .
Our proofs apply this 'overshooting the expectation yields extra terms in the exponent' idea to a set of carefully chosen auxiliary random variables. As the reader can guess, the technical details are, e.g., complicated by the fact that the edges of H p are not independent, and that we may not assume x ≫ µ.
Guide to the paper
In Section 2 we introduce our key probabilistic tools. In Section 3 we give a fairly detailed proof outline, and present our main combinatorial and probabilistic arguments in the random induced subhypergraphs setup. In Section 4 we then extend the discussed arguments to a more general setup. In Section 5 we derive some concrete upper tail inequalities, which in Section 6 are then applied to several pivotal examples. The reader interested in our proof techniques may wish to focus on Section 3, which contains our core ideas and arguments. The reader interested in applications may wish to skip to Section 6, where the 'easyto-apply' concentration inequalities of Section 5.1 are used in several different examples. Finally, the reader interested in comparing our results with the literature may wish to focus on the general setup of Section 4.1 and the concentration inequalities in Section 5.2.
Probabilistic preliminaries 2.1 A Chernoff-type upper tail inequality
In this subsection we state a powerful Chernoff-type upper tail inequality from [36]. It might be instructive to check that, for sums X = i∈A ξ i of independent random variables ξ i ∈ [0, 1], inequality (14) below reduces to the classical Chernoff bound (writing i ∼ j if i = j, for Y i = ξ i , I = A and C = 1 we have X = Z C ). We think of ∼ as a 'dependency relation': α ∼ β implies that the random variables Y α and Y β are independent. For indicator random variables Y α ∈ {0, 1} the condition max β∈J α∈J :α∼β Y α ≤ C essentially ensures that each variable Y β with β ∈ J 'depends' on at most C variables Y α with α ∈ J . Intuitively, Z C defined below thus corresponds to an approximation of X = α∈I Y α with 'bounded dependencies'.
Theorem 7. Given a family of non-negative random variables (Y α ) α∈I with α∈I EY α ≤ µ, assume that ∼ is a symmetric relation on I such that each Y α with α ∈ I is independent of {Y β : β ∈ I and β ∼ α}. Let
Z C = max α∈J Y α , where the maximum is taken over all J ⊆ I with max β∈J α∈J :α∼β Y α ≤ C. Set ϕ(x) = (1 + x) log(1 + x) − x.
Then for all C, t > 0 we have
P(Z C ≥ µ + t) ≤ exp − ϕ(t/µ)µ C = e −µ/C · eµ µ + t (µ+t)/C ≤ min exp − t 2 2C(µ + t/3) , 1 + t 2µ −t/(2C) ≤ 1 + t µ −t/(4C) .(14)
Remark 8. In applications there often is a family of independent random variables (ξ σ ) σ∈A such that each Y α is a function of (ξ σ ) σ∈α . Then it suffices to define α ∼ β if α ∩ β = ∅ (as α ∼ β implies that Y α and Y β depend on disjoint sets of variables ξ σ ).
Remark 9. Theorem 7 remains valid after weakening the independence assumption to a form of negative correlation: it suffices if E( i∈[s] Y αi ) ≤ i∈[s] EY αi for all (α 1 , . . . , α s ) ∈ I s satisfying α i ∼ α j for i = j. For example, writing α ∼ β if α ∩ β = ∅, it is not hard to check that this weaker condition holds for variables
of form Y α = w α ½ {α∈Hm} , where the uniform model H m = H[V m (H)]
is defined as in Section 3.5.
Remark 10.
Replacing the assumption α∈I EY α ≤ µ of Theorem 7 with α∈I λ α ≤ µ and min α∈I λ α ≥ 0, the correlation condition of Remark 9 can be further weakened to
E( i∈[s] Y αi ) ≤ i∈[s] λ αi .
Remark 11. Note that inequality (14) implies ϕ(ε) ≥ ε 2 /[2(1 + ε/3)] ≥ min{ε 2 , ε}/3 for ε ≥ 0.
Remarks 9-10 suggest that the proof of Theorem 7 is fairly robust (it exploits independence only in a limited way; see also the discussion in [36] and the proof of Lemma 4.5 in [34]).
The BK-inequality
In this subsection we state a convenient consequence of the BK-inequality of van den Berg and Kesten [3] and Reimer [21]. As usual in this context, we consider a sample space Ω = Ω 1 × · · · × Ω M with finite Ω i , and write ω = (ω 1 , . . . , ω M ) ∈ Ω. Given an event E ⊆ Ω and an index set I ⊆ [M ] = {1, . . . , M }, we define E| I = ω ∈ E : for all π ∈ Ω we have π ∈ E whenever π j = ω j for all j ∈ I .
In intuitive words, the event E| I occurs if knowledge of the variables indexed by I already 'guarantees' the occurrence of E (note that all other variables are irrelevant for E| I ). Given a collection (E i ) i∈C of events, for the purposes of this paper it seems easiest to introduce the convenient definition
⊡ i∈C E i = there are pairwise disjoint I i ⊆ [M ] such that i∈C E i | Ii occurs .(15)
The event ⊡ i∈C E i intuitively states that all E i 'occur disjointly', i.e., that there are disjoint subsets of variables which guarantee the occurrence of each event E i (the definition of ⊡ sidesteps that the usual box product is, in general, not associative). The general BK-inequality of Reimer [21] implies the following estimate.
Theorem 12. Let P be a product measure on Ω = Ω 1 ×· · ·×Ω M with finite Ω i . Then for any collection (E i ) i∈C of events we have
P ⊡ i∈C E i ≤ i∈C P(E i ).(16)
Remark 13. For increasing events E i , [4] implies that inequality (16) also holds for P assigning equal probability to all outcomes ω ∈ {0, 1} M with exactly m ones (as usual, an event E is called increasing if for all ω ∈ E and π ∈ Ω we have π ∈ E whenever ω j ≤ π j for all j ∈ [M ]).
Core ideas and arguments
In this section we present our core combinatorial and probabilistic arguments in a slightly simplified setup. Our main focus is on the new proof ideas and methods (which we believe are more useful to the reader than the theorems), so we defer applications and concrete upper tail inequalities to Sections 5-6. This organization of the paper also makes the extension to the more general setup of Section 4 more economical. Indeed, similar to the high-level proof strategy discussed in Section 1.2, the main results of this section are Theorem 15 of form P(X ≥ (1 + ε)EX) ≤ i P(¬E i ) and Theorem 18 of form P(¬E i ) ≤ exp(−Ψ i ). Together they yield upper tail inequalities, and in Section 4.2 we adapt both to our more general setup. In Section 3.1 we give a detailed proof overview, and introduce the simpler random induced subhypergraphs setup (where our main arguments and ideas are more natural). As a warm-up, in Section 3.2 we revisit existing inductive concentration methods, and reinterpret some of the underlying ideas. Section 3.3 contains our key combinatorial arguments, which hinge on 'sparsification' ideas and the BK-inequality. In Section 3.4 these arguments are complemented by probabilistic estimates, which rely on the Chernoff-type tail inequality Theorem 7. Finally, in Section 3.5 we demonstrate that our proofs are somewhat 'robust'.
Overview
H p = H[V p (H)].(17)
Given non-negative weights (w f ) f ∈H , for every G ⊆ H we set
w(G) = f ∈G w f ½ {f ∈Hp} ,(18)
where our main focus is on the weighted number of induced edges w(H) = w(H p ). The 'unweighted' case with w f = 1 occurs frequently in the literature (see, e.g., [14,23,16,36,38]), where the random variable w(H) = e(H p ) simply counts the number of edges of H induced by V p (H). Our arguments will also carry over to the uniform variant H m = H[V m (H)] defined in Section 3.5 (see Remark 19).
To formulate our results, we need some more notation and definitions. As usual, we write
Γ U (H) = {f ∈ H : U ⊆ f },(19)∆ j (H) = max U⊆V (H):|U|=j |Γ U (H)|.(20)
In concrete words, Γ U (H) corresponds to the set of all edges f ∈ H that contain the vertex subset U ⊆ V (H), and ∆ j (H) denotes the maximum number of edges that contain j given vertices (which we think of as a 'maximum degree' parameter). Inspired by [15,20,31,33], we now define the following two crucial assumptions (P') and (Pq), where q ∈ N is a parameter:
(P') Assume that max f ∈H |f | ≤ k, max f ∈H w f ≤ L and v(H) ≤ N . Define µ = Ew(H) and µ j = max U⊆V (H):|U|=j f ∈ΓU (H) p |f |−|U| .(21)
(Pq) Assume that ∆ q (H) ≤ D.
Property (P') ensures that every edge f ∈ H has at most k vertices, that the associated edge weights satisfy 0 ≤ w f ≤ L, and that H contains at most v(H) ≤ N vertices. Although we shall not assume this, our main focus is on the common case where k + L = O(1) and N = ω(1) holds. Property (Pq) will be useful when D = O(1) holds for q < k (this is trivial for q = k). The key parameters µ j intuitively quantify the 'dependencies' between the edges, and we think of them as average variants of the 'maximum degree' parameter ∆ j (H p ) from (20). To see this, note that P(f ∈ H p | U ⊆ V p (H)) = p |f |−|U| , so (21) equals
µ j = max U⊆V (H):|U|=j E |Γ U (H p )| U ⊆ V p (H) .(22)
In concrete words, after conditioning on the presence of any vertex subset U ⊆ V p (H) of size |U | = j, the expected number of edges in H p that contain U is at most µ j (for this reason, µ j can be interpreted as the 'maximum average effect' of any j vertices or variables, see also [20,33]). For example, if the edges of the k-uniform hypergraph H = H n correspond to k-term arithmetic progressions, then we can take V (H) = [n], N = n, L = 1, µ = Θ(n 2 p k ) and µ j = Θ(n 2−j p k−j ) for 1 ≤ j ≤ q = 2 (note that ∆ 2 (H) = O(1) holds).
The basic form of our tail estimates
In this subsection we discuss the approximate form of our upper tail estimates. As we shall see in Section 3.2, for hypergraphs H with ∆ q (H) ≤ D the usual inductive concentration of measure methods [20,15,33] yield basic inequalities of the following form (omitting several technicalities). Given positive parameters (R j ) 1≤j≤q with R q ≥ D, for every ε > 0 there are positive constants a = a(ε, k) and b = b(k) such that roughly
P(e(H p ) ≥ (1 + ε)µ) ≤ exp −aµ/R 1 + 1≤j<q µ j R j bRj /Rj+1 ,(23)
say (see (76) of Claim 33; the freedom of choosing the parameters (R j ) 1≤j≤q is part of the method, though one naturally aims at roughly µ/R 1 ≈ R j /R j+1 ). The 'prepackaged versions' of these inequalities usually assume that the parameters satisfy roughly µ/R 1 ≥ λ and R j ≥ max{2µ j , λR j+1 } (see, e.g., Theorem 4.2 in [33] or Theorem 3.10 in [15]). In this case there are positive constants c = c(a, b) and C = C(q) such that
P(e(H p ) ≥ (1 + ε)µ) ≤ C exp −cλ .(24)
The punchline of this paper is that we can often improve the exponential decay of (24) if stronger bounds than R j ≥ 2µ j hold. For example, setting λ ≈ µ 1/q and R j ≈ λ q−j (similar to, e.g., the proof of Corollary 6.3 in [33] or Theorem 2.1 in [32]), in the applications of Section 6.1 we naturally arrive at bounds of form
max 1≤j<q µ j R j ≈ max 1≤j<q µ j µ (q−j)/j = O(p α ).(25)
It might be instructive to check that (25) holds with α = 1/2 for k-term arithmetic progressions with k ≥ 3. Intuitively, replacing R j ≥ 2µ j by the stronger assumption (25) improves the exponential decay of the sumterms in (23) by a factor of roughly log(1/p) for small p. Hence the exp −aµ/R 1 term in (23) is the main obstacle for improving inequality (24). Here our new 'sparsification' based approach is key: after some technical work it essentially allows us to replace R 1 by
Q 1 = max R 1 / log(1/p), B ,
where B ≥ 1 is some constant (of course, we later need to be a bit careful when p ≈ 1 holds, e.g., replacing log(1/p) with log(e/p), say). More concretely, assuming (25), for µ/R 1 ≥ λ, R j ≥ λR j+1 and p = o(1) we eventually arrive (ignoring some technicalities) at a bound that is roughly of the form
P(e(H p ) ≥ (1 + ε)µ) ≤ exp −aµ/Q 1 + 1≤j<q µ j R j bRj /Rj+1 + µ j R j aµ/R1 ≤ C exp −c min µ, λ log(1/p) ,(26)
with c = c(a, b, α, B) > 0 and C = q (see (80) of Theorem 34). In words, (26) essentially adds a logarithmic factor to the exponent of the classical bound (24). This improvement of (23)- (24) is conceptually important, since in several interesting examples the resulting estimate (26) is qualitatively best possible (see Section 6.1).
Sketch of the argument
In this subsection we expand on the high-level proof strategy from Section 1.2, and give a rough sketch of our main combinatorial line of reasoning (the full details are deferred to Sections 3.2-3.4 and 4.2). As we shall argue in Section 3.2, at the conceptual heart of the usual inductive concentration approaches lies the
following combinatorial 'degree' event D j : ∆ j+1 (H p ) ≤ R j+1 implies ∆ j (H p ) ≤ R j . Given a hypergraph H with ∆ q (H) ≤ R q ,
for the induced number of edges e(H p ) the basic idea is that an iterative application of the events D q−1 ∩ · · · ∩ D 1 reduces the upper tail problem to
P(e(H p ) ≥ (1 + ε)µ) ≤ P(e(H p ) ≥ (1 + ε)µ and ∆ q (H p ) ≤ R q ) ≤ P(e(H p ) ≥ (1 + ε)µ and ∆ 1 (H p ) ≤ R 1 ) + 1≤j<q P(¬D j ).(27)
It turns out that all the probabilities on the right hand side of (27) can easily be estimated by the concentration inequality Theorem 7 (see Claim 14 and Theorem 18), which eventually yields a variant of the upper tail estimate (23). As before, the crux is that smaller values of the 'maximum degree' R 1 translate into better tail estimates. To surpass the usual inductive approaches, similar to (26) our plan is thus to reduce the 'degree bound' R 1 down to Q 1 , and here our new 'sparsification idea' will be key, achieving this 'degree reduction' by deleting up to εµ/2 edges. Our starting point is the observation that, via Theorem 7, we can strengthen the degree event D j to all subhypergraphs G ⊆ H p (see Claim 14 and Theorem 18). Namely, let D + j denote the event that ∆ j+1 (G) ≤ Q j+1 implies ∆ j (G) ≤ Q j for all G ⊆ H p . A crucial aspect of our argument is that the events D j , D + j work hand in hand with the following combinatorial 'sparsification' event E q :
∆ 1 (H p ) ≤ R 1 implies existence of a subhypergraph G ⊆ H p with e(H p \ G) ≤ εµ/2 and ∆ q−1 (G) ≤ Q q−1 (tacitly assuming q ≥ 2). Intuitively, E q states that the deletion of 'few' edges reduces the degree ∆ q−1 (H p ) down to ∆ q−1 (G) ≤ Q q−1 .
The basic combinatorial idea of our approach is roughly as follows (see Section 3.3 for the more involved details). We first (i) obtain the coarse degree bound ∆ 1 (H p ) ≤ R 1 via an iterative application of the degree events D q−1 ∩ · · · ∩ D 1 , then (ii) exploit the sparsification event E q to find a subhypergraph G ⊆ H p with e(H p \ G) ≤ εµ/2 and ∆ q−1 (G) ≤ Q q−1 , and finally (iii) deduce the improved degree bound ∆ 1 (G) ≤ Q 1 via an iterative application of the degree events D + q−2 ∩ · · · ∩ D + 1 . Taking into account that we obtain G ⊆ H p by deleting up to εµ/2 edges, for hypergraphs H with ∆ q (H) ≤ R q we eventually arrive at
P(e(H p ) ≥ (1 + ε)µ) ≤ P(e(G) ≥ (1 + ε/2)µ and ∆ 1 (G) ≤ Q 1 for some G ⊆ H p ) + 1≤j<q P(¬D j ) + P(¬E q ) + 1≤j<q−1 P(¬D + j ).(28)
The crux is that we can again obtain good tail estimates for P(e(G) ≥ (1 + ε/2)µ · · · ) and P(¬D j ) + P(¬D + j ) via Theorem 7 (see Claim 14 and Theorem 18), so in (28) it remains to bound P(¬E q ).
To estimate the probability that the sparsification event E q fails, we shall rely on combinatorial arguments and the BK-inequality, developing a 'maximal matching' based idea from [36]. Simplifying slightly (see Section 3.3.1 for the full details), for any vertex set
U ⊆ V (H) with |U | = q − 1 we tentatively call K U ⊆ Γ U (H) = {f ∈ H : U ⊆ f } with |K U | = r an r-star,
where we set r = Q q−1 for brevity. The basic idea is to take a maximal vertex disjoint collection of r-stars in H p , which we denote by M (to clarify: the edges from any two distinct r-stars K U , K W ∈ M are vertex disjoint), and remove all edges f ∈ H p that are incident to M, i.e., which share at least one vertex with some r-star from M. Denoting the resulting subhypergraph by G ⊆ H p , using maximality of M it is not difficult to argue that ∆ q−1 (G) < r = Q q−1 holds (otherwise we could add another r-star to M). Furthermore, by construction the deleted number of edges is at most
e(H p \ G) ≤ KU ∈M f ∈KU v∈f |Γ {v} (H p )| ≤ |M| · r · k · ∆ 1 (H p ).(29)
Since the event E q presupposes ∆ 1 (H p ) ≤ R 1 , we thus see that |M| ≤ εµ/(2rkR 1 ) implies |H p \ G| ≤ εµ/2. It remains to estimate the probability that |M| is big, and here we shall exploit the fact that the r-stars K U ∈ M satisfy two properties: they (i) are pairwise vertex disjoint, and (ii) each 'guarantee' that |Γ U (H p )| ≥ r holds. Intuitively, the point of (i) and (ii) is that |M| events of from |Γ U (H p )| ≥ r 'occur disjointly' in the sense of Section 2.2, which allows us to bring the BK-inequality (16) into play. Indeed, by analyzing a ⊡-based moment of U:|U|=q−1 ½ {|ΓU (Hp)|≥r} , we then eventually obtain sufficiently good estimates for P(¬E q ), as desired (see the proofs of Lemma 16 and inequality (48) of Theorem 18).
As the reader can guess, the actual details are more involved. For example, instead of just E q for ∆ q−1 (·), we also need to consider similar sparsification events for the others degrees ∆ j (·) with 1 ≤ j < q. In fact, analogous to D + j , these events must moreover apply to all subhypergraphs G ⊆ H p simultaneously (see E j,ℓ (x, r, y, z) defined in Section 3.3). Furthermore, due to technical reasons, the decomposition (28) requires some extra bells and whistles (see (33) of Theorem 15). Finally, we have also ignored how Theorem 7 and the BK-inequality (16) eventually allow us to convert the decompositions (27)
Inductive concentration proofs revisited
The goal of this warm-up section is to reinterpret the classical inductive concentration proofs from [15,20,33] using the following 'degree intuition': an (improved) upper bound for ∆ j+1 (H p ) and ∆ 1 (H p ) translates into an improved upper tail estimate for ∆ j (H p ) and w(H p ), respectively. We exemplify this with the following claim, which is usually stated for G = H p only (the proof of is based on routine applications of Theorem 7, and thus deferred to Section 3.4). We find inequalities (30)-(31) below remarkable, since they intuitively yield bounds for all subhypergraphs G ⊆ H p without taking a union bound.
Claim 14.
Given H, assume that (P') holds. Then for all t, x, y > 0 and 1 ≤ j < k we have
P w(G) ≥ µ + t and ∆ 1 (G) ≤ y for some G ⊆ H p ≤ 1 + t µ −t/(4Lky) ,(30)P ∆ j (G) ≥ µ j + x and ∆ j+1 (G) ≤ y for some G ⊆ H p ≤ N j 1 + x µ j −x/(4ky) .(31)
Now, by a straightforward iterative degree argument similar to (27), we obtain the simple estimate
P w(G) ≥ µ + t and ∆ q (G) ≤ R q for some G ⊆ H p ≤ P w(G) ≥ µ + t and ∆ 1 (G) ≤ R 1 for some G ⊆ H p + 1≤j<q P ∆ j (G) > R j and ∆ j+1 (G) ≤ R j+1 for some G ⊆ H p .(32)
Restricting to the special case w(H p ), using Claim 14 it turns out that inequality (32) is essentially equivalent to the basic induction of Janson and Ruciński [15] (see the proof of Theorem 3.10 in [15]), which in turn qualitatively recovers the upper tail part of Kim and Vu [20] (see Section 5 of [15,13]). The iterative point of view (32) is somewhat more flexible than induction, making the arguments subjectively easier to modify (as there is no need to formulate a suitable induction hypothesis). Estimates for all subhypergraphs G ⊆ H p also make room for additional combinatorial arguments, which is crucial for the purposes of this paper.
Combinatorial sparsification: degree reduction by deletion
In this section we introduce our key combinatorial arguments, which eventually allow us to obtain improved upper tail estimates by 'sparsifying' H p , i.e., deleting edges from H p . Loosely speaking, via this sparsification idea we can effectively ignore certain 'exceptional' edges from H p (which contain vertices with extremely high degree, say). For the purpose of this paper, we encapsulate this heuristic idea with the definition below. In intuitive words, for ℓ = 1 the 'sparsification' event E j,1 (x, r, y, z) essentially ensures that every G ⊆ H p with bounded ∆ j+1 (G) and ∆ 1 (G) contains a large subhypergraph J ⊆ G with small ∆ j (J ).
Definition (Sparsification event). Let E j,ℓ (x, r, y, z) denote the event that for every G ⊆ H p with ∆ j+1 (G) ≤ y and ∆ ℓ (G) ≤ z there is J ⊆ G with ∆ j (J ) ≤ x and e(G \ J ) ≤ r.
Here one conceptual difference to the 'deletion lemma' of Rödl and Ruciński [23,14] is that our focus is on 'local properties' such as degrees (somewhat in the spirit of [30]), and not on 'global properties' such as subgraph counts. Furthermore, we are deleting edges from H p = H[V p (H)], whereas the classical approach corresponds to deleting vertices from V p (H) = E(G n,p ), say.
With E j,1 (x, r, y, z) in hand, we now refine 4 the basic estimate (32) via the strategy outlined in Section 3.1.3 (see also (28) therein). We believe that the ideas used in the proof of Theorem 15 below are more important than its concrete statement (which is optimized for the purposes of this paper). Here one new ingredient is the edge deletion of the sparsification events in P j,3,ℓ of (36), which allows us to decrease certain maximum degrees. The total weight of the deleted edges can be as large as t/2, which is the reason why in (33) we need to relax w(G) ≥ µ + t to w(G) ≥ µ + t/2. In later applications we shall use S j ≈ R j /s with s = ω(1), and then the parametrization Q j = max{S j , D j } allows us to easily deal with S j = o(1) border cases. The indicators in (35)- (36) can safely be ignored on first reading (they mainly facilitate certain technical estimates). A key aspect of (33) is that we intuitively replace ∆ 1 (G) ≤ R 1 of (32) with ∆ 1 (G) ≤ min{Q 1 , R 1 }, which by the discussion of Section 3.2 is crucial for obtaining improved tail estimates (see also Theorem 18).
Theorem 15 (Combinatorial decomposition of the upper tail). Given H with 1 ≤ q ≤ k, assume that (P') holds. Suppose that t > 0. Given positive (D j ) 1≤j≤q , (R j ) 1≤j<q and (S j ) 1≤j<q , define R q = Q q = D q and Q j = max{S j , D j } for 1 ≤ j < q. Then we have
P w(G) ≥ µ + t and ∆ q (G) ≤ D q for some G ⊆ H p ≤ P w(G) ≥ µ + t/2 and ∆ 1 (G) ≤ min{Q 1 , R 1 } for some G ⊆ H p + 1≤j<q P j,1 + P j,2 + P j,3,1 ,(33)
where
P j,1 = P ∆ j (G) > R j and ∆ j+1 (G) ≤ R j+1 for some G ⊆ H p ,(34)P j,2 = ½ {Qj <Rj and Qj+1>Dj+1} P ∆ j (G) > Q j and ∆ j+1 (G) ≤ S j+1 for some G ⊆ H p ,(35)P j,3,ℓ = ½ {Qj <Rj and Qj+1=Dj+1} P ¬E j,ℓ (Q j , t/(2Lq), D j+1 , R ℓ ) .(36)
The combinatorial proof proceeds in two sparsification rounds. In the first round we use our usual iterative degree argument to deduce that ∆ q (G) ≤ R q implies ∆ j (G) ≤ R j for all 1 ≤ j ≤ q. We start the second round with the sparsification event, by deleting edges such that J ⊆ G satisfies ∆ q−1 (J ) ≤ Q q−1 (tacitly assuming Q q−1 < R q−1 , say). The idea is that our usual iterative degree argument should then allow us to deduce that ∆ j+1 (J ) ≤ Q j+1 implies ∆ j (J ) ≤ Q j for all 1 ≤ j < q − 1. Unfortunately, our later probabilistic estimates break down if the parameter Q j+1 is 'too small'. With foresight we thus use our alternative 'degree reduction' argument whenever Q j+1 = D j+1 holds, i.e., we again delete edges.
Proof of Theorem 15. Inequality (33) is trivial for q = 1 (since R 1 = Q 1 = D 1 ). For q ≥ 2 the plan is to show that properties (a)-(d) below deterministically imply that w(G) < µ + t for every G ⊆ H p with ∆ q (G) ≤ D q . Using a union bound argument this then completes the proof (it is routine to check that (a)-(d) correspond to the complements of the events on the right hand side of (33), since Q j+1 > D j+1 implies S j+1 = Q j+1 ).
Turning to the details, we henceforth assume that the following properties hold for all G ⊆ H p and 1 ≤ j < q:
(a) ∆ 1 (G) ≤ min{Q 1 , R 1 } implies w(G) < µ + t/2, (b) ∆ j+1 (G) ≤ R j+1 implies ∆ j (G) ≤ R j , (c) if Q j < R j and Q j+1 > D j+1 , then ∆ j+1 (G) ≤ Q j+1 implies ∆ j (G) ≤ Q j , and (d) if Q j < R j and Q j+1 = D j+1 , then ∆ j+1 (G) ≤ Q j+1 and ∆ 1 (G) ≤ R 1 implies existence of J ⊆ G with ∆ j (J ) ≤ Q j and e(G \ J ) ≤ t/(2Lq).
For the remaining deterministic argument we fix G ⊆ H p with ∆ q (G) ≤ D q , and claim that we can construct a hypergraph sequence G = J q ⊇ · · · ⊇ J 1 such that
∆ i (J j ) ≤ R i , if 1 ≤ i < j, min{Q i , R i }, if j ≤ i ≤ q,(37)e(J j+1 \ J j ) ≤ t/(2Lq).(38)
With this sequence in hand, using (38) we have (37) and (a) then yields
w(J j+1 \ J j ) = f ∈Jj+1\Jj w f ≤ max f ∈Jj+1\Jj w f · e(J j+1 \ J j ) ≤ L · t/(2Lq) = t/(2q), which together with ∆ 1 (J 1 ) ≤ min{Q 1 , R 1 } ofw(G) = w(J 1 ) + 1≤j<q w(J j+1 \ J j ) < (µ + t/2) + (q − 1) · t/(2q) ≤ µ + t.(39)
It thus remains to construct G = J q ⊇ · · · ⊇ J 1 with the claimed properties. For the base case G = J q ,
using ∆ q (J q ) = ∆ q (G) ≤ D q = R q repeated applications of (b) yield that ∆ i (J q ) ≤ R i for all 1 ≤ i ≤ q, so (37) holds since ∆ q (J q ) ≤ R q = min{R q , Q q }. Given J j+1 with 1 ≤ j < q, our construction of J j ⊆ J j+1 distinguishes several cases; in view of ∆ i (J j ) ≤ ∆ i (J j+1 ) it clearly suffices to check (37) for ∆ j (J j ) only. If Q j ≥ R j , then we set J j = J j+1 , which satisfies ∆ j (J j ) = ∆ j (J j+1 ) ≤ R j = min{Q j , R j } by (37). If Q j < R j and Q j+1 > D j+1 , then we set J j = J j+1 , which by (37) satisfies ∆ j+1 (J j ) = ∆ j+1 (J j+1 ) ≤ Q j+1 . Hence (c) implies ∆ j (J j ) ≤ Q j = min{Q j , R j }.
Finally, if Q j < R j and Q j+1 = D j+1 , then by (37)
we have ∆ j+1 (J j+1 ) ≤ Q j+1 and ∆ 1 (J j+1 ) ≤ R 1 . Hence (d) implies existence of J j ⊆ J j+1 satisfying ∆ j (J j ) ≤ Q j = min{Q j , R j } and e(J j+1 \ J j ) ≤ t/(2Lq), completing the proof.
The above proof demonstrates that estimates for all subhypergraphs G ⊆ H p are extremely powerful along with combinatorial arguments. It seems likely that the above sparsification approach can be sharpened in specific applications, i.e., that there is room for alternative (ad-hoc) arguments which apply the 'degree reduction' idea differently. For example, in [36] the degrees are iteratively reduced by a factor of two, say (replacing the finite sum in (39) by a convergent geometric series). In [28] the iterative argument also takes 'trivial' upper bounds for the ∆ j (H) into account (which can be smaller than R j or Q j ).
A combinatorial local deletion argument
The goal of this subsection is to estimate P ¬E j,1 (x, r, y, z) , i.e., the probability that our 'sparsification' event fails. As indicated in Section 3.1.3, our proof uses a maximal matching based idea which relies on combinatorial arguments and the BK-inequality. The following auxiliary event D U,x,y intuitively states that, in H p , the vertex set U is the centre of a 'star' with at least x spikes (satisfying some degree constraint).
Definition (Auxiliary degree event). Let D U,x,y denote the event that there is K ⊆ Γ U (H p ) with |K| ≥ x and ∆ |U|+1 (K) ≤ y.
To put this definition into our 'all subhypergraphs' context, note that ¬D U,x,y implies |Γ U (G)| < x for all G ⊆ H p with ∆ |U|+1 (G) ≤ y. It might also be instructive to note that a union bound argument yields
P ∆ j (G) ≥ x and ∆ j+1 (G) ≤ y for some G ⊆ H p ≤ U⊆V (H):|U|=j P(D U,x,y ).(40)
The next result relates the auxiliary event D U,x,y with the sparsification event E j,1 (x, r, y, z). For example, (41). Lemma 16 (Auxiliary result for the sparsification event). Given H, assume that max f ∈H |f | ≤ k holds. Then for all x, r, y, z > 0 and 1 ≤ j < k we have
U P(D U,x,y ) ≤ B −x/y translates into P(¬E j,1 (x, r, y, z)) ≤ B −r/(kyz) by inequalityP ¬E j,1 (x, r, y, z) ≤ U⊆V (H):|U|=j P(D U,x,y ) ⌈r/(k⌈x⌉z)⌉ .
(41)
Remark 17. Inequality (41) remains valid after dividing the right hand side by ⌈r/(k⌈x⌉z)⌉!.
The proof of Lemma 16 develops a combinatorial idea from [36], which in turn was partially inspired by [29,14]. We call (U,
K U ) an (j, x, y)-star in G if U ⊆ V (G) and K U ⊆ Γ U (G) = {f ∈ G : U ⊆ f } satisfy |U | = j, |K U | = ⌈x⌉ and ∆ j+1 (K U ) ≤ y. Note that we allow for overlaps of the edges f, g ∈ K U outside of the 'centre' U . Writing S j,x,y (G) for the collection of all (j, x, y)-stars in G, we define M j,x,y (G) as the size of the largest M ⊆ S j,x,y (G) satisfying V (K U ) ∩ V (K W ) = ∅ for all distinct (U, K U ), (W, K W ) ∈ M.
In intuitive words, M j,x,y (G) denotes the size of the 'largest (j, x, y)-star matching' in G, i.e., vertex-disjoint collection of stars. We are now ready to follow the strategy sketched in Section 3.1.3 (see also (29) therein).
Proof of Lemma 16. Letr = r/(k⌈x⌉z) and R = ⌈r⌉. We first assume that M j,x,y (H p ) ≤r holds, and claim that this implies the occurrence of E j,1 (x, r, y, z). For any G ⊆ H p with ∆ j+1 (G) ≤ y and ∆ 1 (G) ≤ z, it clearly suffices to show that there is J ⊆ G with ∆ j (J ) ≤ x and e(G\J ) ≤ r. Let M ⊆ S j,x,y (G) attain the maximum in the definition of M j,x,y (G). We then remove all edges f ∈ G which overlap some star (U, K U ) ∈ M, where overlap means that f ∩ g = ∅ for some edge g ∈ K U . We denote the resulting subhypergraph by J ⊆ G.
Using ∆ j+1 (J ) ≤ ∆ j+1 (G) ≤ y and maximality of M, we then infer ∆ j (J ) ≤ ⌈x⌉ − 1 < x (because otherwise we could add another (j, x, y)-star to M). Furthermore, since |M| = M j,x,y (G) ≤ M j,x,y (H p ) ≤r and ∆ 1 (G) ≤ z, by construction the number of deleted edges is at most
e(G \ J ) ≤ KU ∈M f ∈KU v∈f |Γ {v} (G)| ≤ |M| · ⌈x⌉ · max f ∈G |f | · ∆ 1 (G) ≤r · ⌈x⌉kz = r.(42)½ ⊡ i∈[R] DU i ,x,y ,(43)
where ⊡ is defined as in (15).
If M j,x,y (H p ) >r, then there is M ⊆ S j,x,y (H p ) of size |M| = ⌈r⌉ = R which satisfies V (K U ) ∩ V (K W ) = ∅ for all distinct (U, K U ), (W, K W ) ∈ M. So, since the disjoint vertex sets V (K U ) ⊆ V p(
H) guarantee the occurrence of each event D U,x,y , it follows that ⊡ (U,KU )∈M D U,x,y occurs. As U ⊆ V (K U ) holds, by vertex disjointness of the V (K U ) we deduce that the corresponding 'star-centres' U are distinct. Since Z R counts ordered R-tuples, we thus infer Z R ≥ R!. Hence, Markov's inequality yields
P(M j,x,y (H p ) >r) ≤ P(Z R ≥ R!) ≤ (EZ R )/R!.(44)
Turning to EZ R , using the BK-inequality (16) we readily obtain
EZ R =(
which together with (44) and R ≥ 1 completes the proof.
The 'star-matching' based deletion argument used in the above proof seems of independent interest. In applications it might be easier to avoid E j,1 (x, r, y, z), and directly work with the random variable M j,x,y (H p ), see also [36,28]. The above estimates (44)-(45) exploit the BK-inequality to relate M j,x,y (H p ) with the simpler events D U,x,y . In H p and other probability spaces one can sometimes also estimate P(M j,x,y (H p ) ≥ z) more directly (see, e.g., the remark after the proof of Lemma 17 in [36], or the proof of Lemma 9 in [28]).
Probabilistic estimates
In this section we introduce our key probabilistic estimates, which complement the combinatorial decomposition of Theorem 15, i.e., allow us to bound the right hand side of (33). A key aspect of inequalities (46)-(47) is that improved degree constraints ∆ i (G) ≤ y translate into improved tail estimates. In our applications (48) below often reduces to P ¬E j,1 (x, r, y, z) ≤ (eµ j /x) −Θ(r/(yz)) , say (see, e.g., the proof of Theorem 34).
Theorem 18 (Probabilistic upper tail estimates). Given H, assume that (P') holds. Set ϕ(x) = (1 + x) log(1 + x) − x. Then for all x, r, y, z, t > 0 and 1 ≤ j < k we have
P w(G) ≥ µ + t/2 and ∆ 1 (G) ≤ y for some G ⊆ H p ≤ exp − ϕ(t/µ)µ 4Lky ,(46)P ∆ j (G) ≥ x and ∆ j+1 (G) ≤ y for some G ⊆ H p ≤ N j eµ j x x/(ky) ,(47)P ¬E j,1 (x, r, y, z) ≤ N j eµ j ⌈x⌉ ⌈x⌉/(ky) ⌈r/(k⌈x⌉z)⌉ .(48)
The proofs of (46)-(47) are based on fairly routine applications of Theorem 7. The crux is that the restrictions ∆ 1 (G) ≤ y and ∆ j+1 (G) ≤ y translate into bounds for the parameter C in (14), which intuitively controls the 'largest dependencies' (∆ 1 (G) ≤ y ensures that every edge f ∈ G overlaps at most |f |·∆ 1 (G) ≤ ky edges e ∈ G).
Y e ≤ max e∈G w e · e∈G:e∩f =∅ ½ {e∈Hp} ≤ L · v∈f |Γ {v} (G)| ≤ L · |f | · ∆ 1 (G) ≤ Lky.
To sum up, if w(G) ≥ µ + t/2 and ∆ 1 (G) ≤ y for some G ⊆ H p , then Z C ≥ µ + t/2 holds with C = Lky, where Z C is defined as in Theorem 7 with I = H. So, applying (14), we deduce
P w(G) ≥ µ + t/2 and ∆ 1 (G) ≤ y for some G ⊆ H p ≤ P(Z C ≥ µ + t/2) ≤ exp − ϕ(t/(2µ))µ Lky .(49)
Using calculus (see, e.g., the proof of Lemma 13 in [36]) it is easy to check that ϕ(t/(2µ) ≥ ϕ(t/µ)/4. In view of (49) and (14), inequality (46) now follows.
Next we turn to (47), which hinges on the union bound estimate (40). Note that v(H) < 1 implies H = ∅, so (47) is trivial for N < 1 (the left hand side is zero). Similarly, (47) is also trivial for x ≤ eµ j and N ≥ 1 (the expression on the right hand side is at least one). To sum up, we henceforth may assume x > eµ j
½ {e∈Hp} ≤ v∈f \U |Γ U∪{v} (K)| ≤ |f \ U | · ∆ |U|+1 (K) ≤ ky.(50)
So, if D U,x,y occurs, then Z C ≥ x holds with C = ky, where Z C is defined as in Theorem 7 with I = Γ U (H).
For f ∈ I, note that U ⊆ V p (H) implies f ∈ H p = H[V p (H)]. Recalling Y f = ½ {f ∈Hp} and ξ σ = ½ {σ∈Vp(H)} ,
using the definition of µ j (see (21)) it follows that
f ∈I E(Y f | (ξ σ ) σ∈U ) = f ∈ΓU (H) P(f ∈ H p | (ξ σ ) σ∈U )½ {U⊆Vp(H)} ≤ f ∈ΓU (H) P(f ∈ H p | U ⊆ V p (H)) = f ∈ΓU (H) p |f |−|U| ≤ µ |U| = µ j .(51)
Furthermore, conditional on (ξ σ ) σ∈U , the independence assumption of Theorem 7 holds by the same reasoning as in Remark 8 (in the conditional space, each Y f is a function of the independent random variables (ξ σ ) σ∈f \U ). So, applying (14) with µ = µ j and µ + t = x > eµ j , we deduce the conditional inequality
P(D U,x,y | (ξ σ ) σ∈U ) ≤ P(Z C ≥ x | (ξ σ ) σ∈U ) ≤ eµ j x x/(ky) .(52)EP(D U,x,y | (ξ σ ) σ∈U ) ≤ N j eµ j x x/(ky) ,(53)
and (47) follows in view of (40). It remains to establish (48). Exploiting integrality of the underlying variables, note in (52) we can strengthen Z C ≥ x to Z C ≥ ⌈x⌉. In (52)-(53) we thus may replace (eµ j /x) x/(ky) by (eµ j /⌈x⌉) ⌈x⌉/(ky) , and so (48) Proof. The proof of Theorem 15 is based on (deterministic) combinatorial arguments, and after replacing H p with H m thus carries over word-for-word to H m .
Turning to Theorem 18, using Remark 9 it is easy to see that the proof of (46) carries over to H m (with minor notational changes).
For (47) more care is needed. To avoid conditional probabilities and expectations, set Y f = ½ {f \U⊆Vm(H)} for all f ∈ I := Γ U (H). Writing α ∼ β if (α ∩ β) \ U = ∅, note that inequality (50) readily carries over. It is folklore (analogous to, e.g., the proof of Theorem 15 in [18]) that (21). Recalling the definition of ∼, it is similarly folklore that the random variables Y f = ½ {f \U⊆Vm(H)} satisfy the negative correlation condition of Remark 9. Mimicking the argument leading to (52), using Theorem 7 we obtain P(D U,x,y ) ≤ P(Z C ≥ x) ≤ (eµ j /x) x/(ky) for H m , which by a simpler variant of (53) then establishes (47).
EY f = P(f \ U ⊆ V m (H)) ≤ p |f |−|f ∩U| for p = m/v(H), so that f ∈I EY f ≤ f ∈ΓU (H) p |f |−|U| ≤ µ j by
As the proof of (47) carries over, for (48) it remains to check that (41) holds for H m . A close inspection of the proof of Lemma 16 reveals that only the usage of the BK-inequality in (45) needs to be justified. But, since D U,x,y is an increasing event, this application of (16) is valid by Remark 13, completing the proof.
More general setup
In this section we introduce our general Kim-Vu/Janson-Ruciński type setup, and show that the combinatorial and probabilistic arguments of Section 3 carry over with somewhat minor changes. Readers only interested in random induced subhypergraphs H p may wish to skip to Section 5 (see Remark 29).
Setup
Our general setup is based on certain independence assumptions, i.e., we do not restrict ourselves to polynomials of independent random variables (and we also do not make any monotonicity assumptions). Given a hypergraph H and non-negative random variables (Y f ) f ∈H , for every G ⊆ H we set
X(G) = f ∈G Y f ,(54)
where our main focus 5 is on the sum X(H) of all the variables Y f (sometimes H is also called the 'supporting' or 'underlying' hypergraph, see [20,33]). Loosely speaking, the plan is to adapt the combinatorial arguments of Sections 3.3-3.4 to the associated random subhypergraph
H p = {f ∈ H : Y f > 0},(55)
which due to X(H) = X(H p ) loosely encodes all 'relevant' variables (recall that Y f ≥ 0). Similar to [15], we shall use the following independence assumption (Hℓ), where ℓ ∈ N is a parameter:
(Hℓ) Let (ξ σ ) σ∈A be a family of independent finite random variables.
= ½ {σ∈Vp(H)} , A f = f and Y f = w f σ∈A f ξ σ .
A key consequence of (Hℓ) is that Y e and Y f are independent whenever |e ∩ f | < ℓ, since by (i) and (iii) then both depend on disjoint sets of variables ξ σ . The 'structural' assumption (i) that each Y f depends only on the variables ξ σ with σ ∈ A f is very common in applications; often A U = U suffices. The 'consistency' assumption (ii) and 'independence' assumption (iii) of the index sets A U are also very natural. For example, in the frequent case
A U = U we have A e ∩ A f = A e∩f , so A e ∩ A f = ∅ if |e ∩ f | < 1. Example 22 in Section 4.1.1 illustrates the case ℓ = 1 with A U = {f ∈ E(K n ) : f ⊆ U }.
We now introduce the modified key parameters µ j , which intuitively quantify the 'dependencies' among the variables Y f (in the spirit of [15,20,31,33]). Recalling Γ U (H) = {f ∈ H : U ⊆ f }, with Section 3.1.1 in mind we now define the following two crucial assumptions (P) and (Pq), where q ∈ N is a parameter:
(P) Assume that max f ∈H |f | ≤ k, max f ∈H sup Y f ≤ L and v(H) ≤ N . Define µ = EX(H) and
µ j = max U⊆V (H):|U|=j sup E |Γ U (H p )| (ξ σ ) σ∈AU ,(56)
where the supremum is over all values of the variables ξ σ with σ ∈ A U . (Pq) Assume that ∆ q (H) ≤ D.
In view of (22), property (P) is a natural extension of (P') from the basic setup of Section 3.1.1. Our general setup lacks monotonicity, and so the conditioning in (56) is with respect to all possible values of the ξ σ .
For the interested reader, we now briefly discuss how our setup and assumptions differ in some (usually irrelevant) minor details from the literature [15,20,31,33]. Firstly, the 'normal' assumption of Vu implies max f ∈H sup Y f ≤ 1 in (P) above (see, e.g., Theorem 1.2 in [31] and Theorem 4.2 in [33]). Secondly, classical variants of the 'maximum average effect' parameter µ j (see, e.g., Sections 3 in [15] and Section 4 in [33]) are roughly defined as the maximum over all sup E( f ∈ΓU (Hp) Y f | (ξ σ ) σ∈AU ) with |U | = j, but in most applications f ∈ΓU (Hp) Y f = Θ(|Γ U (H p )|) holds, so the difference is usually immaterial. Thirdly, in (Hℓ) our assumptions for the index sets A U are slightly simpler than in Section 3 of [15]. Finally, in contrast to [15], we assume that the (ξ σ ) σ∈A are finite random variables, which is very natural in combinatorial applications (this technicality can presumably be removed by approximation arguments, but we have not pursued this).
Examples
The above assumptions (Hℓ) and (P ) might seem a bit technical at first sight, and for this reason we shall below spell out three pivotal examples (see Section 3 of [15] for more examples).
Example 20 (Random induced subhypergaphs). For a given k-uniform hypergraph H, analogous to Sec-
tion 3.1.1 we consider X = e(H p ) = f ∈H ½ {f ∈Hp} . Note that A = H, ξ σ = ½ {σ∈Vp(H)} , A f = f and Y f = σ∈A f ξ σ ∈ {0,
1} satisfy properties (H1) and (Pk). In fact, for (P) we can simplify the definition of µ j . Namely, since
U ⊆ V p (H) implies f ∈ H p = H[V p (H)] for all f ∈ Γ U (H), we have sup E |Γ U (H p )| (ξ σ ) σ∈AU = E |Γ U (H p )| U ⊆ V p (H) = f ∈ΓU (H) P f ∈ H p U ⊆ V p (H) . As H is k-uniform, for any f ∈ Γ U (H) it is easy to see that P f ∈ H p U ⊆ V p (H) = P f \ U ⊆ V p (H) = p k−|U| .
Combining these observations, it follows that (56) simplifies for 1 ≤ j ≤ k to
µ j = max U⊆V (H):|U|=j |Γ U (H)| · p k−j .(57)n v−vJ p e−j .(58)
Note that any q = e − δ + 1 ≤ e edges already determine the vertex set, so (Pq) holds with D = O(1). Finally, a minor variant of the described approach also applies to induced subgraph counts (with k = vH 2 , by letting E(H) correspond to copies of the complete graph K vH , and defining Y f as the indicator for the event that the subgraph of G n,p defined by the edges in f is isomorphic to H).
Example 22 (Subgraph counts in G n,p : vertex exposure approach). Subgraph counts in G n,p can also be treated via a 'vertex exposure' based approach. Given a fixed subgraph H with e = e H edges and v = v H edges, we consider the complete v-uniform hypergraph H with vertex set V (H) = [n], so N = n and k = v. For I ⊆ V (H) with |I| = v the random variable Y I counts the number of copies of H in G n,p that have vertex set I. Note that 0 ≤ Y I ≤ L = O(1). Since X = I∈H Y I , we take A = E(K n ), ξ σ = ½ {σ∈Vp(H)} , and A I = {f ∈ E(K n ) : f ⊆ I}. As A I ∩ A J = A I∩J is empty whenever |I ∩ J| < 2, for ℓ = 2 properties (Hℓ) and (Pk) are satisfied. Conditioning on (ξ σ ) σ∈AU corresponds to conditioning on G n,p [U ], so bounding µ j is conceptually analogous (58). Indeed, by similar reasoning as in Example 21, we arrive for 1 ≤ j ≤ v at
µ j ≤ B induced J⊆H:vJ =j n v−j p e−eJ ,(59)
where B = B(H) > 0. Finally, induced subgraph counts can clearly be treated analogously.
Adapting the arguments of Sections 3.3-3.4
In this section we adapt the key results Theorem 15 and 18 from Sections 3.3-3.4 to our more general setup. The crux is that the random variables (Y f ) f ∈H satisfy Y f = Y f (ξ σ : σ ∈ A f ) by the independence assumption (Hℓ), so that the intersection properties of the index sets A f give us a handle on the dependencies. This allows us to adapt our combinatorial arguments to the auxiliary subhypergraph H p = {f ∈ H : Y f > 0}. We start with a natural analogue of Theorem 15, which is at the heart of our arguments.
Theorem 23 (Combinatorial decomposition of the upper tail: general setup). Given H with 1 ≤ ℓ ≤ q ≤ k, assume that (Hℓ) and (P) hold. Suppose that t > 0. Given positive (R j ) ℓ≤j<q and (D j ) ℓ≤j≤q , define R q = Q q = D q and Q j = max{S j , D j } for ℓ ≤ j < q. Then we have P X(G) ≥ µ + t and ∆ q (G) ≤ D q for some G ⊆ H p ≤ P X(G) ≥ µ + t/2 and ∆ ℓ (G) ≤ min{Q ℓ , R ℓ } for some G ⊆ H p + ℓ≤j<q P j,1 + P j,2 + P j,3,ℓ ,
where P j,1 , P j,2 and P j,3,ℓ are defined as in (34)- (36).
Recalling X(G) = f ∈G Y f and H p = {f ∈ H : Y f > 0}
, the deterministic proof of Theorem 15 carries over to Theorem 23 with minor obvious changes (inequality (60) is trivial if q = ℓ; for q > ℓ it suffices to construct G = J q ⊇ · · · ⊇ J ℓ , with indices of form ℓ ≤ i, j ≤ q in (37)); we omit the routine details.
Next we state an analogue of Lemma 16 for the 'sparsification' event E j,ℓ (x, r, y, z) from Section 3.3.
Lemma 24 (Auxiliary result for the sparsification event: general setup). Given H with 1 ≤ ℓ ≤ k, assume that (Hℓ) and max f ∈H |f | ≤ k hold. Then for all x, r, y, z > 0 and ℓ ≤ j < k we have
P ¬E j,ℓ (x, r, y, z) ≤ U⊆V (H):|U|=j P(D U,x,y ) r/ ( k ℓ )⌈x⌉z .(61)
Remark 25. Inequality (61) remains valid after dividing the right hand side by ⌈r/( k ℓ ⌈x⌉z)⌉!. For the proof of Lemma 24 we adapt the definition of M j,x,y (G) used for Lemma 16. Intuitively, the idea is to replace 'vertex disjoint' by 'depending on disjoint sets of variables'. Namely, here we define M j,x,y (G) as the size of the largest collection M ⊆ S j,x,y (G) of (j, x, y)-stars in G satisfying the following property for all distinct (U, K U ), (W, K W ) ∈ M: we have |e ∩ f | < ℓ for all e ∈ K U and f ∈ K W . The point will be (i) that each Y f is a function of the variables (ξ σ ) σ∈A f , and (ii) that |e ∩ f | < ℓ implies A e ∩ A f = ∅ by (Hℓ).
Proof of Lemma 24. Using the above definition of M j,x,y (G), we shall adapt the proof of Lemma 16. Let r = r/ k ℓ ⌈x⌉z and R = ⌈r⌉. We first assume that M j,x,y (H p ) ≤r holds, and claim that this implies the occurrence of E j,ℓ (x, r, y, z). Fix G ⊆ H p with ∆ j+1 (G) ≤ y and ∆ ℓ (G) ≤ z, and let M ⊆ S j,x,y (G) attain the maximum in the definition of M j,x,y (G). We remove all edges f ∈ G which 'overlap' some star (U, K U ) ∈ M, where overlap means that |f ∩g| ≥ ℓ for some edge g ∈ K U . We denote the resulting subhypergraph by J ⊆ G. Recalling ∆ j+1 (J ) ≤ ∆ j+1 (G) ≤ y, by maximality of M we infer ∆ j (J ) ≤ ⌈x⌉ − 1 < x. Similar to (42), using |M| = M j,x,y (G) ≤ M j,x,y (H p ) ≤r and ∆ ℓ (G) ≤ z it is easy to see that we removed at most
e(G \ J ) ≤ |M| · ⌈x⌉ · max f ∈G |f | ℓ · ∆ ℓ (G) ≤r · ⌈x⌉ k ℓ z = r(62)
edges. It follows that M j,x,y (H p ) ≤r implies E j,ℓ (x, r, y, z), as claimed. For (61) it remains to estimate P(M j,x,y (H p ) >r). Suppose that M j,x,y (H p ) >r occurs. If M ⊆ S j,x,y (H p ) attains the maximum in the definition of M j,x,y (H p ), then we know (i) that |M| ≥ ⌈r⌉ = R holds, and (ii) that (U,KU )∈M D U,x,y occurs. In the following we argue that these events D U,x,y 'occur disjointly' in the sense of Section 2.2. For each (U, K U ) ∈ M, note that the variables indexed by
V (K U ) = f ∈KU A f
guarantee the occurrence of D U,x,y . The crux is now that for all distinct (U, K U ), (W, K W ) ∈ M, by (iii) of (Hℓ) we have A e ∩ A f = ∅ for all e ∈ K u and f ∈ K W (since |e ∩ f | < ℓ), so
V (K U ) ∩ V (K W ) = e∈KU f ∈KW (A e ∩ A f ) = ∅.(63)
It follows that ⊡ (U,KU )∈M D U,x,y occurs (since the disjoint sets of variables indexed by V (K U ) guarantee the occurrence of each D U,x,y ). Next we claim that all the corresponding sets U are distinct. To see this, note that for distinct (U, K U ), (W, K W ) ∈ M we have ℓ > |e ∩ f | ≥ |U ∩ W | by definition of M, which due to |U | = |W | = j ≥ ℓ implies U = W . To sum up, M j,x,y (H p ) >r implies Z R ≥ R!, where Z R is defined as in (43). The arguments of (44) and (45) now carry over unchanged, completing the proof of (61).
Finally, we state a natural analogue of Theorem 18, which contains our core probabilistic estimates (inequalities (64)-(66) allow us to bound the right hand side of (60) from Theorem 23).
Theorem 26 (Probabilistic upper tail estimates: general setup). Given H with 1 ≤ ℓ ≤ k, assume that (Hℓ) and (P) hold. Set ϕ(x) = (1 + x) log(1 + x) − x. Then for all x, r, y, z, t > 0 and ℓ ≤ j < k we have
P X(G) ≥ µ + t/2 and ∆ ℓ (G) ≤ y for some G ⊆ H p ≤ exp − ϕ(t/µ)µ 4L k ℓ y ,(64)P ∆ j (G) ≥ x and ∆ j+1 (G) ≤ y for some G ⊆ H p ≤ N j eµ j x x/(ky) ,(65)P ¬E j,ℓ (x, r, y, z) ≤ N j eµ j ⌈x⌉ ⌈x⌉/(4ky) r/ ( k ℓ )⌈x⌉z .(66)
The proof is based on a minor modification of the proof of Theorem 18. As we shall see, our main task is to adapt the definitions of the dependency relations ∼. To this end recall (i) that each Y f is a function of the independent variables (ξ σ ) σ∈A f , and (ii) that (Hℓ) implies A e ∩ A f = ∅ whenever |e ∩ f | < ℓ.
Proof of Theorem 26. For (64), note that f ∈H EY f = EX(H) = µ. We define α ∼ β if |α ∩ β| ≥ ℓ. In view of properties (i) and (ii) discussed above, the independence assumption of Theorem 7 holds by analogous reasoning as in Remark 8. Furthermore, for any f ∈ G ⊆ H with ∆ ℓ (G) ≤ y we have
e∈G:e∼f Y e ≤ max e∈G sup Y e · e∈G:|e∩f |≥ℓ ½ {f ∈G} ≤ L · U⊆f :|U|=ℓ |Γ U (G)| ≤ L · |f | ℓ · ∆ ℓ (G) ≤ L k ℓ y.
Setting C = L k ℓ y, the remaining proof of (46) readily carries over to (64) with obvious notational changes. Next we turn to (65), which is again based on (40). As before, we may assume that x > eµ j and N ≥ 1 (otherwise the claim is trivial). Furthermore, given U ⊆ V (H) with |U | = j, we set I = Γ U (H). With the random variables ½ {Y f >0} f ∈I in mind, define α ∼ β if (α ∩ β) \ U = ∅. Note that, for any f ∈ K ⊆ I with ∆ |U|+1 (K) ≤ y, analogous to (50) we have e∈K:e∼f ½ {Y f >0} ≤ |f \ U | · ∆ |U|+1 (K) ≤ ky. Furthermore, by definition of I = Γ U (H), H p = {f ∈ H : Y f > 0} and µ j (see (56)) we obtain
f ∈I E ½ {Y f >0} | (ξ σ ) σ∈AU = E |Γ U (H p )| (ξ σ ) σ∈AU ≤ µ |U| = µ j .
Note that, conditional on (ξ σ ) σ∈AU , each ½ {Y f >0} is now a function of the independent random variables
(ξ σ ) σ∈A f \AU . Furthermore, for all e, f ∈ I = {g ∈ H : U ⊆ g} we see that (e ∩ f ) \ U = ∅ implies e ∩ f = U , so that (ii) of (Hℓ) yields A e ∩ A f ⊆ A e∩f = A U .
For all e, f ∈ I we thus infer that e ∼ f implies
(A e \ A U ) ∩ (A f \ A U ) = (A e ∩ A f ) \ A U ⊆ A U \ A U = ∅.
Conditional on (ξ σ ) σ∈AU , it follows (by the reasoning of Remark 8) that the independence assumption of Theorem 7 holds for the variables ½ {Y f >0} f ∈I . The remaining proof of (47) readily carries over to (65).
Finally, for (66) we recall that (48) is based on Lemma 16 and the argument leading to (47). In view of Lemma 24 and the above proof of (65), the same line of reasoning carries over, establishing (66).
Adapting Section 3.5: vertex exposure approach for H m
In this section we partially adapt our arguments to the uniform random induced subhypergraph F m = F [V m (F )]. Generalizing the 'vertex exposure' approach of Example 22, we rely on the following assumption.
(HℓP) Suppose that H, E and F are hypergraphs with V (H) = V (E), V (F ) = {h ∈ E} and min h∈E |h| ≥ ℓ.
Defining
A U = {h ∈ E : h ⊆ U } for all U ⊆ V (E), assume that F = f ∈H F [A f ] is a disjoint union of induced subhypergraphs. Suppose that (w g ) g∈F are non-negative weights. For all f ∈ H, let Y f = g∈F [A f ] w g ½ {g∈Fm} .
(67)
Assume
that max f ∈H |f | ≤ k, max f ∈H Y f ≤ L and v(H) ≤ N . Define µ = EX(H), p = m/v(F ), and µ j = max U⊆V (E):|U|=j f ∈ΓU (H) g∈F [A f ] p |g|−|g∩AU | .(68)
Example 27. Using the 'vertex exposure' setup discussed in Example 22, subgraph counts in G n,m satisfy (HℓP) with ℓ = 2 and k = v H (by setting E = K n , and defining F as the hypergraph H of Example 21). In (68) the modified parameter µ j is again bounded from above by the right hand side of (59).
Y αi = g1∈F [Aα 1 ] · · · gs∈F [Aα s ] E i∈[s] w gi ½ {gi∈Fm} ≤ i∈[s] EY αi ,(69)
and so the proof of (64) carries over (above we used that
α i ∼ α j implies F [A αi ] ∩ F [A αj ] = ∅). For (65) we define α ∼ β if (α ∩ β) \ U = ∅, and replace ½ {Y f >0} f ∈I by ½ {Y f } f ∈I , where Y f denotes the event that g \A U ⊆ V m (F ) for some g ∈ F [A f ]. Let λ f = g∈F [A f ] P(g \A U ⊆ V m (F ))
. It is folklore that Remark 19), so I = Γ U (H) and (68) yield f ∈I λ f ≤ µ |U| = µ j . Since Henceforth, we tacitly set ϕ(x) = (1 + x) log(1 + x) − x for brevity (as in Theorems 7, 18 and 26).
P(g \ A U ⊆ V m (F )) ≤ p |g|−|g∩AU | (see½ {Y f } ≤ g∈F [A f ] ½ {g\AU
Easy-to-apply tail inequalities
In this section we state some simplified upper tail inequalities that suffice for all the applications in Section 6 (we have not optimized the usually irrelevant constants); the proofs are deferred to Section 5.3.
On first reading of the following upper tail inequality for X(H) = f ∈H Y f , the reader may wish to set ℓ = 1 and q = k, so that (72) is of form P(X(H) ≥ 2µ) ≤ exp(−d min{µ, µ 1/k log(e/π)}). Here our main novelty is the log(e/π) term: it allows us to gain an extra logarithmic factor if π ∈ {N −1 , p}, which yields best possible tail estimates in the applications of Section 6.1. We think of (70) as a 'balancedness' condition, and mainly have parameters of form π ∈ {1, N −1 , p} in mind. In fact, for π ∈ {N −1 , p} the technical assumption (71) usually holds automatically for small τ (see Remark 31 and the proof of Theorem 36).
Theorem 30 (Easy-to-apply upper tail inequality). Given H with 1 ≤ ℓ ≤ q ≤ k, assume that (Hℓ), (P) and (Pq) hold. If there are constants A, α, τ > 0 and a parameter π ∈ (0, 1] such that
max ℓ≤j<q µ j max{µ (q−j)/(q−ℓ+1) , 1} ≤ Aπ α ,(70)Aµ 1/(q−ℓ+1) ≥ ½ {π>N −τ } log N,(71)
then for ε > 0 we have
P(X(H) ≥ (1 + ε)µ) ≤ (1 + bN −ℓ ) exp −c min ϕ(ε)µ, min ε 2 , 1 µ 1/(q−ℓ+1) log(e/π) ≤ (1 + bN −ℓ ) exp −d min ε 2 , 1 min µ, µ 1/(q−ℓ+1) log(e/π) ,(72)
where b = 3q, c = c(ℓ, q, k, L, D, A, α, τ ) > 0 and d = c/3.
Remark 31.
If π = N −1 , then (71) is trivially satisfied for τ = 1/2, and log(e/π) ≥ log N holds in (72).
Simple applications of the inductive approaches [20,15,33] often implicitly assume (70) with π = 1, and replace (71) by the stronger assumption min{ε 2 , 1}µ 1/(q−ℓ+1) = ω(log N ), say (see, e.g., the proof of Corollary 6.3 in [33] or Theorem 2.1 in [32]). Their conclusion is then of the form P(X(H) ≥ (1 + ε)µ) ≤ exp(−a min{ε 2 , 1}µ 1/(q−ℓ+1) ), where µ 1/(q−ℓ+1) = min µ, µ 1/(q−ℓ+1) log(e/π) holds by assumption. In other words, our inequality (72) yields an extra logarithmic factor when π ∈ {N −1 , p} in (70). To illustrate this, for subgraph counts in G n,p the setup of Example 21 (with ℓ = 1, q = k = e and N = n 2 ) naturally yields max ℓ≤j<q µ j µ (q−j)/(q−ℓ+1) ≤ max which is well-known to be O(n −β ) for so-called 'strictly balanced' graphs and O(1) for 'balanced' graphs (the details are deferred to (104) and (115) in Section 6.2; see also Section 6.3 in [33]). The next upper tail result assumes that all the parameters µ j are decaying polynomially in N , which typically requires that µ = EX(H) is small (as v(H) ≤ N ). On first reading of Theorem 32 the reader may wish to set ℓ = 1, q = k and K = 1, so that (74) is of form P(X(H) ≥ µ + t) ≤ exp(−a min{t 2 /µ, t 1/k log N }) when t ∈ [1, µ]. Here our main novelty is the t 1/k log N term, which is key for the applications in Section 6.2.1.
Theorem 32 (Easy-to-apply upper tail inequality: the small expectations case). Given H with 1 ≤ ℓ ≤ q ≤ k, assume that (Hℓ), (P) and (Pq) hold. If there are constants A, α > 0 such that
max ℓ≤j<q µ j ≤ AN −α ,(73)
then for t, K > 0 we have
P(X(H) ≥ µ + t) ≤ (1 + bN −q ) exp − min cϕ(t/µ)µ, max ct 1/(q−ℓ+1) , K log N ≤ (1 + bN −q ) exp − min dt 2 /µ, dt, max ct 1/(q−ℓ+1) , K log N ,(74)
where b = 2q, c = c(ℓ, q, k, L, D, A, α, K) > 0 and d = c/3.
The inductive approaches [31,15] yield variants of (74) where max ct 1/(q−ℓ+1) , K is qualitatively replaced by K (see, e.g., Corollary 4.10 in [15]). For K large enough this gives bounds of the form P(X(H) ≥
(1 + ε)µ) ≤ N −β for µ ≥ C(ε, d, β) log n, and P(X(H) ≥ (1 + ε)µ) ≤ exp(−dε 2 µ) for µ ≤ log n and ε ≤ 1, say (see, e.g., Corollaries 4.11-4.12 in [15]). To illustrate assumption (73) which for 'strictly balanced' graphs is well-known to be O(n −σ/2 ) for sufficiently small σ > 0 (the details are deferred to (104) and (107) in Section 6.2; see also Claim 6.2 in [33]).
More general tail inequalities
In this section we state some more general upper tail inequalities which (i) mimic the heuristic discussion of Section 3.1.2, and (ii) are easier to compare with the work of Kim-Vu/Janson-Ruciński [20,31,33,15]; the proofs are deferred to Section 5.3. Readers primarily interested in applications may proceed to Section 6. We start with a rigorous analogue of the basic upper tail inequality (23) from Section 3.1.2, which is inspired by very similar classical results for the special case G = H p with ∆ q (H) ≤ D (see, e.g., Theorem 3.10 in [15] and Theorem 4.2 in [33]). In applications convenient choices of the parameters (R j ) ℓ≤j<q and D are often of form D = Θ(1), R j = λ q−j D and λ = B max{µ 1/(q−ℓ+1) , 1}, so that in (76) we have min{µ/R ℓ = Θ(λ) and R j /R j+1 = λ when µ ≥ 1 (see, e.g., the proof of Corollary 6.3 in [33] or Theorem 2.1 in [32]).
Claim 33 (Basic upper tail inequality).
Given H with 1 ≤ ℓ ≤ q ≤ k, assume that (Hℓ) and (P) hold. Suppose that t > 0. Given positive (R j ) ℓ≤j<q and D, let R q = D. If inequality
eµ j R j Rj /Rj+1 ≤ N −4kj (75)
holds for all ℓ ≤ j < q, then there are a, b > 0 (depending only on ℓ, k, L) such that
P(X(G) ≥ µ + t and ∆ q (G) ≤ D for some G ⊆ H p ) ≤ exp − aϕ(t/µ)µ R ℓ + ℓ≤j<q N −j eµ j R j bRj /Rj+1 .(76)
To familiarize the reader with the form of assumption (75) and inequality (76), it is instructive to briefly relate them to work of Kim and Vu [20,32,33]. Theorem 4.2 in [33] qualitatively sets t = √ λµR ℓ , and (in our notation) its parametrization assumes roughly ∆ q (H) ≤ D = R q , µ/R ℓ ≥ λ = ω(log N ), as well as R j ≥ 2eµ j and R j /R j+1 ≥ λ for all ℓ ≤ j < q, say. In this case (eµ j /R j ) Rj /Rj+1 ≤ 2 −λ = N −ω(1) follows, so assumption (75) holds. We also have t = µ λR ℓ /µ ≤ µ, so that Remark 11 yields ϕ(t/µ)µ/R ℓ ≥ t 2 /(3µR ℓ ) = λ/3, say. Recalling ∆ q (H) ≤ D, for suitable C = C(q) and c = c(a, b) it follows that (76) yields
P(X(H p ) ≥ µ + t) ≤ exp −aλ/3 + ½ {q>ℓ} qN −ℓ 2 −bλ ≤ C exp −cλ ,(77)
which is of similar form as (24) or Theorem 4.2 [33].
We now state our improved variant 6 of Claim 33, which corresponds to a rigorous analogue of the upper tail inequality (26) from Section 3.1.2. Convenient choices of the parameters (R j ) ℓ≤j<q and (D j ) ℓ≤j≤q are often of form D j = B q−j D q = Θ(1), R j = λ q−j D q and λ = B max{µ 1/(q−ℓ+1) , 1}, so that in (80) we have R j /R j+1 = λ and t/R ℓ = Θ(λ) when t = Θ(µ) and µ ≥ 1. One key novelty of (80) is the µ/Q ℓ = min{µs/R ℓ , µ/D ℓ } term, which intuitively allows us to sharpen inequality (76) whenever R j = ω(µ j ) holds (by using s = ω(1) in (78), so that usually µ/Q ℓ = ω(µ/R ℓ ) in (80), say).
Theorem 34 (Extended upper tail inequality). Given H with 1 ≤ ℓ ≤ q ≤ k, assume that (Hℓ) and (P) hold. Suppose that s ≥ 1 and t > 0. Given positive (R j ) ℓ≤j<q and (D j ) ℓ≤j≤q with R j ≥ D j , define
Q j = max{R j /s, D j }(78)
for ℓ ≤ j < q, and R q = Q q = D q . If inequality max eµ j Q j
Rj /Rj+1
, ½ {Qj <Rj and Qj+1=Dj+1}
eµ j Q j Qj /Dj+1 ≤ N −4kj (79)
holds for all ℓ ≤ j < q, then for a = 1/ 4L k ℓ , b = 1/(2k) and d = 1/ 4Lqk k ℓ we have
P(X(G) ≥ µ + t and ∆ q (G) ≤ D q for some G ⊆ H p ) ≤ exp − aϕ(t/µ)µ Q ℓ + 2 ℓ≤j<q N −j eµ j Q j bRj /Rj+1 + ℓ≤j<q ½ {Qj <Rj and Qj+1=Dj+1} N −j eµ j Q j max dt/(R ℓ Dj+1), bQj /Dj+1 .(80)
To illustrate Theorem 34, in the applications of Sections 5.3.2 and 6.1 we have eµ j /R j ≤ p α /e with p ∈ (0, 1], in which case s = log(e/p α/2 ) is a convenient choice. Indeed, x log(e/x) ≤ 1 then implies eµ j /Q j ≤ eµ j s/R j ≤ p α/2 /e = e −s . We thus think of the (79) as a minor variant of the assumption (75) from Claim 33 (note that eµ j /R j ≤ e −s holds, and that Q j < R j implies Q j = R j /s). Using D j = Θ(1) and the additional Kim-Vu type assumptions discussed below Claim 33, we now review inequality (80) of Theorem 34. Since 1/Q ℓ = min{s/R ℓ , 1/D ℓ }, using t/R ℓ = λµ/R ℓ ≥ λ we obtain analogous to (77) an estimate of the form
P(X(H p ) ≥ µ + t) ≤ exp −ã min{t 2 /µ, λs} + ½ {q>ℓ} 3qN −ℓ e −dλs ≤ C exp −c min{t 2 /µ, λ log(e/p)} .(81)
If q > ℓ then t 2 /µ = λR ℓ ≥ λ q−ℓ+1 R q = ω(λ log N ), so (81) usually decays like C exp(−cλ log(e/p)). When λ ≈ µ 1/(q−ℓ+1) or t = εµ we similarly see that (81) decays like C exp(−c min{µ, λ log(e/p)}). In all these cases we thus improve the exponential decay of the classical bound (77) by an extra logarithmic factor. The following upper tail inequality for polynomially small µ j is a minor extension of Theorem 32. Note that (82) decays exponentially in min{t 2 /µ, t 1/(q−ℓ+1) log N } for 1 ≤ t ≤ O(µ), which seems quite informative when µ = Θ(Var X(H)) holds (i.e., in the Poisson range).
Theorem 35 (Upper tail inequality: the small expectations case). Given H with 1 ≤ ℓ ≤ q ≤ k, assume that (Hℓ) and (P) hold. If there are A, α > 0 such that inequality (73) holds, then for t, K > 0 we have
P(X(G) ≥ µ + t and ∆ q (G) ≤ D for some G ⊆ H p ) ≤ exp −aϕ(t/µ)µ + ½ {q>ℓ} 2qN −q exp − max{bt 1/(q−ℓ+1) , K} log N ,(82)
where a, b > 0 depend only on ℓ, q, k, L, D, A, α, K.
Proofs
Proofs of Claim 33 and Theorems 34-35
Combining Theorem 15 and 18, by setting S j = R j /s the proof of Theorem 34 is straightforward.
Proof of Theorem 34. We first consider the special case q = ℓ. Since R q = D q , using s ≥ 1 we thus infer max{R ℓ /s, D ℓ } = D ℓ = R ℓ . Hence (64) of Theorem 26 readily implies (80).
In the remainder we focus on the more interesting case q > ℓ. Analogous to the proof of Theorem 18, inequality (80) is trivial when N < 1 (the left hand side is zero). So we henceforth may assume N ≥ 1, and using the assumption (79) it follows that Q j ≥ eµ j . Let S j = R j /s, and recall that Q j = max{S j , D j } in Theorem 23. Note that s ≥ 1 and R j ≥ D j imply Q j ≤ R j . In view of (60) and (64) of Theorem 23 and 26, it remains to estimate P j,1 , P j,2 and P j,3,ℓ defined in (34)- (36). Starting with P j,1 and P j,2 , using (65) together with R j ≥ Q j , Q j /S j+1 ≥ S j /S j+1 = R j /R j+1 and the assumption (79) we infer
P j,1 + P j,2 ≤ N j eµ j R j Rj /(kRj+1) + N j eµ j Q j Qj /(kSj+1) ≤ 2N j eµ j Q j Rj /(kRj+1) ≤ 2N −j eµ j Q j Rj /(2kRj+1) .(83)
Finally, for P j,3,ℓ of (36) we henceforth tacitly assume Q j < R j and Q j+1 = D j+1 . With an eye on (66), using Q j ≥ eµ j and the assumption (79) we then (with foresight) similarly deduce
Π := N j eµ j ⌈Q j ⌉ ⌈Qj ⌉/(kDj+1) ≤ N j eµ j Q j ⌈Qj ⌉/(kDj+1) ≤ N −j eµ j Q j ⌈Qj ⌉/(2kDj+1)
.
Since ⌈x⌉ ≥ max{x, 1}, by applying (66) with (x, r, y, z) = (Q j , t/(2qL), D j+1 , R ℓ ) it follows that
P j,3,ℓ ≤ (Π) ⌈t/(2Lq( k ℓ )⌈Qj⌉Rℓ)⌉ ≤ N −j eµ j Q j max dt/(R ℓ Dj+1), bQj /Dj+1
. Recalling our tacit assumption for P j,3,ℓ , this completes the proof in view of (60), (64) and (83).
The details of the similar but simpler proof of Claim 33 are omitted (the above proof carries over by setting s = 1 and D j = R j , since Q j = max{R j /s, D j } = R j implies P j,2 = P j,3,ℓ = 0).
For the proof of Theorem 35 we need to define the parameters (R j ) ℓ≤j≤q and (D j ) ℓ≤j≤q of Theorem 15 and 18 in a suitable way. Intuitively, we shall set R j = λ q−j D, λ = max{t 1/(q−ℓ+1) , B} and D j = Q j = B q−j D = Θ(1), and the crux is that the assumption (73) eventually yields eµ j /x ≤ N −Θ(1) in (65)-(66). We shall also exploit the indicators in Theorem 23 for estimating t/R ℓ in (80), see (86) below.
Proof of Theorem 35. With foresight, let B = max 4qk/α, 2kK/α, Ae/D, 1 and λ = max{t 1/(q−ℓ+1) , B}. Define D j = S j = B q−j D and R j = λ q−j D for all ℓ ≤ j ≤ q. Note that Q j = max{S j , D j } = D j and min{Q j , R j } = D j , so that P j,2 = 0 in (35). Combining (60) and (64) of Theorem 23 and 26, we obtain
P(X(G) ≥ µ + t and ∆ q (G) ≤ D for some G ⊆ H p ) ≤ exp − ϕ(t/µ)µ 4L k ℓ D ℓ + ℓ≤j<q P j,1 + P j,3,ℓ .(84)
Tacitly assuming q > ℓ, it remains to estimate P j,1 and P j,3,ℓ defined in (34) and (36). Starting with P j,1 , by inserting (73) into (65), using R j ≥ DB ≥ Ae and R j /R j+1 = λ ≥ B ≥ 4qk/α we infer
P j,1 ≤ N j eµ j R j Rj /(kRj+1) ≤ N q µ j /A λ/k ≤ N q−αλ/k ≤ N −q−αλ/(2k) .(85)
For P j,3,ℓ , using ⌈Q j ⌉ ≥ Ae and Q j /D j+1 ≥ B ≥ 4qk/α we (with foresight) similarly deduce
Π := N j eµ j ⌈Q j ⌉ ⌈Qj ⌉/(kDj+1) ≤ N −q−α⌈Qj ⌉/(2kDj+1 ) . Note that λ = B implies R j = D j = Q j . Hence Q j < R j ensures λ = t 1/(q−ℓ+1) , so that t/R ℓ = t 1/(q−ℓ+1) /D.
Recalling ⌈Q j ⌉/D j+1 ≥ B, by applying (66) with (x, r, y, z) = (Q j , t/(2qL), D j+1 , R ℓ ) we thus infer
P j,3,ℓ ≤ ½ {Qj <Rj } (Π) ⌈t/(2Lq( k ℓ )⌈Qj⌉Rℓ)⌉ ≤ N −q−max βt 1/(q−ℓ+1) /Dj+1, αB/(2k) ,(86)
with β = α/(4Lqk k ℓ D). With the above estimates (85) and (86) for P j,1 and P j,3,ℓ in hand, using B ≥ 2kK/α and D j+1 ≤ D ℓ it follows by definition of λ = max{t 1/(q−ℓ+1) , B} that ℓ≤j<q P j,1 + P j,3,ℓ ≤ ½ {q>ℓ} 2qN −q exp − max bt 1/(q−ℓ+1) , K log N , with b = min α/(2k), β/D ℓ . Recalling (84), this establishes (82) with a = 1/(4L k ℓ D ℓ ).
Proofs of Theorem 30 and 32
The 'easy-to-apply' inequalities from Section 5.1 are convenient corollaries of Theorems 34-35. Indeed, Remark 11 implies ϕ(t/µ)µ ≥ min{t 2 /µ, t}/3, so Theorem 32 follows readily from Theorem 35. For Theorem 30 the basic strategy is to apply Theorem 34 with s = log(e/π α/2 ), R j = λ q−j D, λ = B max{µ 1/(q−ℓ+1) , 1} and D j = B q−j D = Θ(1). The crux is that the assumption (70) eventually yields eµ j /Q j ≤ π α/2 /e = e −s in (79)-(80). As before, the indicators in Theorem 34 facilitate estimating t/R ℓ in (80), see (89) below.
Proof of Theorem 30. The proof is naturally divided into four parts: (i) introducing definitions, (ii) estimating eµ j /Q j , (iii) applying inequality (80) of Theorem 34, and (iv) verifying assumption (79).
Analogous to the proof of Theorem 18 and 34, we may henceforth assume N ≥ 1. Furthermore, by increasing A or D if necessary, we may of course assume A, D ≥ 1. With foresight, let β = α/2 and s = log(e/π β ). Set B = max{e 2 A/D, 4k 2 /(τ β), 4k 2 (4A) q , 1} and λ = B max{µ 1/(q−ℓ+1) , 1}. Define R j = λ q−j D and D j = B q−j D, so that R j ≥ D j and R q = D q = D.
Next we estimate eµ j /Q j , where Q j ≥ R j /s. Using assumption (70) and α = 2β, for ℓ ≤ j < q we have
eµ j Q j ≤ eµ j s R j = eµ j s DB q−j max{µ (q−j)/(q−ℓ+1) , 1} ≤ eAπ 2β log(e/π β ) DB ≤ π β e = e −s ,(87)
where we tacitly used π ∈ (0, 1] and x log(e/x) ≤ 1 for all x ∈ [0, 1]. We now apply inequality (80) of Theorem 34, deferring the proof of the claim that assumption (79) holds. Using (87) and R j /R j+1 = λ, note that X(H) = X(H p ) and ∆ q (H) ≤ D = D q yield
P(X(H) ≥ (1 + ε)µ) ≤ P(X(G) ≥ µ + εµ and ∆ q (G) ≤ D q for some G ⊆ H p ) ≤ exp − aϕ(ε)µ max{R ℓ /s, D ℓ } + qN −ℓ 2e −bλs + max ℓ≤j<q ½ {Qj <Rj } e −dεµs/(R ℓ Dj+1) .(88)Note that λ = B implies R j = D j , in which case s ≥ 1 yields Q j = D j = R j . Hence Q j < R j ensures λ = Bµ 1/(q−ℓ+1) , so that R ℓ = (Bµ 1/(q−ℓ+1) ) q−ℓ D. Noting D j+1 ≤ D ℓ , it follows that max ℓ≤j<q ½ {Qj <Rj } e −dεµs/(R ℓ Dj+1) ≤ exp − d D ℓ B q−ℓ D · εµ 1/(q−ℓ+1) s .(89)
Similarly, using s ≥ 1 we also see that R ℓ /s > D ℓ implies R ℓ = (Bµ 1/(q−ℓ+1) ) q−ℓ D. Hence
exp − aϕ(ε)µ max{R ℓ /s, D ℓ } ≤ exp − min a D ℓ · ϕ(ε)µ, a B q−ℓ D · ϕ(ε)µ 1/(q−ℓ+1) s .(90)
Remark 11 implies min{ϕ(ε), 1, ε} ≥ min{ε 2 , 1}/3. So, combining (88)-(90), using s ≥ min{1, β} log(e/π) and λ ≥ Bµ 1/(q+ℓ−1) our findings thus establish (72) for suitable c = c(ε, k, q, D, L, α) > 0. In the following we verify assumption (79), i.e., the claim omitted above. Note that R j /R j+1 = λ ≥ B and Q j /D j+1 ≥ D j /D j+1 = B. Using (87), for π ≤ N −τ the left hand side of (79) can thus be bounded by
eµ j Q j B ≤ π βB ≤ N −τ βB ≤ N −4k 2 ≤ N −4kj .(91)
For π > N −τ we defer the proof of the claim that for ℓ ≤ j < q we have
min{λ, R j /D j+1 } ≥ 4k 2 log N.(92)
Using (87), s ≥ 1, Q j ≥ R j /s and (92) we see that the left hand side of (79) can be bounded by
max e −1 Rj /Rj+1 , e −s Rj /(sDj+1) ≤ max e −λ , e −Rj /Dj+1 ≤ N −4k 2 ≤ N −4kj .
To sum up, we have verified (79), assuming that (92) holds for π > N −τ . Turning to the remaining claim (92), using assumption (71) we see that π > N −τ implies
λ ≥ Bµ 1/(q−ℓ+1) ≥ B(log N )/A ≥ 4k 2 log N.
Similarly, π > N −τ , ℓ ≤ j < q and N ≥ 1 imply
R j /D j+1 = λ q−j /B q−j−1 ≥ Bµ 1/(q−ℓ+1) q−j /B q−j−1 ≥ B (log N )/A q−j ≥ 4k 2 log N,
establishing (92). As discussed, this completes the proof of (72).
Applications
In this section we illustrate our concentration techniques, by applying the basic inequalities from Section 5.1 to several pivotal examples. In Section 6.1 we improve previous work of Janson and Ruciński [16] on random induced subhypergraphs, and derive sharp upper tail inequalities for several quantities of interest in additive combinatorics. In Section 6.2 we answer a question of Janson and Ruciński [13] on subgraph counts in binomial random graphs, and improve the main applications of Wolfovitz [38] andŠileikis [26].
Random induced subhypergraphs
In probabilistic combinatorics, random induced subhypergraphs H p are a standard test-bed for upper tail inequalities (see, e.g., Section 3 in the survey [14]). Janson and Ruciński studied the number of randomly induced edges in [16], and one of their principle results concerns k-uniform hypergraphs with v(H) = N vertices, e(H) ≥ γN q edges and ∆ q (H) ≤ D (for easier comparison with Theorem 2.1 in [16], note that ∆ j (H) ≤ N max{q−j,0} ∆ q (H) holds). Writing X = e(H p ) and µ = EX, they obtained bounds of form
exp −C(ε)µ 1/q log(1/p) ≤ P(X ≥ (1 + ε)µ) ≤ exp −c(ε)µ 1/q ,(93)
determining log P(X ≥ (1 + ε)µ) up to a missing logarithmic factor (in fact, their lower bound needs an extra assumption). For 2 ≤ q < k the following corollary of Theorem 30 improves the exponential rate of decay of (93) in the more general weighted case. Noteworthily, inequality (94) below closes the log(1/p) gap left open by Janson and Ruciński [16] (for the special case q = 2 this was already resolved in [36]). 1) it is routine to see that P(w(H p ) ≥ (1 + ε)µ) = exp −Θ(N p) = exp −Θ(µ 1/q ) holds, i.e., that there is no logarithmic term.
Concerning sharpness of (94), in applications we usually do not consider a single hypergraph H, but sequences of hypergraph (H N ) N ∈N which are nearly monotone, i.e., where H N ⊆ H N +1 holds up to some minor 'defects' (arising, e.g., due to boundary effects). The following remark states that, in this frequent case, the upper tail inequality (94) is best possible up to the value of the parameter c (for 2 ≤ q < k).
Remark 38 (Matching lower bound). Let 2 ≤ q < k and γ, D, a, L, n 1 , n 2 > 0. Let (H N ) N ≥n1 be a sequence of k-uniform hypergraphs such that all H = H N satisfy the assumptions of Theorem 36. Assume that there is β ∈ (0, 1] such that e(H N ∩ H M ) ≥ βe(H N ) for all M ≥ N ≥ n 2 . Then for all ε > 0 there are n 0 = n 0 (k, γ, D, a, L, β, n 1 , n 2 ) > 0 and C = C(ε, γ, k, q, D, a, L, β, n 1 , n 2 , ) > 0 such that for all H = H N with N ≥ n 0 , setting X = w(H p ) and µ = EX, for all p ∈ (0, 1] we have
P(X ≥ (1 + ε)µ) ≥ ½ {1≤(1+ε)µ≤w(H)} exp −C min µ, µ 1/q log(1/p) .(95)
We omit the proof of Remark 38, which mimics the lower bound techniques from [36] in a routine way.
Proof of Theorem 36. Let δ = aγ, and note that µ ≥ e(H)p k · min f ∈H w f ≥ δN q p k (we never use w f ≥ a again, i.e., we could weaken our assumptions). Inequality (94) holds trivially whenever N < k (since then 0 ≤ w(H p ) ≤ L · e(H) = 0), so we may henceforth assume N ≥ k. Our main task is to verify the assumptions of Theorem 30. Let ℓ = 1 and τ = q/(2k). As N 1/2 ≥ log N for all N > 0, for p ≥ N −τ we have µ 1/(q−ℓ+1) = µ 1/q ≥ δ 1/q N p k/q ≥ δ 1/q N 1−kτ /q ≥ δ 1/q N 1/2 ≥ δ 1/q log N.
As discussed in Example 20, using (57) and |Γ U (H)| ≤ v(H) q−j · ∆ q (H), for 1 ≤ j < q we thus have
µ j ≤ N q−j · D · p k−j .(97)
Recalling ℓ = 1, (96) and q < k, there thus is a constant A = A(D, δ) > 0 such that for 1 ≤ j < q we have
µ j µ (q−j)/(q−ℓ+1) ≤ DN q−j p k−j (µ 1/q ) q−j ≤ Dδ j/q−1 p j(k/q−1) ≤ Ap 1/q .(98)
Hence assumptions (70)-(71) hold with π = p and α = 1/q. Using (72) of Theorem 30 it follows that
P(w(H p ) ≥ (1 + ε)µ) ≤ (1 + 3qN −1 )e −Π ,(99)
where Π = c ′ min ε 2 , 1 min{µ, µ 1/(q−ℓ+1) log(e/p)} and c ′ = c ′ (ℓ, q, k, L, D, A, δ) > 0. The author finds (99) quite satisfactory, but in the literature the usually irrelevant prefactor 1 + 3qN −1 is often suppressed for cosmetic reasons. Below we shall achieve this by inflating the constant in the exponent (without assuming that n, p or Π are large). If Π ≥ 6, then N ≥ k ≥ q implies 3qN −1 ≤ 3 ≤ Π/2, so that
P(w(H p ) ≥ (1 + ε)µ) ≤ e −Π+3qN −1 ≤ e −Π/2 .
Otherwise 1 ≥ Π/6 holds, in which case ε/(1 + ε) ≥ min{1, ε}/2 and Markov's inequality yield
P(w(H p ) ≥ (1 + ε)µ) ≤ 1 1 + ε = 1 − ε 1 + ε ≤ e −ε/(1+ε) ≤ e − min{1,ε}Π/12 , establishing (94) for suitable c = c(ε, c ′ ) > 0.
Combining Theorem 36 and Remark 38, we obtain the following convenient upper tail result (see [36] for a similar result in the special case q = 2). It applies to many widely-studied objects in additive combinatorics and Ramsey theory, each time closing the logarithmic gap present in previous work, see (93) and [16].
Corollary 39. Let 2 ≤ q < k and γ, D, a, L, n 1 > 0. Let (H n ) n≥n1 be k-uniform hypergraphs such that H n ⊆ H n+1 , v(H n ) ≤ n, e(H n ) ≥ γn q , ∆ q (H n ) ≤ D, and w f ∈ [a, L] for all f ∈ H n . Then for all ε > 0 there are n 0 = n 0 (k, γ, D, a, L, n 1 ) > 0 and c, C > 0 (depending only on ε, k, γ, D, a, L, n 1 ) such that for all H = H n with n ≥ n 0 , setting X = w(H p ) and µ = EX, for all p ∈ (0, 1] we have
½ {1≤(1+ε)µ≤w(H)} exp −CΨ q,µ ≤ P(X ≥ (1 + ε)µ) ≤ exp −cΨ q,µ ,(100)
where Ψ q,µ = min{µ, µ 1/q log(1/p)}.
In particular, letting the edges of the k-uniform hypergraphs H n with vertex-set V (H) = [n] encode the relevant objects, it is not difficult to check that Corollary 39 with uniform weights w f = 1 implies 7 all the upper tail bounds presented in Examples 2-5 of Section 1.1.1 (using q = 2 for k-term arithmetic progressions, (k, q) = (3, 2) for Schur triples, (k, q) = (4, 3) for additive quadruples, and (k, q) = (r + s, r + s − 1) for (r, s)sums). Motivated by Section 2.1 in [16], we now record a further common generalization of these examples.
Example 40 (Integer solutions of linear homogeneous systems). Let 1 ≤ r ≤ k − 2. Let A be a r × k integer matrix. Following [16], we assume that every r × r submatrix B of A has full rank, i.e., rank(B) = r = rank(A). We also assume that there exists a distinct-valued positive integer solution to Ax = 0, where x = (x 1 , . . . , x k ) is a column vector and 0 = (0, . . . , 0) is an r-dimensional column vector. Let the edges of the k-uniform hypergraph H n with V (H n ) = [n] encode solutions {x 1 , . . . , x k } ⊆ [n] of the system Ax = 0 with distinct x i . The discussion of Section 2.1 in [16] implies that (H n ) n≥n1 satisfies the assumptions of Corollary 39 with q = k − r, so the upper tail inequality (100) holds for X = e(H p ), say.
Small expectations case
Note that inequality (100) does not guarantee a similar dependence of c, C > 0 on ε. Of course, we can also ask for finer results, which determine how the exponential decay of the upper tail depends on ε. The following corollary of Theorem 32 provides a partial answer for small p (see [36] for results which for q = 2 cover all p).
Theorem 41. Let k ≥ 2. Let 1 ≤ q ≤ k and D, L > 0. Assume that H is a k-uniform hypergraph with v(H) ≤ N , ∆ q (H) ≤ D and max f ∈H w f ≤ L, where N ≥ 1. Set X = w(H p ) and µ = EX. For all σ, Λ > 0 there are c = c(σ, Λ, k, D, L) > 0 and d = d(q) ≥ 1 such that for all p ≤ ΛN −(q−1)/(k−1)−σ and t > 0 we have P(X ≥ µ + t) ≤ d exp −c min ϕ(t/µ)µ, t 1/q log N .
Furthermore, setting p = m/v(H), inequality (101) also holds with H p replaced by H m .
Assume that H = H N also satisfies e(H N ) ≥ γN q , the monotonicity conditions of Remark 38, w f = 1 and 2 ≤ q < k. Mimicking the lower bound arguments from [36], inequality (101) can then shown to be best possible up to the values of d, c for some range of small p (we leave the details to the interested reader).
Proof of Theorem 41. Our main task is to verify assumption (73) of Theorem 32. To this end we exploit that
q − 1 k − 1 = max 1≤j<q q − j k − j .
Indeed, using (97) and N ≥ 1 there thus is a constant A = A(D, Λ) > 0 such that we have
max 1≤j<q µ j ≤ 1≤j<q DN q−j p k−j ≤ D 1≤j<q Λ k−j N (q−j)−(k−j)(q−1)/(k−1)−(k−j)σ ≤ AN −σ .
Applying Theorem 32 (with σ = α and K = 1) now readily establishes inequality (101).
Subgraph counts in random graphs
In this section we consider subgraph counts in the binomial random graph G n,p , which are pivotal examples for illustrating various concentration methods (see, e.g., [20,32,33,14,15,12] and Examples 21-22 in Section 4.1.1). We shall discuss two qualitatively different upper tail bounds in Sections 6.2.1 and 6.2.2. We henceforth tacitly write X = X H for the number of copies of H in G n,p , and set µ = EX = Θ(n vH p eH ). Let us recall some definitions from random graph theory.
Small deviations: sub-Gaussian type bounds
We first consider sub-Gaussian type P(X ≥ µ + t) ≤ C exp(−ct 2 / Var X) upper tail inequalities. Our main focus is on the Poisson range, where Var X ∼ EX = µ holds, which according to Kannan [19] is the more difficult range. For small p the following simple corollary of Theorem 32 extends/sharpens several results from [31,15,26,38,19,37], and implies Theorem 6. (For balanced and 2-balanced graphs H it is folklore that δ H ≥ 1. Furthermore, with the exception of perfect matchings, all 2-balanced graphs are strictly balanced.)
Theorem 42 (Subgraph counts in random graphs: small expectations case). Let H be a graph with v = v H vertices, e = e H edges and minimum degree δ = δ H . Let X = X H and µ = EX. Define s = min{v − 1, e − δ + 1}. If H is strictly balanced, then for every Λ > 0 there are c = c(Λ, H) > 0 and C = C(H) ≥ 1 such that for all n ≥ v, ε ∈ (0, Λ] and p ∈ [0, 1] satisfying µ (s−1)/s ≤ Λ log n we have P(X ≥ (1 + ε)µ) ≤ C exp −cε 2 µ .
If H is 2-balanced, then for all σ, Λ > 0 there are c = c(σ, Λ, H) > 0 and C = C(H) ≥ 1 such that for all n ≥ v, 0 ≤ p ≤ Λn −(v−2)/(e−1)−σ and 0 < t ≤ Λ min{(µ log n) 1/(2−1/s) , µ} we have P(X ≥ µ + t) ≤ C exp −ct 2 /µ .
Remark 43. It is well-known that in (102)-(103) we have µ = EX ∼ Var X when p = o(1). The proof shows that the constants C can be replaced by 1 + o(1), and that (102)-(103) both carry over to G n,m . Furthermore, [27] demonstrates that the sub-Gaussian type tail inequality (102) can already fail for balanced graphs H.
To put Theorem 42 into context, in the year 2000 Vu [31] showed that the sub-Gaussian inequality (102) holds for strictly balanced graphs as long as ε = O(1) and µ ≤ log n (note that ε 2 µ ∼ (εµ) 2 / Var X by Remark 43). Shortly afterwards, this result was reproved via a different method by Janson and Ruciński [15], who also raised the question whether the restriction µ = O(log n) is necessary (see Section 6 in [13]). For the special case ε = Θ(1) the aforementioned results were yet again reproved byŠileikis [26] in 2012. Our methods allow us (i) to go beyond all these three approaches from 2000-2012, and (ii) to answer the aforementioned question of Janson and Ruciński: inequality (102) still holds in the wider range µ = O((log n) 1+ξ ).
Wolfovitz demonstrated the applicability of his sub-Gaussian concentration result [38] via the complete graph K r and the complete bipartite graph K r,r , showing that inequality (103) holds for both strictly 2balanced graphs in certain ranges of the parameters p, t. Theorem 42 generalizes these main applications from [38] to all 2-balanced graphs (for a slightly wider parameter range). For n −1 ≤ p ≤ n −1/2−σ inequality (103) also slightly extends the t-range of two K 3 -specific results of Kannan [19] and Wolfovitz [37].
Proof of Theorem 42. The proofs of (102)-(103) are very similar: each time we shall apply Theorem 32 twice, using the two different setups of Examples 21-22. Hence our main task is to check assumption (73).
For (102) we assume that H is strictly balanced, in which case δ = δ H ≥ 1 is folklore. By assumption there is a constant β = β(H) > 0 such that for all subgraphs J H with v J ≥ 1 we have v J · e v ≥ e J + β and e J · v e ≤ v J − β.
Armed with (107), we now apply Theorem 32 with K = 1, A = B 5 and α = β/4, using the setup of Example 21 (with ℓ = 1, k = e, q = e − δ + 1 and N = n 2 ) and Example 22 (with ℓ = 2, k = q = v and N = n). So, applying (74) twice, there is a constant c 1 > 0 such that for t = εµ we have P(X ≥ µ + t) ≤ 1 + 2 max{v H , e H }n −1 exp −c 1 min t 2 /µ, t, t 1/s log n .
Since t = εµ ≤ Λµ, we infer t ≥ t 2 /(Λµ). Hence, after adjusting the constant c 1 , the t-term is irrelevant for the exponent of (108). As t 2−1/s ≤ (Λµ) 1+(s−1)/s = O(µ log n) by assumption, this establishes (102). For (103) we proceed similarly, assuming that H is 2-balanced. In this case, for all subgraphs J H with 2 ≤ v J < v, the assumption that H is 2-balanced (and noting that (109) is trivial when v J = 2) implies
e − e J v − v J = (e − 1) − (e J − 1) (v − 2) − (v J − 2) ≥ e − 1 v − 2 .(109)
Applying Theorem 30 and Remark 31 with A = B 2 and α = β/2, there thus is c 1 > 0 such that P(X ≥ (1 + ε)µ) ≤ (1 + 3e H n −2 ) exp −c 1 min ε 2 , 1 min{µ, µ 1/e log n} .
Next we use the setup of Example 22 with ℓ = 2, k = q = v and N = n. We distinguish several cases. If p ≤ n −v/e , then using the bound (58) for µ j and the density result (104), we infer for 2 ≤ j < v = v H that
Otherwise p ≥ n −v/e , so n v p e ≥ 1. Note that for j < v we have
(v − j)/(v − 1) ≥ (v − j)/v + 1/v 2 .
Recalling ℓ = 2 and q = v, using (59), µ = Θ(n v p e ) and (104) we infer for 2 ≤ j < v = v H that
µ j µ (q−j)/(q−ℓ+1) ≤ µ j B 4 (n v p e ) (v−j)/v+1/v 2 ≤ B 5 J⊆H:vJ =j p vJ e/v−eJ (n v p e ) 1/v 2 ≤ B 6 p β (n v p e ) 1/v 2 .(116)
Distinguishing n −v/e ≤ p ≤ n −v/(2e) and n −v/(2e) ≤ p ≤ 1, we see that µ j µ (q−j)/(q−ℓ+1) ≤ B 6 max{n −βv/(2e) , n −1/(2v) }.
(117)
Applying Theorem 30 and Remark 31 with A = max{B 3 , B 6 } and α = min{β, βv/(2e), 1/(2v)}, we deduce P(X ≥ (1 + ε)µ) ≤ (1 + 3v H n −1 ) exp −c 2 min ε 2 , 1 min{µ, µ 1/(v−1) log n} .
Finally, we combine the two upper bounds (114) and (118), and then remove (for cosmetic reasons) the multiplicative prefactor 1 + O(n −1 ) analogous to the proof of Theorem 36, which establishes (112).
For Remark 46 the point is that for balanced graphs H the density condition (104) only holds with β = 0, so in (116) we need p ≥ ξn −v/e+σ to establish (117) with ≤ O(n −eσ/v 2 ), say.
setup: random induced subhypergraph H p Our basic setup concerns random induced subhypergraphs. For a hypergraph H with vertex set V (H), let V p (H) denote the binomial random vertex subset where each v ∈ V (H) is included independently with probability p. We define the subhypergraph of H induced by V p (H) as
-(28) into concrete upper tail inequalities of form (23) and (26); see Sections 3.3.1, 3.4, 4.2 and 5.3 for these technical calculations.
U1,...,UR): Ui⊆V (H)s and |Ui|=j P ⊡ i∈[R] D Ui,x,y ≤ (U1,...,UR): Ui⊆V (H) and |Ui|=j i∈[R] P(D Ui,x,y ) ≤ U⊆V (H):|U|=j P(D U,x,y ) R ,
For verifying the independence assumption of Theorem 7, we use the following simple observation: e ∩ f = ∅ implies that ½ {e∈Hp} = ½ {e⊆Vp(H)} and ½ {f ∈Hp} = ½ {f ⊆Vp(H)} are independent, since both depend on disjoint sets of independent variables ξ σ = ½ {σ∈Vp(H)} . Assuming (e ∩ f ) \ U = ∅, we below exploit that an analogous (conditional independence) reasoning works after conditioning on U ⊆ V p (H). Proof of Theorem 18. With an eye on Theorem 7, inspired by Remark 8 we set ξ σ = ½ {σ∈Vp(H)} .We first prove(46).Let Y f = w f ½ {f ∈Hp} , which satisfies Y f = w f σ∈f ξ σ and f ∈H EY f = Ew(H) = µ.Furthermore, w(G) = w∈G Y f for any G ⊆ H p . Defining α ∼ β if α ∩ β = ∅, the independence assumption of Theorem 7 holds by Remark 8. Observe that for any f ∈ G ⊆ H with ∆ 1 (G) ≤ y we have e∈G:e∼f
and N ≥ 1. Given U ⊆ V (H) with |U | = j, set I := Γ U (H) = {f ∈ H : U ⊆ f }. Let Y f = ½ {f ∈Hp} , and define α ∼ β if (α ∩ β) \ U = ∅. Note that for any f ∈ K ⊆ I with ∆ |U|+1 (K) ≤ y we have e∈K:e∼f Y e = e∈K:(e∩f )\U =∅
Taking expectations, by summing over all relevant U ⊆ V (H) we thus infer U⊆V (H):|U|=j P(D U,x,y ) = U⊆V (H):|U|=j
follows from (41) of Lemma 16, with room to spare. The proof of Claim 14 (only used in our informal discussion) is very similar, and thus left to the reader.
3. 5
5Extension: uniform random induced subhypergraph H m The proofs in Sections 3.3-3.4 exploited the independence of H p = H[V p (H)] in a limited way. In this section we record that they extend to the uniform model H m = H[V m (H)], where the vertex subset V m (H) ⊆ V (H) of size |V m (H)| = m is chosen uniformly at random (this is a natural variant of H p with mild dependencies). Remark 19. Theorems 15 and 18 carry over to H m after setting p = m/v(H) in (21).
Example 21 (
21Subgraph counts in G n,p : induced subhypergaphs approach). Subgraph counts in G n,p can be viewed as a special case of Example 20, i.e., random induced subhypergaphs. Given a fixed subgraph H with e = e H edges, v = v H vertices and minimum degree δ = δ H ≥ 1, we consider the e-uniform hypergraph H with vertex set V (H) = E(K n ), where edges correspond to copies of H. Clearly, k = e and N = n 2 suffice. Note that for the copy of H counted by Y f , any subset of the edges U ⊆ f ∩ E(K n ) ⊆ V (H) is isomorphic to some subgraph J ⊆ H. So, taking all subgraphs of H with exactly |U | = j edges into account, using (57) with k = e and V (H) = E(K n ) there is universal constant B = B(H) > 0 such that for 1 ≤ j ≤ e we have µ j ≤ J⊆H:eJ =j max U⊆E(Kn): U ∼ =J |Γ U (H)| · p e−j ≤ B J⊆H:eJ =j
Remark 28 .
28Theorems 23 and 26 remain valid after replacing the assumptions (Hℓ),(P) with (HℓP). Proof. With the ideas of Remark 19 in mind, we only sketch the key modifications for (64)-(65) of Theorem 26.For (64) it suffices to verify the negative correlation condition of Remark 9, writing α ∼ β if |α ∩ β| ≥ ℓ. Using (67) and the negative correlation properties of F m (see Remark 9), it is not hard to check that E i∈[s]
⊆Vm(F )} , analogous to (69) we infer E( i∈[s] ½ {Yα i } ) ≤ i∈[s] λ αi , establishing the correlation condition of Remark 10. Mimicking Remark 19, the proof of (47) then carries over to (65).5 Corollaries: upper tail inequalitiesThe main results of Sections 3-4 are Theorems 15,23 of form P(X ≥ (1 + ε)EX) ≤ i P(¬E i ) and Theorems 18,26 of form P(¬E i ) ≤ exp(−Ψ i ). In this section we derive upper tail inequalities that are convenient for the applications of Section 6, and briefly compare some of our more general estimates with the literature.Remark 29 (Random induced subhypergraph setup). The results in Sections 5.1-5.2 are stated for the general setup of Section 4.1. But, with minor changes, they remain valid in the simpler random induced subhypergraph setup of Section 3.1.1. Indeed, setting ℓ = 1 and replacing the assumptions (Hℓ),(P) with (P'), all results carry over to H p by defining X(J ) := w(J ). After setting p = m/v(H) in (21), these results for H p then also carry over to the uniform variant H m defined in Section 3.5. Finally, after replacing the assumptions (Hℓ),(P) with (HℓP), all results in Sections 5.1-5.2 also remain valid in the setup of Section 4.3.
1≤j<e O(
1≤j<eJ⊆H:eJ =j n v−vJ p e−j ) Θ((n v p e ) (e−j)/e ) ≤ O( J⊆H:1≤eJ <e n eJ v/e−vJ ),
, for subgraph counts in G n,p with p = O(n −v/e+σ ), the setup of Example 21 (with ℓ = 1, q = k = e and N = n 2 ) yields µ = O(n
Theorem 36 (
36Weighted edge-count of random induced subhypergraphs). Let 1 ≤ q < k and γ, D, a, L > 0.Assume that H is a k-uniform hypergraph with v(H) ≤ N , e(H) ≥ γN q , ∆ q (H) ≤ D, and w f ∈ [a, L] for all f ∈ H. Set X = w(H p ) and µ = EX. For any ε > 0 there is c = c(ε, k, γ, D, a, L) > 0 such that for all p ∈ (0, 1] we have P(X ≥ (1 + ε)µ) ≤ exp −c min µ, µ 1/q log(e/p) .(94)Remark 37. Setting p = m/v(H), inequality (94) also carries over to H m as defined in Section 3.5. Inequality (94) does not always hold in the excluded case q = k. A concrete counterexample is the complete k-uniform hypergraph H = H N with V (H) = [N ] and w f = 1. Then q = k, X = |[N ]p| k ≈ |[N ] p | k /k! and µ = N k p k ≈ (N p) k /k!. For µ = ω(1), p ≤ 1/2 and ε = Θ(
Writing d(J) = e J /v J , a graph H is called balanced if e H ≥ 1 and d(H) ≥ d(J) for all J H with v J ≥ 1. If this holds with d(H) > d(J), then H is called strictly balanced. Writing d 2 (J) = (e J − 1)/(v J − 2), a graph H is called 2-balanced if e H ≥ 2 and d 2 (H) ≥ d 2 (J) for all J H with v J ≥ 3. If this holds with d 2 (H) > d 2 (J), then H is called strictly 2-balanced.
setup of Example 21, by (58) there is a constant B 1 > 0 such that the corresponding µ j satisfy max 1≤j<e−δ+1 µ j ≤ B 1 J⊆H:1≤eJ <e−δ+1 n v−vJ p e−eJ . (105) Similarly, using the setup of Example 22, by (59) there is a constant B 2 > 0 such that max 2≤j<v µ j ≤ B 2 J⊆H:2≤vJ <v n v−vJ p e−eJ . (106) Recalling s = min{v − 1, e − δ + 1}, in our further estimates of (105)-(106) we may assume s > 1 (otherwise H = K 2 and (105)-(106) are both equal to zero). Recalling µ = Θ(n v p e ), we now pick S = S(Λ, H) ≥ 1 large enough such that the assumption µ (s−1)/s ≤ Λ log n implies p ≤ Sn −v/e+β/(2e) for all n ≥ v. Using δ = δ H ≥ 1 and the density condition (104), it follows that there are constants B 3 , B 4 , B 5 > 0 such that (105) + (106) ≤ B 3 J⊆H:vJ ≥2,eJ <e n v−vJ p e−eJ ≤ B 4 J⊆H:vJ ≥2,eJ <e n eJ v/e−vJ +β/2 ≤ B 5 n −β/2 .
v/e−vJ ≤ B 3 n −β .
It follows that M j,x,y (H p ) ≤r implies E j,1 (x, r, y, z), as claimed.For (41) it remains to estimate P(M j,x,y (H p ) >r). Similar to the proof of Theorem 11 in[36], we setZ R =
(U1,...,UR):
Ui⊆V (H) and |Ui|=j
Suppose that there are families of subsets A U ⊆ A such that (i) each non-negative random variable Y f with f ∈ H is a function of the variables (ξ σ ) σ∈A f , (ii) we have A e ∩ A f ⊆ A e∩f for all e, f ∈ H, and (iii) we have A e ∩ A f = ∅ for all e, f ∈ H with |e ∩ f | < ℓ.The setup of Section 3.1.1 corresponds to the special case ξ σ
Note that P(X ≥ (1 + ε)µ) = 0 when (1 + ε)µ > e(H), and that P(X ≥ (1 + ε)µ) = 1 − P(X = 0) when (1 + ε)µ < 1. 2 A phenomenon not relevant for the qualitative accuracy of (3)-(4) is that |Vp(H)| can also be somewhat 'bigger' than E|Vp(H)|, which in some range yields sub-Gaussian type tail behaviour, see also[36,28].
For subgraph counts lower tail estimates of sub-Gaussian type follow from Janson's inequality (see, e.g.,[18]).
Note that by setting D j = R j = S j the indicators in (35)-(36) are zero, so (33) qualitatively reduces to(32).
Usually we have X = f ∈H w f I f in mind, for random variables I f ∈ {0, 1} and constants w f ∈ (0, ∞). All examples and applications in[20,31,33,15,14,16] are of this form, with w f = 1 (possibly after rescaling X by a constant factor).
Note that by setting s = 1 and D j = R j we have Q j = R j in (78), so the indicators in (79)-(80) are zero and Theorem 34 recovers Claim 33 up to irrelevant constant factors.
Note that using weights w f = 1 we count unordered objects, i.e., treat the objects as k-sets (if desired, we could also treat them as ordered k-vectors by using non-uniform weights w f > 0, say).
Acknowledgement.We are grateful to the referees for helpful suggestions concerning the presentation.A Proofs omitted from Section 6.2.2In this appendix we give the proof of Theorem 44, which proceeds similar to Theorem 36 and 42. Namely, we prove (112) by two applications of Theorem 30 and Remark 31 (using the setups of Examples 21-22).Proof of Theorem 44. We first use the setup of Example 21 with ℓ = 1, q = k = e and N = n 2 . Using the bound (58) for µ j , the expectation µ = Θ(n v p e ) and the density result (104), for 1 ≤ j < e = e H we infer
J⊆H:vJ ≥2,eJ <e n (v−vJ )−(e−eJ )(v−2)/(e−1)−(e−eJ )σ ≤ B 7 n −σ. J⊆H:vJ ≥2,eJ <e n (v−vJ )−(e−eJ )(v−2)/(e−1)−(e−eJ )σ ≤ B 7 n −σ .
but it turns out that our methods can routinely sharpen results based on classical inductive approaches (which might potentially be useful in other contexts). Indeed, for balanced graphs Kim and Vu used two different inductions (see Sections 6.3 and 6.6 in [33]), which together establish the following tail estimate: if ε ≤ C and ε 2 max{µ 1/(v−1) , µ 1/e } = ω. 12, 7, 6. log n), then P(X ≥ (1 + ε)µ) ≤ exp −cε 2 max µ 1/(v−1) , µ 1/eas [12, 7, 6], but it turns out that our methods can routinely sharpen results based on classical inductive approaches (which might potentially be useful in other contexts). Indeed, for balanced graphs Kim and Vu used two different inductions (see Sections 6.3 and 6.6 in [33]), which together establish the following tail estimate: if ε ≤ C and ε 2 max{µ 1/(v−1) , µ 1/e } = ω(log n), then P(X ≥ (1 + ε)µ) ≤ exp −cε 2 max µ 1/(v−1) , µ 1/e .
Using Theorem 30, we shall go beyond both approaches for strictly balanced graphs: (i) we improve the exponential rate of decay by an extra logarithmic factor, and (ii) we remove the restriction to 'large' expectations µ. Theorem 44. Let H be a strictly balanced graph with v = v H vertices and e = e H edges. Let X = X H and µ = EX. For any ε > 0 there is c = c(ε, H) > 0 such that for all n ≥ v and p ∈ [0, 1] we have P(X ≥ (1 + ε)µ) ≤ exp −c min µ. This inequality was reproved by Janson and Ruciński [15] via their alternative inductive method. max µ 1/(v−1) , µ 1/e log nThis inequality was reproved by Janson and Ruciński [15] via their alternative inductive method. Using Theorem 30, we shall go beyond both approaches for strictly balanced graphs: (i) we improve the exponential rate of decay by an extra logarithmic factor, and (ii) we remove the restriction to 'large' expectations µ. Theorem 44. Let H be a strictly balanced graph with v = v H vertices and e = e H edges. Let X = X H and µ = EX. For any ε > 0 there is c = c(ε, H) > 0 such that for all n ≥ v and p ∈ [0, 1] we have P(X ≥ (1 + ε)µ) ≤ exp −c min µ, max µ 1/(v−1) , µ 1/e log n .
the proof shows that c = c ′ min{ε 2 , 1} with c ′ = c ′ (H) > 0 suffices when min{ε 2 , 1}Ψ ≥ 1. Furthermore, inequality (112) also carries over to G n,m . Remark 46. For balanced graphs H, the proof yields the following variant: for all n ≥ v, p ≥ ξn −v/e+σ and ε > 0 we have P(X ≥ (1 + ε)µ) ≤ exp(−cµ 1/(v−1) log n). Remark 45. Writing the exponent of (112) in the form exp(−cΨ. where c = c(σ, ξ, ε, H) > 0Remark 45. Writing the exponent of (112) in the form exp(−cΨ), the proof shows that c = c ′ min{ε 2 , 1} with c ′ = c ′ (H) > 0 suffices when min{ε 2 , 1}Ψ ≥ 1. Furthermore, inequality (112) also carries over to G n,m . Remark 46. For balanced graphs H, the proof yields the following variant: for all n ≥ v, p ≥ ξn −v/e+σ and ε > 0 we have P(X ≥ (1 + ε)µ) ≤ exp(−cµ 1/(v−1) log n), where c = c(σ, ξ, ε, H) > 0.
For r-armed stars H = K 1,r inequality (112) yields an exp −Ω(min{µ, µ 1/r log n}) exponential decay, which by [28] is best possible for p ≤ n −1/r and ε = Θ(1). However, for general graphs H other approaches such as. 12, 7, 6] yield better estimates (as mentioned before. so we defer the proof of Theorem 44 to Appendix A. ReferencesFor r-armed stars H = K 1,r inequality (112) yields an exp −Ω(min{µ, µ 1/r log n}) exponential decay, which by [28] is best possible for p ≤ n −1/r and ε = Θ(1). However, for general graphs H other approaches such as [12, 7, 6] yield better estimates (as mentioned before), so we defer the proof of Theorem 44 to Appendix A. References
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arxiv |
High Q 2 HERA Events and pQCD at High x
20 Jun 1997 June 1997
Stephen Rock
American University
Washington D.C. 20016
Peter Bosted
American University
Washington D.C. 20016
High Q 2 HERA Events and pQCD at High x
20 Jun 1997 June 1997
We compare data on F p 2 from SLAC experiments in the range 0.7 ≤ x ≤ 0.97 in and near the resonance region with previous empirical fits to Deep Inelastic Scattering and with calculations from parton distribution functions. The data is in rough agreement with the empirical fits, but is an order of magnitude higher than the pQCD calculations at the highest x. This compares with the two orders of magnitude increase in the quark distributions at high x which seem to be necessary to explain the HERA high Q 2 events.
Recent interest in the anomalous high Q 2 , high x events at HERA [1] has generated interest in the accuracy of the pQCD evolution to these regions. This evolution depends on 1 x of the quark distributions, thus making the high x quark distributions important. pQCD fits have ignored data in the region x ≥ 0.75 because of fear of possible higher twist contamination. However, there is a wealth of SLAC data in the region up to x = 0.97 [2,3], albeit in the resonance region (W 2 ≤ 4GeV 2 ) and at relatively low Q 2 . Bloom-Gilman [4] duality states that the average of the data in the resonance region approximates the DIS structure. With that in mind, we compare pQCD and various fits to the DIS region with the SLAC high-x resonance data. Figure 1 shows a sample of the SLAC measurements of F p 2 for x ≥ 0.7 and 7 ≤ Q 2 ≤ 30. F p 2 decreases by approximately 3 orders of magnitude over this x range and there is a Q 2 dependence of approximately a factor of 2 to 4 between the various bands of data. Figure 2 shows the ratio the same SLAC data to the NMC DIS fit [5]. This fit used only data with x ≤ 0.75. The data in Figure 2. span the range 7 ≤ Q 2 ≤ 30. In the range of validity of the fit ( x ≤ 0.75), the ratio is similar to unity, as expected. The ratio raises to about 2, at the highest x. This fit is dominated by a (1 − x) 3 behavior at high x as predicted by the quark counting rules [6]. The agreement is remarkable considering that the data at the highest x is in the region of the first resonance, far from the normal 1 DIS region. The SLAC global fits used data up to x=0.85. As seen in Figure 3, similar good agreement is obtained when using the SLAC-Λ 12 [3] as the denominator. This fit is also dominated by a (1 − x) 3 behavior at high x. As observed in Figure 4, the ratio of data to the SLAC-Ω 9 fit is close to unity for the high Q 2 data, but the ratio is less than unity for the lower Q 2 , high-x points. Thus previous fits to world DIS data can be used to estimate the F p 2 structure function in the resonance region to within a factor of about two. Figure 5 shows the ratio of SLAC data to a typical pQCD evolution fit, CTEQ 4M [7]. Again, there is good agreement for x ≈ 0.7. However, the ratio raises sharply and is about a factor of 10 at the highest x. Note that since the data shown in Figure 1 decreases by a factor of 1000 over this range in x, this indicates that the pQCD fits decrease by 4 orders of magnitude. The difference between the pQCD fit and the data could be explained by "higher twist" effects, but they would have to be an order of magnitude greater than the pQCD evolution and also behave similar to the quark counting rule prediction of (1 − x) 3 . A typical higher twist form is [8]. Figure 6 shows the ratio of the data to this higher twist form with C HT = 4. While the rapid rise at high x from pQCD alone is considerably suppressed, the DIS region of excellent pQCD fits (x ≈ 0.7) is suppressed by an unacceptably large 30%. A previous higher twist analysis [9] for x ≤ 0.75 determined that the equivalent of C HT is ≈ 0.3, an order of magnitude smaller than that used in Figure 6.. Of course, more complex higher twist terms could be included with higher powers of Q 2 (1−x) which would eventually fit the data.
F pQCD 2 [1 + C HT /(Q 2 (1 − x))]
Recently Kuhlmann, Lai and Tung [10] have tried to explain the HERA events using a pQCD toy model which enhances the u quark probability beyond the CTEQ4M fit by approximately two orders of magnitude at x=0.95. This pQCD fit would overestimate the SLAC data by an order of magnitude and thus require a huge negative "higher twist effect" of approximately the form discussed above.
In conclusion, the SLAC data in the kinematic region .75 ≤ x ≤ .97 which is mostly in the resonance region, is an order of magnitude higher than current pQCD parton distributions , and an order of magnitude less than the toy models created created to explain the high Q 2 HERA events. Existing DIS fits with quark counting rule behavior at high x are a reasonable approximation of the data, consistant with Bloom-Gilman duality. A dedicated experiment with the SLAC 50 GeV electron beam could measure F p 2 in the kinematic region 0.80 ≤ x ≤ 0.95 in the region of Q 2 ≥ 50GeV 2 2 with the possibility of tagging charm. This data could be used to disentangle higher twist effects from pQCD evolution, and determine the u and c quark distributions at high x.
Figure 1: Selected SLAC data for F p 2 in the range 0.7 ≤ x ≤ 0.97 and 7 ≤ Q 2 ≤ 30. Most of the data at the higher x values is in the resonance region.Figure 1for F p 2 divided by the pQCD NLO fit 4M done by CTEQ with an additional empirical higher twist term described in the text.
DESY-97-024. C Adloff, H1 CollaborationDESY-97-025ZEUS Collaboration. H1 Collaboration (C. Adloff et. al., DESY-97-024, Feb 1997. ZEUS Col- laboration (J.Breitweg et. al., DESY-97-025, Feb. 1997
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S Kuhlmann, H L Lai, W K Tung, CTEQ-705. 97043382MSU-HEP-70316S. Kuhlmann, H. L. Lai, and W. K. Tung, MSU-HEP-70316, CTEQ-705, hep-ph/9704338 v2.
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arxiv |
GYRO-GROUPS, GYRO-SPLITTINGS AND CO-HOMOLOGY
18 Feb 2023
Ramji Lal
Vipul Kakkar
GYRO-GROUPS, GYRO-SPLITTINGS AND CO-HOMOLOGY
18 Feb 2023arXiv:2302.09366v1 [math.GR]Gyro-groupsGyro-splittingsCo-homologySchur Multipliers
In this paper, we study gyro-groups associated to groups, group extensions admitting gyro-sections, and corresponding co-homologies. We also describe the obstructions in terms of co-homomology. The notion of gyro-Schur Multiplier and that of gyro-Milnor K 2 group are introduced.Gyro-groups, Gyro-splittings, Co-homology, Schur Multipliers.
Introduction
Let G be a group. We have an associated right loop (G, • 1 ), where the binary operation • 1 is given by x • 1 y = y −1 xy 2 . The study of groups G with prescribed properties on the associated right loop (G, • 1 ) was initiated by Foguel and Ungar [3,4]. Indeed, they studied groups with prescribed properties on the associated left loop (G, •) given by x • y = x 2 yx −1 . However, for our convenience, we shall study it through the right loop structure (G, • 1 ). It can be seen that (G, • 1 ) is a right gyro-group [7,8]. Foguel and Ungar [4] showed that (G, • 1 ) is a gyro-group if and only if G is central by 2-Engel group. Gyro-groups have deep intrinsic relationship with twisted subroups, near subgroups [1], and in turn, with the group theoretic subclass of constraint satisfaction problems [2]. The twisted version of right gyrogroups and subgroups has been studied in [8]. A group G is said to be weakly isomorphic or gyro-isomorphic to a group K if (G, • 1 ) is isomorphic (K, • 1 ). A weak classification program was initiated in [6]. More generally, a map f from G to K will be termed as a gyro-homomorphism if f (a • 1 b) = f (a) • 1 f (b) for all a, b ∈ G. The main purpose of this paper is to introduce and study the extensions admitting sections which are gyro-homomorphisms. We also study the resulting co-homologies, obstructions, and an analogue of Schur multiplier which will be termed as Gyro-Schur multiplier. In turn, we introduce the notion of gyro-Milnor K 2 -group.
Preliminaries
This section is devoted to some basic notions, definitions and results. A magma (S, •) with identity e is called a right loop if the equation X • a = b has a unique solution in S for all a, b ∈ S.
Let (S, •) be a right loop with identity e. For each x, y, z ∈ S, the unique solution to the equation X • (y • z) = (x • y) • z will be denoted by xθf (y, z). The map f (y, z) from S to S defined by f (y, z)(x) = xθf (y, z) is a member of the symmetric group Sym(S) on S which fixes e ∈ S. Thus, f (y, z) is a member of Sym(S − {e}) ⊂ Sym(S) and which is termed as an inner mapping of (S, •) determined by the pair (y, z) ∈ S × S. Since we shall be dealing with right loops and right transversals, for convenience, we shall adopt the convention (p • q)(x) = q(p(x)) for the product in Sym(S). The subgroup of Sym(S) generated by the set {f (y, z) | y, z ∈ S} of all inner mappings is termed as the inner mapping group (also termed as the group torsion) of the right loop (S, •). We will denote the inner mapping group of the right loop (S, •) by G S . For each y ∈ S, let R y denote the right multiplication map on S defined by R y (x) = x • y. Clearly, R y ∈ Sym(S) for each y ∈ S and the map R from S to Sym(S) defined by R(y) = R y is an injective map. Let R(S) denote the subgroup of Sym(S) generated by the set {R y | y ∈ S} of all right multiplications. This is called the right multiplication group of (S, •). Since (f (y, z)oR y•z )(x) = f (y, z)(x) • (y • z) = (x • y) • (z) = (R y oR z )(x) for all x, y, z ∈ S, R y oR z = f (y, z)oR y•z for all y, z ∈ S. Again, (xθf (y ′ , y) −1 • y ′ ) • y = x • (y ′ • y) = y for all x, y ∈ S, where y ′ denotes the left inverse of y. This means that R −1 y = f (y ′ , y) −1 oR y ′ for all y ∈ S. In turn, it follows that G S S is a subgroup of R(S), where S has been identified with the set {R y | y ∈ S} through the map R. Consequently, R(S) = G S S. Since G S S = {I S }, S is a right transversal to G S in G S S. The group G S S is called the group extension ( also called the right multiplication group) of S. Finally, G S S is universal in the sense that if G is any group in which (S, •) appears as a right transversal to a subgroup of G, then there is a unique group homomorphism from G S S to G which is identity on S (see Theorem 3.4 [5]). Definition 2.1. ( [3,7]) A magma (S, •) with a right identity e is termed as a right gyro-group if the following four conditions hold:
(i) For each element a ∈ S, there is a right inverse a ′ ∈ S with respect to e in the sense that a • a ′ = e. (ii) For each x, y, z ∈ S, there is a unique element xθf (y, z) ∈ S such that (x • y) • z = xθf (y, z) • (y • z).
(iii) The map f (y, z) from S to S given by f (y, z)(x) = xθf (y, z) is an automorphism of (S, •). (iv) For all y ∈ S, f (y, y ′ ) = I S , where I S is the identity map on S.
The following proposition gives us a necessary and sufficient condition for a magma to be a right gyro-group. [3,7]) A right transversal S to a subgroup H of the group G containing the identity e of G is called a gyro-transversal if S = S −1 = {x −1 | x ∈ S} and h −1 xh ∈ S for all x ∈ S and h ∈ H.
The following proposition relates right gyro-groups and gyro-transversals. For all the undefined terms of the cohomology theory in this paper, we refer [9, Chapter 10].
Gyro-groups and Gyro-transversals
Consider a group G and the semidirect productĜ = G ⋊ Inn(G) of G with Inn(G), where Inn(G) denotes the group of inner automorphisms of G. An element ofĜ is uniquely expressible in the form (x, α), where x ∈ G and α ∈ Inn(G). The product · is given by (x, α) · (y, β) = (xα(y), αβ). Every element (x, α) is uniquely expressible as (x, α) = (e, α)(α −1 (x), I G ). Thus, S = G × {I G } is a right transversal to {e} × Inn(G) inĜ. The induced right loop structure on S is the group structure on S. Since S is a normal subgroup ofĜ, it is a gyrotransversal. Further, an arbitrary right transversal to {e} × Inn(H) inĜ is of the
form S g = {(e, g(x)) · (x, I G ) = (g(x)(x), g(x)) | x ∈ G}, where g is a map from G to Inn(G) with g(e) = I G . Further, (g(x)(x), g(x))(g(y)(y), g(y)) = (e, α)(g(z)(z), g(z)),
where z = g(y) −1 (x)y and α = g(x)g(y)g(z) −1 . Hence, the induced right loop operation • g on S g is given by
(g(x)(x), g(x)) • g (g(y)(y), g(y)) = (g(z)(z), g(z)),
where z = g(y) −1 (x)y. Clearly, the bijective map x → (g(x)(x), g(x)) from G to S g induces a right loop structure• g on G which is given by
x• g y = g(y) −1 (x)y.
Evidently, (S g , • g ) is isomorphic to (G,• g ). It follows from [7,Lemma 5.11] that S g is a gyro-transversal if and only if g(x −1 ) = g(x) −1 and g is equivariant in the sense that g(α −1 (x)) = α −1 g(x)α for all x ∈ G and α ∈ Inn(G). In turn, it also follows [7, Proposition 5.10] that (S g , •) and so also (G,• g ) is a right gyro-group if and only if g(x −1 ) = g(x) −1 and g is equivariant in the sense that g(α −1 (x)) = α −1 g(x)α for all x ∈ G and α ∈ Inn(G). Now, every map g from G to Inn(G) is determined by a map λ from G to G with λ(e) = e such that g(x) = i λ(x) , where i a denotes the inner automorphism defined by i a (x) = axa −1 . To say that (S g , • g ) is a right gyro-group is to say that i λ(
x −1 ) = i (λ(x) −1 ) and i λ(i b −1 (x)) = i b −1 i λ(x) i b for all x, b ∈ G. This, in turn, is equivalent to say that λ(x −1 )λ(x) and λ(b −1 xb)b −1 λ(x) −1 b belong to the center Z(G) for all x, b ∈ G.
In particular, if a map λ satisfies the conditions (i) λ(
x −1 ) = λ(x) −1 , and (ii) λ is equivariant in the sense that λ(b −1 xb) = b −1 λ(x)b
for all x, b ∈ G, then S g is a gyro-transversal and (S g , • g ) is a right gyro-group. In turn, (G,• g ) is a right gyro-group, where• g is given by
x• g y = i λ(y) −1 (x)y = λ(y −1 )xλ(y)y,
x, y ∈ G. For each n ∈ Z, the map λ n from G to G given by λ(x) = x n satisfies the above two conditions. Consequently, for each n, we get a right gyro-group structure • n on G which is given by
x • n y = i y −n (x)y = y −n xy n+1 .
We shall be interested in right gyro-groups (G, • 1 ).
Definition 3.1. A right loop (S, •) will termed as a group based right loop if it is isomorphic to a sub right loop of (G, • 1 ) for some group G.
The category of group based right loops will be denoted by GR. Note that a group need not be a group based right loop. Indeed, a 3-group G is a group based right loop if and only if all elements of order 3 lie in the center of G [6, Corollary 5.4]. Thus, a group of exponent 3 is group based right loop if and only if it is abelian. In particular, the non abelian group of order 3 3 which is of exponent 3 is not a group based right loop.
Definition 3.2. A map f from a group G to a group G ′ is said to be a gyro- homomorphism if f is a homomorphism from (G, • 1 ) to (G ′ , • 1 ). More explicitly, f is said to be a gyro-homomorphism if f (y −1 xy 2 ) = f (y) −1 f (x)f (y) 2 for all x, y ∈ G. A bijective gyro-homomorphism is called a gyro-isomorphism.
Evidently, a group homomorphism is a gyro-homomorphism. However, a gyrohomomorphism need not be a group homomorphism. For example, consider the extra special 3-group G of exponent 3. Then (G, • 1 ) is an abelian group and the identity map I G is a gyro-homomorphism from the group G to the group (G, • 1 ) which is not a group homomorphism. It also follows that gyro-isomorphic groups need not be isomorphic. We have a categoryĜP whose objects are groups and morphisms are gyro-homomorphisms. Evidently, the category GP of groups is a subcategory ofĜP which is faithful but not full, and the categoryĜP is a faithful subcategory of GR which is not full. The proof of the following proposition is straight forward. Proposition 3.3. Let f be a gyro-homomorphism from a group G to a group G ′ . Then the following hold:
(i) f (e) = e.
(ii) The power of an element considered as an element of (G, • 1 ) is the same as that considered as an element of the group G. (iii) f (a n ) = f (a) n for all a ∈ G and n ∈ Z. (iv) Image of a sub right loop of (G, • 1 ) under f is a sub right loop of (G ′ , • 1 ).
(v) Inverse image of a sub right loop (normal sub right loop) of (G ′ , • 1 ) under f is a sub right loop (normal sub right loop) of (G, • 1 ). (vi) The fundamental theorem of gyro-homomorphisms hold in the categoryĜP.
The proof of the fundamental theorem of gyro-group homomorphism can be found in [12,Theorem 30,p. 418]. Inverse image of a subgroup under f need not be a subgroup. Consider the 3-exponent non-abelian group G of order 3 3 . The identity map from G to the elementary abelian 3-group (G, • 1 ) is a gyro-isomorphism. The number of subgroups of (G, • 1 ) is 13 whereas the number of subgroups of G is 4.
Proposition 3.4. A map f from G to G ′ is a gyro-homomorphism if and only if f preserves identity and f (y −1 xy 2 ) = f (y −1 )f (x)f (y 2 ) for all x, y ∈ G.
Proof. Let f be a gyro-homomorphism. From the previous proposition f preserves identity and powers. Consequently,
f (y −1 xy 2 ) = f (y) −1 f (x)f (y) 2 = f (y −1 )f (x)f (y 2 ) for all x, y ∈ G.
Conversely, suppose that f preserves the identity and f (y −1 xy 2 ) = f (y −1 )f (x)f (y 2 ) for all x, y ∈ G. Putting x = y, we get that f (y 2 ) = f (y −1 )f (y)f (y 2 ). This shows that f (y −1 ) = f (y) −1 for all y ∈ G. Further, putting x = y −1 , we get that 1 = f (y −1 )f (y −1 )f (y 2 ). This shows that f (y 2 ) = f (y) 2 for all y ∈ G. ✷ Proposition 3.5. An identity preserving map t from G to G ′ is a gyro-homomorphism if and only if ∂t(y −1 , x)∂t(y −1 x, y 2 ) = 1, where the boundary map ∂t is given by ∂t(x, y) = t(x)t(y)t(xy) −1 .
Proof. Let t be a map from G to G ′ which preserves identity. Then
∂t(y −1 , x)∂t(y −1 x, y 2 ) =t(y −1 )t(x)t(y −1 x) −1 t(y −1 x)t(y 2 )t(y −1 xy 2 ) −1 =1
for all x, y ∈ G if and only if t(y −1 xy 2 ) = t(y −1 )t(x)t(y 2 ) for all x, y ∈ G. The result follows from Proposition 3.4. ✷
Some Universal Constructions
Let X be a set and F (X) be the free group on X consisting of the freely reduced words in X. LetF (X) denote the free group on F (X) consisting of freely reduced words in F (X). Usually, Ω will denote forgetful functors from a category to another category which forgets some structure. Proof. We construct the adjoint functor Σ from RL to GR. Let (S, •) be a right loop. Consider the free group F (S) on S consisting of freely reduced words in S. LetF (S) denote the group having the presentation S; R where R = {(x • y) −1 y −1 xy 2 }. Let Σ(S) denote the subset {y −1 xy 2 R | x, y ∈ S} = {(x • y) R | x, y ∈ S}. Evidently Σ(S) is a sub right loop of (F (S), • 1 ), and hence it is a group based right loop. Clearly, the map i S from S to Σ(S) given by i S (x) = x R is a homomorphism between right loops.
Let f be a homomorphism from (S, •) to a group based right loop (T, • 1 ) ⊂ (G, • 1 ). From the universal property of a free group, we have a unique group homomorphism 2 for all x, y ∈ S. This means that R is contained in the kernel off . In turn, we have a unique group homomorphism f fromF (S) to G. Evidently, f (Σ(S)) ⊆ T and f | Σ(S) is the unique homomorphism from Σ(S) to (T,
f from F (S) to G such thatf (x) = f (x) for each x ∈ S. Since f (x • y) = f (y) −1 f (x)f (y) 2 ,f (x • y) =f (y) −1f (x)f (y)• 1 ) such that f | Σ(S) • i S = f .
Next, let (S ′ , • ′ ) be a right loop and f be homomorphism from (S, •) to (S ′ , • ′ ). Then i S ′ • f is a homomorphism from (S, •) to the group bases right loop Σ(S ′ ), where i S ′ is the universal map described in the above paragraph. Again from the universal property of Σ(S) as described above, we have a unique homomorphism Σ(f ) from Σ(S) to Σ(S ′ ) such that i S ′ • f = Σ(f ) • i S . Thus, Σ defines a functor from the category RL to GR. Finally, we need to show that the bifunctors M or(−, Ω(−)) and M or(Σ(−), −) from RL × GR to the category SET of sets are naturally isomorphic. It follows from the above discussions that for each (S, T ) ∈ RL × GR, we have the bijective map η S,T from M or(S, Ω(T )) to M or(Σ(S), T ) given by η S,T (f ) = f | Σ(S) . The fact that η = {η S,T | (S, T ) ∈ Obj(RL) × Obj(GR)} is a natural isomorphism is an easy observation. ✷ Now, we construct free objects in the category GR of group based right loops. Let X be a set. Consider the free group F (X) on the set X consisting of freely reduced words in X. If W is a word in X, then W denotes the word in X obtained by freely reducing W . We define subsets A n , n ≥ 0 of F (X) inductively as follows. Put A 0 to be the singleton {∅ = 1} consisting of the empty word representing the identity. Let A 1 = {x ±1 | x ∈ X} be the set consisting of reduced words of length 1. Supposing that A n has already been defined, define
A n+1 = {U −1 V U 2 | U , V ∈ n i=0 A i }. Evidently, F R(X) = ∞ i=1 A i is a sub right loop of (F (X), • 1 ) generated by X. The map i from X to F R(X) given by i(x) = x is injective and the pair (F R(X), i) is universal in the sense that if j is a map from X to a group based right loop (T, • 1 ) ⊆ (G, • 1 ), then there is a unique homomorphism j from F R(X) to T such that j • i = j.
It follows that F R defines a functor from the category SET of sets to the category GR which is adjoint to the forgetful functor Ω. We shall term the (F R(X), i) as the free group based right loop on X. A pair X; R together with a surjective homomorphism f from F R(X) to (T, • 1 ) will be termed as a presentation of T if the kernel of f is the normal sub right loop of F R(X) generated by R. Every group based right loop (S, • 1 ) has the standard multiplication presentation induced by the obvious surjective homomorphism from F R(S) to S. The cyclic group x considered as a group based right loop has a presentation {x}; ∅ and it is the universal free object in GR. If S and T are group based right loops having presentations X; R and Y ; S , then the group based right loop having the presentation X Y ; R S is called the free product of S and T , where X Y is taken as the disjoint union of X and Y . Clearly, free objects in GR are free products of certain copies of universal free objects.
Let K be a group. Let K; R K denote the standard multiplication presentation of K andǨ denotes the group having the presentation K;Š K , whereŠ K is the set of words in K of the type
(y −1 xy 2 ) −1 ⋆ y −1 ⋆ x ⋆ y 2 , x, y ∈ K − {e}.
Here the juxtaposition denotes the operation in the group K and ⋆ denotes the operation in the free group F (K) on K. More explicitly, K ≈ F (K)/ R K , where R K is the normal subgroup of F (K) generated by the set
R K = {(xy) −1 ⋆x⋆y | x, y ∈ K} andǨ ≈ F (K)/ Š K whereŠ K = {(y −1 xy 2 ) −1 ⋆ y −1 ⋆ x ⋆ y 2 | x, y ∈ K}. Clearly, Š K ⊆ R K and hence we have the surjective group homomorphism ν K fromǨ to K given by ν K (x Š K ) = x R K . The map t K from K toǨ given by t K (x) = x Š K is an injective gyro-homomorphism and t K (x n ) = (t K (x)) n . If f is a gyro-homomorphism from K to a group G, then the mapf fromǨ to G given byf (x Š K ) = f (x) is the unique group homomorphism fromǨ to G such thatf • t K = f . Thus, the pair (Ǩ, t K )
is universal in the sense that given any group G and a gyro-homomorphism f from K to G, there is a unique group homomorphismf fromǨ to G such thatf
• t K = f . Note that f • ν K • t K = f but f • ν K need not bef as it need not be a group homomorphism (see Example 4.3)
. It also follows that the association K →Ǩ defines a functor from the category GP toĜP which is adjoint to the forgetful functor, whereĜP is a category whose objects are groups and the morphisms are gyro-homomorphisms.
LetŘ K = R K / Š K andǨ = F (K)/ Š K .
Then, we have the following short exact sequence
(1) 1 −→Ř K −→Ǩ −→ K −→ 1
of groups having a section t K which is a gyro-homomorphism.
More generally, let X; S be an arbitrary presentation of K. Consider the free group F (F (X)) on F (X). We have a surjective group homomorphism η from F (F (X)) to F (X) given by η(
W 1 ⋆ W 2 ⋆ · · · ⋆ W r ) = W 1 W 2 · · · W r , and F (X);Ŝ is also a presentation of K, whereŜ = {W 1 ⋆ W 2 ⋆ · · · ⋆ W r | W 1 W 2 · · · W r ∈ S}. LetŤ denote the subset {(η(U −1 ⋆ V ⋆ U 2 ) −1 ⋆ U −1 ⋆ V ⋆ U 2 | U, V ∈ F (X)} of F (F (X)
). It can be observed that Ť ⊆ Ŝ . Consequently, we obtain a short exact sequence
(2) 1 −→ Ŝ / Ť −→ F (F (X))/ Ť −→ K −→ 1
of groups which is equivalent to (1). Indeed, if µ is the surjective homomorphism from F (X) to K given by the presentation X; S of K, then it further induces a surjective group homomorphismμ from F (F (X)) to F (K). It can be easily observed thatμ( Ť ) = Š K . In turn,μ induces an isomorphism ρ from F (F (X))/ Ť toǨ such that (ρ −1 |Ř K , ρ −1 , I K ) is an equivalence from (1) to (2). In particular, K ≈ F (F (X))/ Ť and Ŝ / Ť ≈Ř K . It follows that F (F (X))/ Ť and Ŝ / Ť are independent (up to isomorphism) of the presentation and they depend only on the group K. The associations K →Ǩ and K →Ř K define functors from GP to itself which are universal in the sense already described.
The groupŘ K can be thought of as the obstruction for gyro-homomorphisms from K to be group homomorphisms. We also term it as a gyro-multiplier of K.
Example 4.2. If G is a cyclic group, then it is evident thatǦ ≈ G. Let G be an elementary abelian 2-group. ThenǦ has the presentation G;Š G , wherě
S G = {(y −1 xy 2 ) −1 ⋆ y −1 ⋆ x ⋆ y 2 | x, y ∈ G − {e}} = {(yx) −1 ⋆ (y ⋆ x) | x, y ∈ G} = R G . Thus, in this case alsoǦ ≈ G. Consider the quaternion group Q 8 = {±1, ±i, ±j, ±k}. Evidently, (j −1 ij 2 ) −1 ⋆ j −1 ⋆ i ⋆ j 2 = (ji) −1 ⋆ (j ⋆ i) and so on. Indeed,Š Q8 = R Q8 . Consequently,Q 8 ≈ Q 8 and Q 8 is gyro-isomorphic to itself. Example 4.3. Consider G = Z 3 × Z 3 × Z 3 .
Since G is of exponent 3,Ǧ is also of exponent 3. SinceǦ is finitely generated, it is finite. We show thatǦ is non-abelian group. Let E denote the non-abelian group of order 3 3 which is of exponent 3. Since E is nilpotent group of class 2 and of exponent 3, (E, • 1 ) is an abelian group of exponent 3 and so it is isomorphic to Z 3 × Z 3 × Z 3 as a group. In particular, we have a gyro-isomorphism η from G to E. From the universal property of (Ǧ, t G ), we get a surjective group homomorphismη fromǦ to E such thatη
• t G = η. Since E is non-abelian,Ǧ is non-abelian. Again, since G is abelian,Ř G contains the commutator [Ǧ,Ǧ] ofǦ. Evidently, η • ν G is not a group homomorphism as (η • ν G ) −1 ({1}) = ν −1 G ({1}) =Ř G ⊇ [Ǧ,Ǧ] and E is non-abelian. Note that η • ν G • t G = η.
Remark 4.4. From the Example 4.3, one observes that for the groups G 1 and G 2 , (G 1 × G 2 ) need not be isomorphic toǦ 1 ×Ǧ 2 . One can also observe that if G 1 is gyro-isomorphic to G 2 , thenǦ 1 is isomorphic toǦ 2 as groups. Even ifǦ 1 is isomorphic toǦ 2 as groups, then G 1 need not be gyro-isomorphic to G 2 .
Example 4.5. If K is a free group on at least two generators, then it can be easily observed that the gyro-multiplierŘ K of K is non-trivial, and t K is gyrohomomorphism which is not a group homomorphism.
Gyro-split extensions
Definition 5.1. A short exact sequence 1 −→ H α −→ G β −→ K −→ 1
of groups is called a gyro-split extension if there is a section t, also called a gyrosplitting, from K to G which is a gyro-homomorphism.
Evidently, a split extension is a gyro-split extension. However, a gyro-split extension need not be a split extension.
Example 5.2. Consider the non-abelian group E of order 3 3 which is of exponent 3. Then (E, • 1 ) is an elementary abelian 3-group and the identity map from E to
(E, • 1 ) is a gyro-isomorphism. Consider the central extension 0 −→ Z(E) i −→ E ν −→ Z 3 × Z 3 −→ 0 of Z 3 by Z 3 × Z 3 . Evidently, it is not a split extension. However, there is a sub right loop L of (E, • 1 ) of order 3 2 such that E = Z(E)L, and the map ν| L is an isomorphism from (L, • 1 ) to Z 3 × Z 3 . Indeed, there are 3 2 + 3 + 1 = 13 subgroups of (E, • 1 ) ≈ Z 3 3 of order 3 2 , whereas there are 4 subgroups of E of order 3 2 . If L is a subgroup (E, • 1 ) of order 3 2 which is not a subgroup of E, then L Z(E) = {1}. Consequently, E = Z(E)L and the map ν| L is an isomorphism from (L, • 1 ) to Z 3 × Z 3 . Evidently, (ν| L ) −1 is a gyro-splitting.
Example 5.3. Let K be an arbitrary field. Consider the unipotent group U (3, K) of unipotent upper triangular 3 × 3 matrices with entries in the field K. Then U (3, K) is a nilpotent group of class 2. Thus, (U (3, K), • 1 ) is a nilpotent group of class at most 2. Let U (a 1 , a 2 , a 3 ) denote the unipotent upper triangular 3 × 3 matrix for which a 12 = a 1 , a 13 = a 2 and a 23 = a 3 . It can be easily observed that
U (b 1 , b 2 , b 3 ) −1 U (a 1 , a 2 , a 3 )U (b 1 , b 2 , b 3 ) 2 = U (a 1 +b 1 , b 2 +2a 1 b 3 −b 1 a 3 +a 2 , b 3 +a 3 ). Thus, (U (3, K), • 1 ) is isomorphic to the group (K 3 , ·), where the product · is given by (a 1 , a 2 , a 3 ) · (b 1 , b 2 , b 3 ) = (a 1 + b 1 , b 2 + 2a 1 b 3 − b 1 a 3 + a 2 , b 3 + a 3 ). Evidently, (U (3, K), • 1 ) is an algebraic group defined over the prime field of K. Further, (U (3, K), • 1 ) is abelian if and only if the characteristic of K is 3. Consider U (3, Z p ), where p is an odd prime different from 3. Then U (3, Z p ) is a non abelian group of order p 3 and (U (3, Z p ), • 1 )
is also a non abelian group of order p 3 whose exponent is the same as that of
U (3, Z p ). It follows that U (3, Z p ) is isomorphic to (U (3, Z p ), • 1 ).
In other words U (3, Z p ) is gyro-isomorphic to itself. Consequently, any gyro-split extension by U (3, Z p ) is a split extension. Further, note that
0 −→ Z(U (3, Z p )) i −→ U (3, Z p ) ν −→ Z p × Z p −→ 0 is not gyro-split.
Using the universal property of the functor G →Ǧ, we can easily establish the following proposition:
Proposition 5.4. To each short exact sequence of groups
E ≡ 1 −→ H α −→ G β −→ K −→ 1,
we have the following commutative diagram
1 1 1 1 Kerβ RŘGŘK 1 KerβǦǨ 1 1 H G K 1 1 1 i i iGβ R iK ν i νGβ νK α β
where the rows and the columns are exact. Further, if the bottom row is gyrosplit, then the middle row is split exact sequence.
Proof. Consider the right most gyro-split vertical exact sequence. We have the gyrosplitting t K from K toǨ, and t K • β is a gyro-homomorphism from G toǨ. From the universal property of the pair (Ǧ, t G ), we have a unique group homomorphism β fromǦ toǨ such thatβ • t G = t K • β. In turn,
ν K •β • t G = ν K • t K • β = β = β • ν G • t G .
Since ν K •β and β • ν G are group homomorphisms fromǦ to K and β is a gyrohomomorphism (being a group homomorphism), it follows from the universal property of (Ǧ, t G ) that ν K •β = β • ν G . Thus the lower right square is commutative. Further, since t K (K) generatesǨ as a group and β is surjective, it follows thatβ is surjective. Evidently, the diagram is commutative, all the rows and the last two columns are exact. The exactness of the first column also follows by chasing the diagram. Note that ν andβ R need not be surjective.
Finally, suppose that the bottom row is gyro-split with t as gyro-splitting. Then t G •t is a gyro-homomorphism from K toǦ. From the universal property of (Ǩ, t K ), we have a unique group homomorphismť fromǨ toǦ such thatť • t K = t G • t. In turn,β
•ť • t K =β • t G • t = t K • β • t = t K = IǨ • t K .
It follows from the universal property of (Ǩ, t K ) thatβ •ť = IǨ. ✷ Remark 5.5. Since t G | Kerβ is a gyro-homomorphism from Kerβ = im(α) to t G (Kerβ) ⊆ Kerβ, we have a unique group homomorphismα fromȞ toǦ such thatα • t Kerβ = t G | Kerβ . Evidently, im(α) ⊆ Kerβ. However, the equality need not hold. In turn, we get a natural invariant inv(E) = Kerβ/im(α) associated to the extension E.
Let GEXT denote the category whose objects are gyro-split extensions and a morphism from a gyro-split extension
E ≡ 1 −→ H α −→ G β −→ K −→ 1
to a gyro-split extension
E ′ ≡ 1 −→ H ′ α ′ −→ G ′ β ′ −→ K ′ −→ 1
is a triple (λ, µ, ν), where λ is a group homomorphism from H to H ′ , µ is a group homomorphism from G to G ′ and ν is a gyro-homomorphism from K to K ′ such that the corresponding diagram is commutative. The composition of morphisms is obvious. Observe that in this context the short five lemma also holds. Thus, (λ, µ, ν) is an equivalence if and only if λ and ν are bijective.
Theorem 5.6. The gyro-split extension described in (1), section 4 is a free gyrosplit extension by K in the sense that if
E ≡ 1 −→ H α −→ L β −→ K ′ −→ 1
is a gyro-split extension by K ′ and η a group homomorphism from K to K ′ , then there is a unique pair (λ, µ) of group homomorphisms such that the triple (λ, µ, η) is a morphism from the extension (1) to E.
Proof. Let s be a gyro-splitting of E. Then s • η is a gyro-homomorphism from K to L. From the universal property of (Ǩ, t K ) we get a unique group homomorphism µ fromǨ to L such that
µ • t K = s • η. Hence β • µ • t K = β • s • η = η = η • ν K • t K .
Since η • ν K is a group homomorphism, it follows from the universal property of (
Ǩ, t K ) that β • µ = η • ν K . Also β • µ • i = η • ν K • i = 0, where i is the inclusion fromŘ K toǨ. Consequently, there is a unique group homomorphism λ fromŘ K to H such that (λ, µ, η) is a morphism in GEXT. ✷ Let E ≡ 1 −→ H α −→ G β −→ K −→ 1
be a gyro-split extension and t be a gyro-splitting of E. We have the corresponding factor system (K, H, σ t , f t ), where f t is the map from K × K to H given by t(x)t(y) = α(f t (x, y))t(xy) and σ t is the map from K to Aut(H) given by α(σ t (x)(h)) = t(x)α(h)t(x) −1 . We denote σ t (x) by σ t x . Further, since t is a gyro-homomorphism, σ t is a gyro-homomorphism (note that it need not be a group homomorphism) and
(3)
f t (y −1 , x)f t (y −1 x, y 2 ) = 1 = σ t y −1 (f t (x, y 2 ))f t (y −1 , xy 2 )
for all x, y ∈ K. In particular f t (y, y −1 ) = 1 for all y ∈ K. This prompts us to have the following definition:
Definition 5.7. A factor system (K, H, σ, f ) will be called a gyro-factor system if σ is a gyro-homomorphism from K to Aut(H) and f satisfies (3) with f t replaced by f . Such a map f is also called a gyro-pairing.
Let (λ, µ, ν) be a morphism from a gyro-split extension
E ≡ 1 −→ H α −→ G β −→ K −→ 1
to a gyro-split extension
E ′ ≡ 1 −→ H ′ α ′ −→ G ′ β ′ −→ K ′ −→ 1.
Let t be a gyro-splitting of E and t ′ be a gyro-splitting of E ′ . Since β ′ (µ(t(x))) = ν(β(t(x))) = ν(x) = β ′ (t ′ (ν(x))) for x ∈ K, there is a unique map g from K to H ′ with g(1) = 1 such that
(4) µ(t(x)) = α ′ (g(x))t ′ (ν(x))
for all x ∈ K. Since t is a gyro-homomorphism,
(5) µ(t(y −1 )t(x)t(y 2 )) = µ(t(y −1 xy 2 )) = α ′ (g(y −1 xy 2 ))t ′ (ν(y −1 xy 2 ))
for all x, y ∈ K. Now,
µ(t(y −1 )t(x)t(y 2 )) =µt(y −1 )µt(x)µt(y 2 ) =α ′ (g(y −1 ))t ′ (ν(y −1 ))α ′ (g(x))t ′ (ν(x))α ′ (g(y 2 ))t ′ (ν(y 2 )) by (5.2) =α ′ (g(y −1 ))α ′ (σ t ′ ν(y −1 ) (g(x)))t ′ (ν(y −1 ))t ′ (ν(x))α ′ (g(y 2 ))t ′ (ν(y 2 )) =α ′ (g(y −1 )σ t ′ ν(y −1 ) (g(x))f t ′ (ν(y −1 ), ν(x)))t ′ (ν(y −1 )ν(x))α ′ (g(y 2 ))t ′ (ν(y 2 )) =α ′ (g(y −1 )σ t ′ ν(y −1 ) (g(x))f t ′ (ν(y −1 ), ν(x))σ t ′ ν(y −1 )ν(x) (g(y 2 ))) t ′ (ν(y −1 )ν(x))t ′ (ν(y 2 )) =α ′ (g(y −1 )σ t ′ ν(y −1 ) (g(x))f t ′ (ν(y −1 ), ν(x))σ t ′ ν(y −1 )ν(x) (g(y 2 )) f t ′ (ν(y −1 )ν(x), ν(y 2 )))t ′ (ν(y −1 xy 2 )) =α ′ (g(y −1 )σ t ′ ν(y −1 ) (g(x))σ t ′ ν(y −1 ) (σ t ′ ν(x) (g(y 2
)))f t ′ (ν(y −1 ), ν(x)) f t ′ (ν(y −1 )ν(x), ν(y 2 )))t ′ (ν(y −1 xy 2 )) =α ′ (g(y −1 )σ t ′ ν(y −1 ) (g(x))σ t ′ ν(y −1 ) (σ t ′ ν(x) (g(y 2 )))t ′ (ν(y −1 xy 2 )) (by (3)) for all x, y ∈ K. Thus, comparing the both sides of Equation (5), we obtain (6) g(y −1 xy 2 ) = g(y −1 )σ t ′ ν(y −1 ) (g(x)σ t ′ ν(x) (g(y 2 ))) for all x, y ∈ K. Further,
α ′ (λ(σ t x (h))) =µ(α(σ t x (h))) =µ(t(x)α(h)t(x) −1 ) =µ(t(x))α ′ (λ(h))µ(t(x −1 )) =α ′ (g(x))t ′ (ν(x))α ′ (λ(h))α ′ (g(x −1 ))t ′ (ν(x −1 )) =α ′ (g(x)σ t ′ ν(x) (λ(h)g(x −1 )
)) since t ′ and ν are gyro-homomorphisms. Thus, (7) λ(σ t x (h)) = g(x)σ t ′ ν(x) (λ(h)g(x −1 )) for all x ∈ K and h ∈ H. Let (λ 1 , µ 1 , ν 1 ) be a morphism from a gyro-split extension
E 1 ≡ 1 −→ H 1 α1 −→ G 1 β1 −→ K 1 −→ 1 to E 2 ≡ 1 −→ H 2 α2 −→ G 2 β2 −→ K 2 −→ 1,
and (λ 2 , µ 2 , ν 2 ) be a morphism from E 2 to a gyro-split extension
E 3 ≡ 1 −→ H 3 α3 −→ G 3 β3 −→ K 3 −→ 1.
Let t 1 , t 2 and t 3 be the corresponding choice of gyro-splittings. Then µ 1 (t 1 (x)) = α 2 (g 1 (x))t 2 (ν 1 (x)) for all x ∈ K 1 and µ 2 (t 2 (x)) = α 3 (g 2 (x))t 3 (ν 2 (x)) for all x ∈ K 2 , where g 1 is the uniquely determined map from K 1 to H 2 and g 2 is the uniquely determined map from K 2 to H 3 . In turn, µ 2 (µ 1 (t 1 (x))) = α 3 (g 3 (x))t 3 (ν 2 (ν 1 (x))), where g 3 (x) = λ 2 (g 1 (x))g 2 (ν 1 (x)) for each x ∈ K 1 . This introduces a category GFAC of gyro-factor systems whose objects are gyro-factor systems and a morphism from a gyro-factor system (K 1 , H 1 , σ 1 , f 1 ) to (K 2 , H 2 , σ 2 , f 2 ) is a triple (ν, g, λ), where ν is a gyro-homomorphism from K 1 to K 2 , λ a group homomorphism from H 1 to H 2 , and g is a map from K 1 to H 2 such that
(i) g(1) = 1, (ii) g(y −1 xy 2 ) = g(y −1 )σ 2 ν(y −1 ) (g(x)σ 2 ν(x) (g(y 2 ))) and (iii) λ(σ 1 x (h)) = g(x)σ 2 ν(x) (λ(h)g(x −1 )
), for all x, y ∈ K 1 and h ∈ H 1 . The composition of a morphism (ν 1 , g 1 , λ 1 ) with (ν 2 , g 2 , λ 2 ) is (ν 2 • ν 1 , g 3 , λ 2 • λ 1 ), where g 3 (x) = λ 2 (g 1 (x))g 2 (ν 1 (x)) for all x ∈ K 1 .
Using the axiom of choice, we have a choice t E of a gyro-splitting of a gyro-split extension E. Evidently, the association GF AC which associates to each gyroextension E the gyro-factor system GF AC(E, t E ) associated to the section t E gives an equivalence between GEXT and GFAC.
Let us fix a pair H and K of groups. We try to describe the equivalence classes of gyro-split extensions of H by K. Let G be a gyro-split extension of H by K given by the exact sequence
E ≡ 1 −→ H α −→ G β −→ K −→ 1.
Let (λ, µ, ν) be an equivalence from E to a gyro-split extension G ′ of H ′ by K ′ which is given by the exact sequence
E ′ ≡ 1 −→ H ′ α ′ −→ G ′ β ′ −→ K ′ −→ 1.
Then it is clear that G ′ is also a gyro-split extension of H by K given by the exact sequence
E ′′ ≡ 1 −→ H α ′ •λ −→ G ′ β•µ −1 −→ K −→ 1.
such that E is equivalent to E ′′ and E ′′ is equivalent to E ′ . As such there is no loss of generality in restricting the concept of equivalence on the class GE(H, K) of all gyro-split extensions of H by K by saying that
E 1 ≡ 1 −→ H α1 −→ G 1 β1 −→ K −→ 1.
and
E 2 ≡ 1 −→ H α2 −→ G 2 β2 −→ K −→ 1.
in GE(H, K) are equivalent if there is an isomorphism φ from G 1 to G 2 such that (I H , φ, I K ) makes the corresponding diagram commutative.
Proposition 5.8. An abstract kernel ψ from K to Out(H) is realizable from a gyro-split extension if and only if the obstruction Obs(ψ) ∈ H 3 σ (K, Z(H)) is 0 and ψ has a lifting from K to Aut(H) which is a gyro-homomorphism. Here σ is a group homomorphism from K to Aut(Z(H)) induced by ψ.
Proof. We already know that ψ is realizable from an extension if and only if Obs(ψ) = 0 (see [9,Proposition 10.2.1,p. 392]). Further, then, it is realizable from a gyro-split extension
1 −→ H α −→ G β −→ K −→ 1
if and only if there is a gyro-splitting t such that ψ(x) = σ t
x Inn(H) for each x ∈ K. Since t is a gyro-splitting, σ t is a lifting of ψ which is a gyro-homomorphism. ✷
The following two corollaries are immediate.
Corollary 5.9. An abstract kernel ψ from K to Out(H) is realizable from a gyrosplit extension if and only if the obstruction Obs(ψ) ∈ H 3 σ (K, Z(H)) is 0 and the short exact sequence
0 −→ Inn(H) i1 −→ Aut(H) × (ν,ψ) K p2 −→ K −→ 1
is a gyro-split extension, where Aut(H) × (ν,ψ) K is pull-back of the pair (ν, ψ) and ν : Aut(H) → Out(H) is the natural group homomorphism. For all finite simple groups H, the above sequence splits except when H = A 6 . For H = A 6 , the above sequence is not even a gyro-split extension.
A group is an internal semidirect product of its two subgroups if and only if the corresponding extension splits, that is the splitting is a group homomorphism. We now observe that the same is true in the case of gyro-splitting.
Definition 5.11. Let G be a group. We shall say that G is internal gyro-semi direct product of a normal subgroup H with a sub right loop S of (G,
• 1 ) if S is a right transversal to H in G.
Thus, the exponent 3 non-abelian group G of order 3 3 is a gyro-semi direct product of its center with a sub loop of order 3 2 of (G, • 1 ). Evidently, a semidirect product is also a gyro-semi direct product. However, a gyro-semi direct product need not be a semidirect product. Proof. Suppose that G is internal gyro-semi direct product of a normal subgroup H with a sub right loop S of (G, • 1 ). Since S is a right transversal, G = HS. Given y ∈ S, since S is a sub right loop of (G, • 1 ), y 2 ∈ S and since S is a right transversal, Hy 2 S = {y 2 }. Conversely, let H be a normal subgroup of G, and S be a sub right loop of (G, • 1 ) such that the conditions (i) and (ii) hold. We need to show that S is a right transversal. Already, G = HS. Suppose that y −1 x ∈ H, x, y ∈ S. Then y −1 xy 2 ∈ Hy 2 S = {y 2 }. This means that y −1 x = 1 and so S is a right transversal to H in G. ✷ Remark 5.13. Unlike semidirect product, if G is an internal gyro-semi direct product of H with S and it is also a gyro-semi direct product of H with T , then S need not be conjugate to T .
The following proposition is immediate.
Proposition 5.14. G is internal gyro-semi direct product of H with a sub right loop of (G, • 1 ) if and only if the exact sequence
1 −→ H i −→ G ν −→ G/H −→ 1 is gyro-split.
Next, let H be an abelian group and K σ → Aut(H) be an abstract kernel. Let GEXT σ (K, H) denote the set of equivalence classes of gyro-split extensions of H by K with abstract kernel σ. Obviously, GEXT σ (H, K) is non-empty, as the split extension exists which is also a gyro-split extension. Let GZ 2 σ (K, H) denote the set of gyro-factor systems associated to σ. Evidently, GZ 2 σ (K, H) is a subgroup of Z 2 σ (K, H). We shall term GZ 2 σ (K, H) as the group of gyro-cycles. Denote B 2 σ (K, H) GZ 2 σ (K, H) by GB 2 σ (K, H) and call it the group of gyro-co-boundaries. We shall also term GH 2 σ (K, H) = GZ 2 σ (K, H)/GB 2 σ (K, H) the second gyro-cohomology of K with coefficients in H. From the proof of [9, Proposition 10.1.11, p. 373], one can observe that given (K, H, σ, f ) ∈ GZ 2 σ (K, H) there is the corresponding gyro-split extension of H by K. The following proposition is easy to establish.
Proposition 5.15. The map η which associates to (K, H, σ, f ) ∈ GZ 2 σ (K, H) the corresponding gyro-split extension induces a bijective map from GH 2 σ (K, H) to GEXT σ (K, H) which in turn, induces a group structure on GEXT σ (K, H).
Further, it can be easily seen that the Baer sum in EXT σ (K, H) induces a sum in GEXT σ (K, H) with respect to which it is a subgroup isomorphic to GH 2 σ (K, H).
Example 5.16. GH 2 σ (Z 3 × Z 3 , Z 3 ) ≈ Z 2 , whereas H 2 σ (Z 3 × Z 3 , Z 3 ) ≈ V 4 .
Here σ is trivial.
Given groups H and K, GHom(K, H) will denote the set of all gyro-homomorphisms from K to H. If H is an abelian group, then GHom(K, H) is also an abelian group. Further, if α is a group homomorphism (gyro-homomorphism) from a group G to a group K and A is an abelian group, then α ⋆ is a homomorphism from GHom(K, A) to GHom(G, A). Clearly, GHom(K, A) is naturally isomorphic to Hom(Ǩ, A). Consequently, we have the following proposition.
Proposition 5.17. Let 1 −→ H α −→ G β −→ K −→ 1
be an exact sequence of groups. Let A be an abelian group. Then the sequence
1 −→ GHom(K, A) β ⋆ → GHom(G, A) α ⋆ → GHom(H, A)
is exact.
Gyro-split central extensions and Gyro-Schur Multiplier
Let GRXT(−, K) denote the category of gyro-split extensions by K. More explicitly, the objects of GEXT(−, K) are gyro-split short exact sequences
E ≡ 1 −→ H α −→ G β −→ K −→ 1 and a morphism from E to E ′ ≡ 1 −→ H ′ α −→ G ′ β −→ K −→ 1 is a pair (λ, µ) such that the triple (λ, µ, I K ) is a morphism from E to E ′ in GEXT. Let E ≡ 1 −→ H α −→ G β −→ K −→ 1
be a gyro-split extension by K. Let s be a gyro-splitting of E. Then s is a gyrohomomorphism from K to G. From the universal property of the pair (Ǩ, t K ), there is a unique group homomorphism µ fromǨ to G such that
µ • t K = s. In turn, β • µ • t K = β • s = I K = ν K • t K , where ν K :Ǩ → K is the natural homomorphism. Since t K (K) generatesǨ, β • µ = ν K . Thus, we get a group homomorphism λ fromŘ K to H such that (λ, µ, I K ) is a morphism from E K to E, where E K ≡ 1 −→Ř K iK −→Ǩ νK −→ K −→ 1
More generally, E K is a free gyro-split extension in the sense that given any gyrosplit extension
E ′ ≡ 1 −→ H ′ α −→ G ′ β −→ K ′ −→ 1
and a gyro-homomorphism ν from K to K ′ , there is a pair (λ, µ) (not necessarily unique) such that (λ, µ, ν) is a morphism from E K to E ′ .
The abstract kernel σ associated to a central extension is trivial. In this case, we shall denote Z 2 σ (K, H) by Z 2 (K, H), B 2 σ (K, H) by B 2 (K, H), H 2 σ (K, H) by H 2 (K, H) and GH 2 σ (K, H) by GH 2 (K, H). Let A be an abelian group. We define a connecting group homomorphism δ from Hom(H, A) to GH 2 (K, A) as follows: Let t be a gyro-splitting of E and f t the corresponding gyro pairing in GZ 2 (K, H). Let η ∈ Hom(H, A). Then η • f t is a map from K × K to A. Since η is a group homomorphism, η • f t ∈ GZ 2 (K, A). If s is another gyro-splitting of E, then f t and f s differ by a member of GB 2 (K, H) and in turn, η • f t and η • f s differ by a member of GB 2 (K, A). This defines a group homomorphism δ from Hom(H, A) to GH 2 (K, A) which is given by δ(η) = η • f t + GB 2 (K, A). Proposition 6.1. For any abelian group A, we have the following natural fundamental exact sequence
0 −→ Hom(K, A) β ⋆ → Hom(G, A) α ⋆ → Hom(H, A) δ → GH 2 (K, A)
associated to a gyro-split central extension
E ≡ 1 −→ H α −→ G β −→ K −→ 1.
Proof. Since Hom is a left exact functor, it is sufficient to prove the exactness at Hom (H, A). Let χ ∈ Hom(G, A). By the definition, δ(α ⋆ (χ)) = (χ • α • f t ) + GB 2 (K, A). Already, t(x)t(y) = α(f t (x, y))t(xy) for all x, y ∈ K and since t is a gyro-splitting, f t (y −1 , x) + f t (y −1 x, y 2 ) = 0 for all x, y ∈ K. Since χ is a group homomorphism, χ(t(x)) + χ(t(y)) = χ(α(f t (x, y))) + χ(t(xy)). Thus, we have a map g = χ • t from K to A with g(1) = 0 and (χ • α) • f t = ∂g, where ∂g(x, y) = g(y) − g(x, y) + g(x). This means that δ • α ⋆ = 0. It follows that im(α ⋆ ) ⊆ Kerδ. Next, let η ∈ Kerδ. Then η • f t ∈ GB 2 (K, A). Hence there is a map g from K to A with g(1) = 0 such that
η(f t (x, y)) = g(y) − g(xy) + g(x)
for all x, y ∈ K. Every element of G is uniquely expressible as α(a)t(x), a ∈ H, x ∈ K. Define a map χ from G to A by χ(α(a)t(x)) = η(a) + g(x). It can be easily seen that χ ∈ Hom(G, A) such that η = χ • α = α ⋆ (χ). It follows that Kerδ ⊆ im(α ⋆ ). ✷ In particular, for an abelian group H, we have the following exact sequence:
0 −→ Hom(K, H) β ⋆ → Hom(G, H) α ⋆ → Hom(H, H) δ → GH 2 (K, H). Remark 6.2. The sequence 0 −→ GHom(K, A) β ⋆ → GHom(G, A) α ⋆ → GHom(H, A) δ → GH 2 (K, A).
need not be exact. Indeed, δ • α ⋆ need not be 0. However, Kerδ ⊆ im(α ⋆ ). Proposition 6.3. The extensioň
E K ≡ 1 −→Ř K /[Ř K ,Ǩ] iK →Ǩ/[Ř K ,Ǩ] νK → K −→ 1
is a free gyro-split central extension of K in the sense that given any gyro-split central extension
E ′ ≡ 1 −→ H ′ α ′ −→ G ′ β ′ −→ K ′ −→ 1
and a gyro-homomorphism γ from K to K ′ , there is a pair (ρ, η) (not necessarily unique) of homomorphism such that (ρ, η, γ) is a morphism fromĚ K to E ′ .
Proof. Evidently,Ě K is a gyro-split central extension. Again since E K is a free gyro-split extension, there is a morphism (λ, µ, γ) from E K to E ′ . Since E ′ is a central extension, (λ, µ) induces a pair (ρ, η) such that (ρ, η, γ) is a morphism from
E K to E ′ . ✷ Proposition 6.4. Let E ≡ 1 −→ H α −→ G β −→ K −→ 1
be a free gyro-split central extension and A be an abelian group. Then the map δ from Hom(H, A) to GH 2 (K, A) is surjective. More explicitly,
0 −→ Hom(K, A) β ⋆ → Hom(G, A) α ⋆ → Hom(H, A) δ → GH 2 (K, A) −→ 0 is exact.
Proof. Let f ∈ GZ 2 (K, A). Then (K, A, σ, f ) is a gyro-factor system with σ being trivial. The corresponding associated extension
E ′ ≡ 0 −→ A α ′ −→ G ′ β ′ −→ K −→ 1
is a gyro-split central extension with a gyro-splitting t ′ such that t ′ (x)t ′ (y) = α ′ (f (x, y))t ′ (xy) for all x, y ∈ K. Since E is a free gyro-split central extension, we have a group homomorphism λ from H to A and a group homomorphism µ from G to G ′ such that (λ, µ, I K ) is a morphism from E to E ′ . Let t be a gyro-splitting of E. Then β ′ (µ(t(x))) = β(t(x)) = x for all x ∈ K.
Hence t ′′ = µ • t is a gyro-splitting of E ′ . Thus, f t ′′ + GB 2 (K, A) = f + GB 2 (K, A). Now, t(x)t(y) = α(f t (x, y))t(xy) for all x, y ∈ K. Further, α ′ (f t ′′ (x, y))t ′′ (xy) = t ′′ (x)t ′′ (y) = µ(t(x))µ(t(y)) = µ(t(x)t(y)) = µ(α(f t (x, y)))µ(t(xy)) = µ(α(f t (x, y)))t ′′ (xy) = α ′ (λ(f t (x, y)))t ′′ (xy).
This shows that A). This shows that δ is surjective. ✷ Proposition 6.5. Let Since GH 2 (K, C ⋆ ) is a subgroup of H 2 (K, C ⋆ ), the following corollary is a consequence of the Schur-Hopf Formula.
α ′ (λ(f t (x, y))) = α ′ (f t ′′ (x, y)). Since α ′ is injective, λ(f t (x, y)) = f t ′′ (x, y). By the definition δ(λ) = f t ′′ + GB 2 (K, A) = f + GB 2 (K,E ≡ 1 −→ H α −→ G β −→ K −→ 1
Corollary 6.7. If K is finite, then
GH 2 (K, C ⋆ ) ≈ [F (F (X)), F (F (X))] Ŝ /[F (F (X)),Ŝ].
We shall term GH 2 (K, C ⋆ ) and also ([Ǩ,Ǩ] Ř K )/[Ǩ,Ř K ] as gyro-Schur Multipliers of K. Note that they are same provided that K is finite. Also observe that K → ([Ǩ,Ǩ] Ř K )/[Ǩ,Ř K ] defines a functor from GP to itself.
The proof of the following proposition is an easy verification. Proposition 6.8. Let K be a group. Then the right gyro-group operation • 1 on K satisfies the following relations:
(i) (xy) • 1 z = x z (y • 1 z), and also (ii) x • 1 (yz) = (x y • 1 z)y z . for each x, y, z ∈ K, where x y = y −1 xy.
The relations described in the above propositions will be termed as trivial relations for • 1 . Recall that the Schur multiplier of a group K has description as the group of non-trivial commutator relations of K [9,10]. We describe the gyro-Schur multiplier ([Ǩ,Ǩ] Ř K )/[Ǩ,Ř K ] also as the group of non-trivial relations of the right gyrogroup operation • 1 of G.
Let K be a group. Let K ⊠ K denote the abelian group generated by the set {x ⊠ y | x, y ∈ K} subject to the relations
(i) 1 ⊠ x = 1 = x ⊠ 1, (ii) (x ⊠ y)((xy) ⊠ z) = (y ⊠ z)((x ⊠ (yz))) and (iii) (y −1 ⊠ x)((y −1 x) ⊠ y 2 ) = 1,
for all x, y, z ∈ K. We shall term K ⊠ K as gyro-square of K. Theorem 6.9. We have a free gyro-split central extension
U ≡ 1 −→ K ⊠ K i1 −→ (K ⊠ K) ⋊ K p2 −→ K −→ 1,
where (K ⊠ K) ⋊ K is a group with respect to the operation given by (a, x)(b, y) = (ab(x ⊠ y), xy).
Proof. Let
E ′ ≡ 1 −→ H ′ α ′ −→ G ′ β ′ −→ K ′ −→ 1
be a gyro-split central extension, and ν be a gyro-homomorphism from K to K ′ . Let t be a gyro-splitting of E ′ , and (K ′ , H ′ , σ t , f t ) be the corresponding factor system. Then σ t is trivial. Further, f t (x, y)f t (xy, z) = f t (y, z)f t (x, yz) and since t(y −1 xy 2 ) = t(y) −1 t(x)t(y) 2 for all x, y ∈ K ′ , f t (y −1 , x)f t (y −1 x, y 2 ) = 1. Thus, we have a group homomorphism λ from K ⊠ K to H ′ given by λ(x ⊠ y) = f t (x, y). In turn, we have a map µ from (K ⊠ K) ⋊ K to G ′ given by µ(a, x) = α ′ (λ(a))t(ν(x)). It can be seen that µ is a group homomorphism and (λ, µ, ν) is a morphism. ✷ Corollary 6.10. The extensionĚ K as described in the Proposition 6.3 is equivalent to U .
Proof. Since the map x → (1, x) is a gyro-homomorphism from K to (K ⊠K)⋊K, it induces a group homomorphism µ fromǨ to (K⊠K)⋊K given by µ(xŠ K ) = (1, x). It can be easily observed that [Ř K ,Ǩ] is contained in the kernel of µ. This in turn induces a morphism fromĚ K to U . Further, Theorem 6.9 gives the inverse of this morphism. ✷ Let K σ → Aut(H) be an abstract kernel, where H is an abelian group. Let
1 −→ H i −→ G ν −→ K −→ 1
be a gyro-split extension of H by K which is associated to σ. Note that it is central extension if and only if σ is trivial. We denote the image σ(x) by σ x . Consider the subset A = {h ∈ H | σ x (h) = h, ∀x ∈ K}. Evidently, A is a central subgroup of G and we have the following commutative diagram.
1 A G G/A 1 1 H G K 1 i i IGνν i ν
where the top row is a gyro-split central extension of A by G/A and the maps are the obvious maps. Indeed, if t is a gyro-splitting of the bottom row, then t •ν is a gyro-splitting of the top row. From the proof of the Theorem 6.9, we have a morphism from the extension U to the extension given in the top row and in turn, we have a morphism (χ, ψ, I K ) from U to the given gyro-split extension
1 −→ H i −→ G ν −→ K −→ 1 with χ(K ⊠ K) ⊆ A.
Conversely, let χ be a group homomorphism from K ⊠ K to A ⊆ H. Then (K, H, σ,χ) is a factor system, whereχ is a map from K × K to H givenχ(x, y) = χ(x ⊠ y). The corresponding extension
E χ ≡ 1 −→ H i1 −→ L = H × K p2 −→ K −→ 1
is a gyro-split extension of H by K with x → (1, x) as a gyro-splitting. Thus, we have a surjective map λ from Hom(K ⊠ K, A) to GEXT σ (H, K) given by λ(χ) = [E χ ]. Clearly, λ is also a group homomorphism. We describe the Kerλ. Now, χ ∈ Kerλ if and only if the corresponding factor system is equivalent to the trivial factor system. In other words, there is a map g from K to H with g(1) = 0 such that χ(x ⊠ y) = ∂g(x, y) = σ x (g(y)) − g(xy) + g(x) belongs to A for all x, y ∈ K. Evidently, (K, H, σ, ∂g) is a gyro-factor system. Let us call such a map g to be a gyro-crossed homomorphism relative to σ. Thus an identity preserving map g from K to H is a gyro-crossed homomorphism if σ x (σ y (g(z)) − g(yz) + g(y)) = σ y (g(z)) − g(yz) + g(y), and σ y −1 (g(x)) + g(y −1 ) + σ y −1 x (g(y 2 )) − g(y −1 xy 2 ) = 0 for all x, y, z ∈ K. Evidently, every crossed group homomorphism is a gyro-crossed homomorphism. However, a gyro-crossed homomorphism need not be a crossed group homomorphism. For example, if K is the exponent 3 non-abelian group of order 3 3 , then the map g from K to K ⊠K given by g(x) = x⊠x can be easily seen to be a gyro-crossed homomorphism which is not a crossed group homomorphism. Let GC σ (K, H) denote the group of all gyro-crossed homomorphisms from K to H. The above discussion establishes the following proposition.
Proposition 6.11. A map g with g(1) = 0 is a gyro-crossed homomorphism from K to H relative to σ if and only if (K, H, σ, ∂g) is a gyro-factor system and ∂g(K × K) ⊆ A. In turn, ∂g induces a homomorphism ∂ from GC σ (K, H) to Hom(K ⊠ K, A) ⊆ Hom(K ⊠ K, H) given by ∂g(x ⊠ y) = ∂g(x, y), and we have the exact sequence
0 → C σ (K, H) i → GC σ (K, H) ∂ → Hom(K ⊠ K, A) λ → GEXT σ (K, H) → 0,
where C σ (K, H) denotes the group of crossed homomorphisms.
In case σ is trivial or equivalently, it is a central extension, then we omit σ in the notation. In particular, we have the following exact sequence:
0 → Hom(K, H) i → GC(K, H) ∂ → Hom(K ⊠ K, H) λ → GEXT (K, H) → 0.
7. Universal free gyro-split central extension, Milnor gyro-K 2 group Definition 7.1. A gyro-split central extension
Ω K ≡ 1 −→ H i −→ U j −→ K −→ 1
will be termed as a universal free gyro-split central extension by K if given any gyro-split central extension
E ≡ 1 −→ L α −→ G β −→ K −→ 1 by K, there is a unique group homomorphism φ from U to G inducing a morphism (ξ, φ, I K ) from Ω K to E.
Evidently, a universal free gyro-split central extension by K (if exists) is unique up to equivalence.
Proposition 7.2.
If
Ω K ≡ 1 −→ H i −→ U j −→ K −→ 1
is a universal free gyro-split central extension by K, then U is perfect. In particular, K is perfect.
Proof. Suppose that U is not perfect. Then U/[U, U ] is a non-trivial abelian group. Consider the direct product extension
1 −→ U/[U, U ] i1 −→ U/[U, U ] × K p2 −→ K −→ 1.
Clearly, this extension is a gyro-split (indeed, a split) central extension. Further, the map (ν, j) from U to U/[U, U ] × K defined by (ν, j)(u) = (u[U, U ], j(u)) and (0, j) given by (0, j)(u) = ([U, U ], j(u)) are two group homomorphisms inducing morphisms from Ω K to this extension. This is a contradiction. This shows that U is perfect. Consequently, K is perfect. ✷ Let us call a gyro-homomorphism f from a group G to a group K to be a strong gyro-homomorphism if f preserves the commutator operation in the sense that f ([a, b]) = [f (a), f (b)] for all a, b ∈ G. An extension E is said to be a strong gyrosplit extension if it has a section t which is a strong gyro-homomorphism. We have a category SGP whose objects are groups and morphism between groups are strong gyro-homomorphisms. Obviously, the category GP of groups is a faithful (but not full) subcategory of SGP. We construct the adjoint to the inclusion functor from GP to SGP.
Let K be a group. Consider the free group F (K) on K and standard group homomorphism ρ from F (K) to K which is the identity map on K. Let SG(K) denote the setŠ K {(xyx −1 y −1 ) −1 ⋆ x ⋆ y ⋆ x −1 ⋆ y −1 | x, y ∈ K} of words in F (K), anď SG(K) denote the group having the presentation K; SG(K) . More explicitly, SG(K) = F (K)/ SG(K) . It follows from the construction that the association K →ŜG(K) defines a functor from GP to SGP, which is adjoint to the forgetful functor from SGP to GP. Clearly, R K ⊇ SG(K) . Further, we have a strong gyro-split extensioñ
E K ≡ 1 −→ R K / SG(K) i −→ŠG(K) ν −→ K −→ 1.
Evidently,Ẽ K is a free strong gyro-split extension by K. We may term R K / SG(K) as a strong gyro-multiplier. Note again that if K is free on a set having at least two elements, the strong gyro-multiplier is non-trivial. Proposition 7.3. Let K be a perfect group in which every element is a commutator. Then K admits a universal free gyro-split central extension.
Proof. Let K be a perfect group in which every element is a commutator. Consider the strong gyro-split extensioñ
E K ≡ 1 −→Ř K = R K / SG(K) i −→ŠG(K) ν −→ K −→ 1.
having a strong gyro-splitting t given by t(x) = x R K / SG(K) . Since every element of K is a commutator, image of t is contained in [ŠG(K),ŠG(K)]. In turn, we get a gyro-split central extensioň We show thatĚ K is universal free gyro-split central extension. Let
E ≡ 1 −→ H i −→ G β −→ K −→ 1
be a gyro-split central extension by K. Since E K is a free gyro-split extension by K, there is a homomorphism φ fromǨ to G which induces a morphism (φ|Ř K , φ, I K ) from E K to E. Further, since K is perfect, β | [G,G] is a surjective group homomorphism. In turn, we get a central extension
E ′ ≡ 1 −→ H [G, G] i −→ [G, G] β −→ K −→ 1.
It follows from the construction that φ induces a group homomorphism from [ŠG(K),ŠG(K)]/[Ř K ,ŠG(K)] to [G, G] which, in turn, induces a morphism fromĚ K to E ′ . Since K is perfect, [ŠG(K),ŠG(K)]/[Ř K ,ŠG(K)] is also perfect. Consequently, the induced morphism is unique (see [9,Proposition 10.4.2]). ✷ Corollary 7.4. (i) Every finite simple group admits a universal free gyro-split central extension.
(ii) SU (n) admits a universal free gyro-split central extension.
Proof. The proof of the Ore's conjecture [10] implies (i), while the fact that every element of SU (n) is a commutator [13] implies (ii). ✷ Remark 7.5. It is not clear if every perfect group admits a universal free gyro-split central extension.
We have the following gyro analogues of non-abelian exterior square, Steinberg group, and Milnor K 2 .
Definition 7.6. We shall term [ŠG(K),ŠG(K)]/[Ř K ,ŠG(K)] as a non-abelian gyro-exterior square of K and denote it by K G K. If K is perfect, we have the universal free gyro-split central extension We have the exact sequence
1 −→ M G (K) i −→ K G K ν −→ K −→ 1,1 −→ K G 2 (R) −→ St G (R) −→ E(R) −→ 1.
Proposition 2. 2 .
2( [7]) A magma (S, •) is a right gyro-group if and only if (S, •) is a right loop with identity such that all inner mappings f (x, y) ∈ Aut(S, •) and f (x ′ , x) = I S , where x ′ denotes the left inverse of x. ♯ Definition 2.3. (
Proposition 2.4. ( [7]) (Representation Theorem for Right Gyro-groups) A right loop (S, •) is a right gyro-group if and only if it is a gyro-transversal to the right inner mapping group (group torsion) G S of S in its group extension (right multiplication group) G S S. ♯
Theorem 4 . 1 .
41Let Ω denote the forgetful functor from the category GR of group based right loops to the category RL of right loops. Then there is a left adjoint to Ω.
Out(H) −→ 1is a gyro-split exact sequence, then every extension of H is a gyro-split extension. If in addition to this, H has trivial center, then there is a unique (up to equivalence) such extension.
Theorem 5 .
512. A group G is internal gyro-semi direct product of a normal subgroup H with a sub right loop S of (G, • 1 ) if and only if (i) G = HS, and (ii) Hy 2 S = {y 2 } (equivalently, H Sy 2 = {1}) for all y ∈ S.
→
be a gyro-split central extension by K, and D be a divisible abelian group. Then the image of δ in the fundamental exact sequence 0 −→ Hom(K, D) GH 2 (K, D) is isomorphic to Hom([G, G] α(H), D). In particular, if the extension E is a free gyro-split central extension, then GH 2 (K, D) is isomorphic to Hom([G, G] α(H), D). Proof. By the fundamental theorem of homomorphism, im(δ) ≈ Hom(H, D)/Kerδ = Hom(H, D)/im(α ⋆ ). The map α induces an injective group homomorphism α from H/(H α −1 ([G, G]) to G/[G, G]. Since D is divisible, α ⋆ is a surjective group homomorphism from Hom(G/[G, G], D) to Hom(H/(H α −1 ([G, G]), D)). Also, since D is abelian, ν ⋆ from Hom(G/[G, G], D) to Hom(G, D) is an isomorphism, where ν : G → G/[G, G] is the quotient map. Further, ρ ⋆ oα ⋆ = α ⋆ oν ⋆ , where ρ is the quotient map from H to H/(H α −1 ([G, G])). It follows that the image of α ⋆ is that of ρ ⋆ . Again, since D is divisible, the following sequence is exact: 0 −→ Hom(H/(H α −1 ([G, G]), D)) Hom(H, D)/im(ρ ⋆ ) ≈ Hom((H α −1 ([G, G])), D) ≈ Hom(([G, G] α(H)), D).The last assertion follows from the proposition 6.4. ✷ Corollary 6.6. GH 2 (K, C ⋆ ) ≈ Hom(([Ǩ,Ǩ] Ř K )/[Ǩ,Ř K ], C ⋆ ). More generally, if X; S is a presentation of K, then GH 2 (K, C ⋆ ) ≈ Hom(([F (F (X)), F (F (X))] Ŝ )/[F (F (X)),Ŝ], C ⋆ ).
E
K ≡ 1 −→ (Ř K [ŠG(K),ŠG(K)])/[Ř K ,ŠG(K)] i → [ŠG(K),ŠG(K)]/[Ř K ,ŠG(K)] ν → K −→ 1.
where M G (K) = (Ř K [ŠG(K),ŠG(K)])/[Ř K ,ŠG(K)] is gyro-Schur multiplier of K. Further, for any ring R with identity, we have the invariant St G (R) = E(R) G E(R) termed as gyro-Steinberg group over R and the group K G 2 (R) = M G (E(R)) termed as gyro-Milnor group.
Acknowledgment: Authors are extremely grateful to the reviewer for his/her fruitful comments.
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| {'fraction_non_alphanumeric': 0.11274962629536942, 'fraction_numerical': 0.02109541644972203, 'mean_word_length': 3.040993620198181, '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': 51, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper, we study gyro-groups associated to groups, group extensions admitting gyro-sections, and corresponding co-homologies. We also describe the obstructions in terms of co-homomology. The notion of gyro-Schur Multiplier and that of gyro-Milnor K 2 group are introduced.Gyro-groups, Gyro-splittings, Co-homology, Schur Multipliers.', 'arxivid': '2302.09366', 'author': ['Ramji Lal ', 'Vipul Kakkar '], 'authoraffiliation': [], 'corpusid': 257038097, 'doi': '10.1142/s0219498824501378', 'github_urls': [], 'n_tokens_mistral': 23477, 'n_tokens_neox': 21443, 'n_words': 12781, 'pdfsha': '11a7b9e4187cb98a705319291aa0f69ed8e6c425', 'pdfurls': ['https://export.arxiv.org/pdf/2302.09366v1.pdf'], 'title': ['GYRO-GROUPS, GYRO-SPLITTINGS AND CO-HOMOLOGY', 'GYRO-GROUPS, GYRO-SPLITTINGS AND CO-HOMOLOGY'], 'venue': []} |
arxiv |
Causal-Discovery Performance of ChatGPT in the context of Neuropathic Pain Diagnosis
Ruibo Tu
KTH Royal Institute of Technology
Microsoft Research
Microsoft Research
Chao Ma [email protected]
KTH Royal Institute of Technology
Microsoft Research
Microsoft Research
Cheng Zhang [email protected]
KTH Royal Institute of Technology
Microsoft Research
Microsoft Research
Causal-Discovery Performance of ChatGPT in the context of Neuropathic Pain Diagnosis
Introduction.ChatGPT[3]has demonstrated exceptional proficiency in natural language conversation, e.g., it can answer a wide range of questions while no previous large language models can. Thus, we would like to push its limit and explore its ability to answer causal discovery questions by using a medical benchmark[5]in causal discovery.
Introduction.
ChatGPT [3] has demonstrated exceptional proficiency in natural language conversation, e.g., it can answer a wide range of questions while no previous large language models can. Thus, we would like to push its limit and explore its ability to answer causal discovery questions by using a medical benchmark [5] in causal discovery.
Causal discovery aims to uncover the underlying unknown causal relationships based purely on observational data [2]. In contrast, applying ChatGPT to answer the questions about causal relationships is fundamentally different. With the current version of ChatGPT, we can only use the names (meta information) instead of observational data of variables to answer causal questions. The answers to the causal questions given by ChatGPT are based on a trained large language model, which can be viewed as an approximation for existing knowledge in the training natural language data. Nevertheless, such investigations still provide us valuable insights into ChatGPT and raise more thoughts about how to leverage its ability. But we need to exercise great caution in the conclusion as benchmarks [4,5] utilizing known knowledge are set for evaluation purposes instead of the goal of the causal discovery. Results and Insights. The ground-truth causal relationships in neuropathic pain diagnosis are obtained from both a domain expert and known medical literature [5]. As the number of all possible cause-effect pairs in this context is huge (more than 10000 pairs), we cannot test all of them manually. Thus, we sub-sampled 50 positive pairs (ground-true causal relationships) and 50 negative pairs (wrong causal relationships) from the dataset and generated the question in the format of "X causes Y. Answer true or false", where X and Y are sampled pairs from the full causal map of the neuropathic pain dataset. The full test results can be found at shorturl.at/amBX1. Many individual answers are reasonable, such as in Figure 2, but the performance is still flowed currently. As shown in Table 1 and 2, ChatGPT tends to make false negative mistakes. We inspected the results qualitatively and quantitatively and observed that:
It only understands the languages that are typically used to describe the situations but not the underlying knowledge. We provide two examples to demonstrate it. The first example is shown in Figure 3. It cannot identify the lower abdominal discomfort that can be caused by T12 radiculopathy. The explanation identifies the lower back, hip, and leg region only, while T12 nerve goes through these regions shown in Figure 1, and the lower abdominal region is part of it. Thus, it indicates that it provides the answer based on the trained content but does not understand the human body's nervous system. The second example is shown in Figure 4, which demonstrates a lack of understanding of how regional discomfort is described. The region around the key bone is the upper shoulder region. ChatGPT can identify shoulder discomfort as an effect but not the discomfort around the key bone.
Its performance is not yet consistent and not stable. Firstly, we observe that it provides different answers to the same question. We have tested some of the queries twice on different days. As shown in Table 3, the answers on the first day differ from the ones on the second day significantly. The answers on the second day are much more conservative to claim a causal relationship. This may be due to internal model updates.
Such inconsistent performance is a major concern for answering causal questions. As the later results have very few positive answers, the final results that we used considered the earlier results when available for the table 1 and 2. Secondly, as the original dataset is associated with terms in Swedish, we found that ChatGPT can correctly identify Swedish in some cases, such as in Figure 2, but fails in some other cases, such as in Figure 5. This may contribute further to a large number of false negatives.
Conclusion.
Based on the observations, we find:
• There are some limitations for the current ChatGPT in terms of understanding new concepts and knowledge beyond the existing corpus of text training data. Moreover, the consistency and stability of its performance need to be improved. Such improvements can happen without a paradigm shift in the models.
• We need to be extremely cautious about using causal claims made by ChatGPT as causal discovery results. This is because causal discovery and causal question answering with large language models are fundamentally different tasks. Causal benchmarks may be biased towards utilizing existing knowledge for evaluation [5,4], which is against the goal of causal discovery.
• In some situations, ChatGPT does give correct answers that can be non-trivial to obtain from a domain expert, which could serve as a good complementary for causal discovery methods to resolve corner cases. This might open up new research opportunities for the causal community on utilizing the recent developments of large language models to complement, improve and develop better causal machine learning tools.
Although there are existing limitations, we believe that opportunities for ChatGPT can help to improve the causality research is huge. With deep integration with ChatGPT type of models and interface. We can also imagine a future where ChatGPT can answer different causal questions. R S1 Radikulopati causes R Lårbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false.
TRUE 1 0 L T5 Radikulopati causes L Bröstbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false.
TRUE 0 0 L C5 Radikulopati causes R Interskapulära besvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false.
TRUE 0 0 R C6 Radikulopati causes R Under armsbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. TRUE 0 0 L L1 Radikulopati causes L Mediala ljumskbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false.
TRUE 0 0 L L1 Radikulopati causes L Adduktortendalgi."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false.
TRUE 0 0 L T10 Radikulopati causes IBS."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false.
TRUE 0 0 R L5 Radikulopati causes L Bakhuvudvärk."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. TRUE 0 0 Table 3: Results demonstrate lack of consistency using ChatGPT.
Figure 1 :
1Dermatome map [1] as a reference for this benchmark.
Figure 2 :
2Example showing that ChatGPT can correctly answer the question and provide reasonable explanations.
Figure 3 :
3The lower abdominal is the region where T12 nerve passes. If looking at the dermatome map 1, it is easy to identify lower back, hip, and lower abdominal discomfort can all be caused by T12 radiculopathy.
Figure 4 :
4Example showing that ChatGPT fails to understand the region on the body. The area around the key bone is largely overlapping with the front shoulder area especially when the patient describes the symptoms.
Figure 5 :
5Example showing that ChatGPT can identify foreign language time by time and that it is not very reliable. causes R Höftkamsbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. causes L Nedre bukbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. causes L Ljumskbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. causes L Laterala armsbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true Radikulopati causes R Armbågsbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. Radikulopati causes IBS."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. Radikulopati causes Nackbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. Radikulopati causes R Laterala armbågsbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. Radikulopati causes R Handbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false. Radikulopati causes L Laterala vadbesvär."R" and "L" refer to the right and left sides of the body, respectively).Answer with true or false.
Table 1 :
1Test results demonstrate high precision and low recall.Negative Positive
Negative
50
44
Positive
0
6
Table 2 :
2Confusion matrix showing
that there were no false positives.
Rows are predictions and columns
are ground truth.
Review of causal discovery methods based on graphical models. C Glymour, K Zhang, P Spirtes, Frontiers in genetics. 10524C. Glymour, K. Zhang, and P. Spirtes. Review of causal discovery methods based on graphical models. Frontiers in genetics, 10:524, 2019.
. Openai, Chatgpt, OpenAI. Chatgpt. https://chat.openai.com/chat/.
Chatgpt causality pairs. A Sharma, A. Sharma. Chatgpt causality pairs. https://github.com/amit-sharma/chatgpt-causality-pairs.
Neuropathic pain diagnosis simulator for causal discovery algorithm evaluation. R Tu, K Zhang, B Bertilson, H Kjellstrom, C Zhang, Advances in Neural Information Processing Systems. 32R. Tu, K. Zhang, B. Bertilson, H. Kjellstrom, and C. Zhang. Neuropathic pain diagnosis simulator for causal discovery algorithm evaluation. Advances in Neural Information Processing Systems, 32, 2019.
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arxiv |
The non-planar contribution to the four-loop universal anomalous dimension in N = 4 Supersymmetric Yang-Mills theory
26 Feb 2009
V N Velizhanin
Theoretical Physics Department Petersburg Nuclear Physics Institute Orlova Roscha
188300Gatchina, St. PetersburgRussia
The non-planar contribution to the four-loop universal anomalous dimension in N = 4 Supersymmetric Yang-Mills theory
26 Feb 2009
We present the result of a full direct component calculation for the non-planar contribution to the four-loop anomalous dimension of the Konishi operator in N = 4 Supersymmetric Yang-Mills theory. The result contains only ζ(5) term and proportional to ζ(5) contribution in the planar case, which comes purely from wrapping corrections. We have extended also our previous calculations for the leading transcendental contribution arXiv:0811.0607 on non-planar case and have found the same results up to a common factor. It allows us to suggest that the non-planar contribution to the four-loop universal anomalous dimension for the twist-2 operators with arbitrary Lorentz spin is proportional to S 2 1 (j) ζ(5). This result gives unusual double-logarithmic asymptotic ln 2 j for large j.In our previous papers [1, 2] we have calculated the planar four-loop anomalous dimension of the Konishi operator and the leading transcendental contribution to the four-loop universal anomalous dimension of twist-2 operators in the N = 4 supersymmetric Yang-Mills (SYM) theory. Calculations have been performed in component as a full direct computation of the anomalous dimension of the twist-2 operator. The advantages of our method are the full automation of process of calculations and the absence of any suggestion about specific properties of operators, so these results can serve as "experimental" test for the similar results obtained with the help of integrability[3]in the framework of AdS/CFT-correspondence[4]. The planar four-loop anomalous dimension of the Konishi operator were computed earlier by two different ways from the both sides of AdS/CFTcorrespondence. In the N = 4 SYM theory the calculations were performed in the superfield formalism and take into account only diagrams did not included in the asymptotic Bethe-ansatz [5], following Ref.[6]. From superstring side [7] the finite size effects were take into account using Lüscher formulas[8]. The results of both computations are in agreement after corrections from the perturbative side. Our result of the full direct calculation is the same and confirms correctness all suggestions of the computations from Refs.[5,7], including the correctness of the asymptotic Bethe-ansatz up to the four loops. Our result for leading transcendental contribution to the four-loop universal anomalous dimension for the twist-2 operators with arbitrary Lorentz spin was used together with the predictions from the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation[9]to check the corresponding part of the finite size corrections [10] to the four-loop anomalous dimension of BMN-operators [11] from the sl(2) sector obtained from the asymptotic Bethe ansatz[12].One more advantage of our method of the full direct calculation is the possibility to obtain full color structure for the four-loop Konishi. In this paper we present from the first time the result for the non-planar (color subleading) contribution to the four-loop Konishi and for the first three even moments for the leading transcendental contribution to the four-loop universal anomalous dimension of the twist-2 operators with arbitrary Lorentz spin in N = 4 SYM theory.The calculation of the non-planar contribution, which is proportional to the quartic Casimir operator d 44 (see Ref.[13]), can be split in two part according to the basic parent topology for master-integrals. For the four-loop tadpoles with equal mass lines we have two parent topologies presented inFig. 1: planar topology A and non-planar topology B. These two topologies correspond to the master-integrals PR12 and PR0 fromTable 1of Ref.[14]and all other topologies from this table can be obtained by cancelling one and more lines inside parent topologies. For the diagrams from the first, planar class we already have the well-tested program for the four-loop calculations which based on our own implementation of the Laporta's algorithm[15]for the resolution of the integration by part (IBP) identities in the form of MATHEMATICA package BAMBA and using the method from Refs.[16]. With this program we produced database for all necessary integrals expressed through master-integrals from Ref.[14]. For the second, non-planar class of diagrams in principal we should generate and resolve all sets of new IBP identities. But we have a great simplification if noting that the cancellation any one line in the non-planar topology B gives topology C, which can be obtained with the cancellation of any horizontal
line from the planar topology A and for topology C we already have the recurrence relations for all necessary integrals. So, for the non-planar topology B we should go only on one step in the resolution of the IBP identities, then change momentum according to topology C and reexpand denominators if it is necessary. In this way we extend our database only with integrals with different positive powers for all nine propagators of the non-planar topology B. Thus, we have repeated our previous calculations for the Konishi operator [17]
O K = tr A i A i + B i B i , i = 1, 2, 3 ,(1)
where A i and B i are the real adjoint scalar and pseudoscalar fields correspondingly, keep quartic Casimir operator d 44 (did not substitute its leading color value d 44 = N 2 c /24 as before) and using additional FORM procedure for the diagrams with the non-planar topology B from Fig. 1. To check, that our program work correctly we reproduce both planar (∼ C 4 A ) and non-planar (∼ d 44 ) parts for the anomalous dimension of the gluon field [14] coming from the pure Yang-Mills gauge theory. All diagrams were produced with DIANA [18] , which call QGRAF [19] to generate all diagrams and obtained code were calculated with FORM [20], using FORM package COLOR [13] for evaluation of the color traces.
The final result after subtraction of the anomalous dimension for the scalar fields is:
γ 4−loop K = 12 g 2 − 48 g 4 + 336 g 6 + −2496 + 576 ζ(3) − 1440 1 + 12 N 2 c ζ(5) g 8 ,(2)
with
g 2 = g 2 Y M N c (4π) 2 , d 44 = N 2 c (N 2 c + 36) 24(3)
and the following non-planar contribution to the four-loop anomalous dimension for the (pseudo)scalar fields (see Ref. [1] for the planar part):
γ 4−loop, non−planar φ = − 42 − 177 ζ(3) + 555 ζ(5) g 8 N 2 c .(4)
An important check of our result (2) is the absence of higher poles and some special numbers such as ζ (2), ζ(4), S 2 and other, which enter in the scalar master integrals from Ref. [14]. Also we have repeated our previous calculations for the first three moments of the leading transcendental contribution to the four-loop universal anomalous dimension of twist-2 operators [2]. As in planar case we have produced the database for the scalar integrals 1 containing ζ 5 for non-planar topology B from Fig. 1 with the MATHEMATICA package FIRE [21]. Surprisingly, but the non-planar leading transcendental contribution to the four-loop universal anomalous dimension is modified in the same way as the Konishi in Eq. (2)
γ (3) uni (j) = − 640 S 2 1 (j − 2) 1 + 12 N 2 c ζ 5 .(5)
Based on the fact, that the non-planar contribution to the four-loop anomalous dimension of the Konishi operator (2) contains only ζ(5) and take into account Eq. (5) it is reasonable to suggest that the full non-planar (color subleading) contribution to the four-loop universal anomalous dimension of twist-2 operators with arbitrary Lorentz spin has the following form:
γ (3) uni, np (j) = −640 S 2 1 (j − 2) 12 N 2 c ζ(5) ,(6)with 2 γ uni (j) = γ (0) uni (j) g 2 + γ (1) uni (j) g 4 + γ (2) uni (j) g 6 + γ (3) uni (j) g 8 + ...(7)
from Refs. [23,24,25,12,10]. Note, that really, we have calculated only first three even moments j = 2, j = 4 and j = 6 for this equation and it can be modified with other harmonic sums (see Ref. [2]). Moreover, it is possible, that for higher values of moments j the non-planar contribution to the four-loop universal anomalous dimension will contain the terms which is proportional to ζ(3) and rational number to give such combination of the harmonic sums that for j = 4, i.e. for the Konishi, they will give zero.
However, let's us suggest that Eq. (6) is correct. There are two important consequence of this result. First of all it will gives the unusual behavior for the large spin j limit, which will proportional to ln 2 j instead of expected ln j [26]. Another consequence comes from the analytical continuation to j → 1 + ω, where the predictions from the BFKL equation exist. The analytical continuation of Eq.(6) gives 1/ω 2 pole which comes from the (jet unknown) next-to-next-to-leading logarithmic approximation (NNLLA or third order corrections) to the kernel of the BFKL equation and, then, it should contain corresponding non-planar contribution at this order. As the universal anomalous dimension of the twist-2 operators and the kernel of the BFKL equation in N = 4 SYM is the most complicated part of the corresponding QCD results these futures should be hold also in QCD.
In spite of planar AdS/CFT system well studied from the both sides of gauge/string duality and seems to be solved [27], non-planar results in N = 8 SYM theory have been obtained only for gluon scattering amplitudes, which are related with N = 8 Supergravity [28]. Our results show that the non-planar contribution at least for the anomalous dimension of the twist-2 operators has a rather simple form and relation with the wrapping corrections and probably can be studied with other methods widely used for the planar case.
Figure 1 :
1Basic parent four-loop planar (A) and non-planar (B) topologies and topology (C) obtained by cancelling one horizontal line in topology A or any one line in topology B.
Results for integrals can be obtained under request. 2 Similar result was obtained for the twist-3 operator up to the five loops[12,22].
AcknowledgmentsWe would like to thank Lev Lipatov, Andrei Onishchenko and Matthias Staudacher for useful discussions. This work is supported by RFBR grants 07-02-00902-a, RSGSS-3628.2008.2. We thank the Max Planck Institute for Gravitational Physics at Potsdam for hospitality while working on parts of this project.
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| {'fraction_non_alphanumeric': 0.08589183211332876, 'fraction_numerical': 0.07272631523513613, 'mean_word_length': 3.548502994011976, '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': 4, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We present the result of a full direct component calculation for the non-planar contribution to the four-loop anomalous dimension of the Konishi operator in N = 4 Supersymmetric Yang-Mills theory. The result contains only ζ(5) term and proportional to ζ(5) contribution in the planar case, which comes purely from wrapping corrections. We have extended also our previous calculations for the leading transcendental contribution arXiv:0811.0607 on non-planar case and have found the same results up to a common factor. It allows us to suggest that the non-planar contribution to the four-loop universal anomalous dimension for the twist-2 operators with arbitrary Lorentz spin is proportional to S 2 1 (j) ζ(5). This result gives unusual double-logarithmic asymptotic ln 2 j for large j.In our previous papers [1, 2] we have calculated the planar four-loop anomalous dimension of the Konishi operator and the leading transcendental contribution to the four-loop universal anomalous dimension of twist-2 operators in the N = 4 supersymmetric Yang-Mills (SYM) theory. Calculations have been performed in component as a full direct computation of the anomalous dimension of the twist-2 operator. The advantages of our method are the full automation of process of calculations and the absence of any suggestion about specific properties of operators, so these results can serve as "experimental" test for the similar results obtained with the help of integrability[3]in the framework of AdS/CFT-correspondence[4]. The planar four-loop anomalous dimension of the Konishi operator were computed earlier by two different ways from the both sides of AdS/CFTcorrespondence. In the N = 4 SYM theory the calculations were performed in the superfield formalism and take into account only diagrams did not included in the asymptotic Bethe-ansatz [5], following Ref.[6]. From superstring side [7] the finite size effects were take into account using Lüscher formulas[8]. The results of both computations are in agreement after corrections from the perturbative side. Our result of the full direct calculation is the same and confirms correctness all suggestions of the computations from Refs.[5,7], including the correctness of the asymptotic Bethe-ansatz up to the four loops. Our result for leading transcendental contribution to the four-loop universal anomalous dimension for the twist-2 operators with arbitrary Lorentz spin was used together with the predictions from the Balitsky-Fadin-Kuraev-Lipatov (BFKL) equation[9]to check the corresponding part of the finite size corrections [10] to the four-loop anomalous dimension of BMN-operators [11] from the sl(2) sector obtained from the asymptotic Bethe ansatz[12].One more advantage of our method of the full direct calculation is the possibility to obtain full color structure for the four-loop Konishi. In this paper we present from the first time the result for the non-planar (color subleading) contribution to the four-loop Konishi and for the first three even moments for the leading transcendental contribution to the four-loop universal anomalous dimension of the twist-2 operators with arbitrary Lorentz spin in N = 4 SYM theory.The calculation of the non-planar contribution, which is proportional to the quartic Casimir operator d 44 (see Ref.[13]), can be split in two part according to the basic parent topology for master-integrals. For the four-loop tadpoles with equal mass lines we have two parent topologies presented inFig. 1: planar topology A and non-planar topology B. These two topologies correspond to the master-integrals PR12 and PR0 fromTable 1of Ref.[14]and all other topologies from this table can be obtained by cancelling one and more lines inside parent topologies. For the diagrams from the first, planar class we already have the well-tested program for the four-loop calculations which based on our own implementation of the Laporta\'s algorithm[15]for the resolution of the integration by part (IBP) identities in the form of MATHEMATICA package BAMBA and using the method from Refs.[16]. With this program we produced database for all necessary integrals expressed through master-integrals from Ref.[14]. For the second, non-planar class of diagrams in principal we should generate and resolve all sets of new IBP identities. But we have a great simplification if noting that the cancellation any one line in the non-planar topology B gives topology C, which can be obtained with the cancellation of any horizontal', 'arxivid': '0902.4646', 'author': ['V N Velizhanin \nTheoretical Physics Department Petersburg Nuclear Physics Institute Orlova Roscha\n188300Gatchina, St. PetersburgRussia\n'], 'authoraffiliation': ['Theoretical Physics Department Petersburg Nuclear Physics Institute Orlova Roscha\n188300Gatchina, St. PetersburgRussia'], 'corpusid': 18338319, 'doi': '10.1134/s0021364009120017', 'github_urls': [], 'n_tokens_mistral': 7619, 'n_tokens_neox': 6206, 'n_words': 3154, 'pdfsha': '88dcf21b6b2a06f926a200006b6f77627b09d37e', 'pdfurls': ['https://arxiv.org/pdf/0902.4646v1.pdf'], 'title': ['The non-planar contribution to the four-loop universal anomalous dimension in N = 4 Supersymmetric Yang-Mills theory', 'The non-planar contribution to the four-loop universal anomalous dimension in N = 4 Supersymmetric Yang-Mills theory'], 'venue': []} |
arxiv |
Comment on "Novel Superfluidity in a Trapped Gas of Fermi Atoms with Repulsive Interaction Loaded on an Optical Lattice"
2004. 2003. 2004. 2004. 1992. 1993. 1990
M Rigol
Physics Department
University of California
95616DavisCAUSA
S R Manmana
Institut für Theoretische Physik III
Universität Stuttgart
70550StuttgartGermany
Fachbereich Physik
Philipps-Universität Marburg
35032MarburgGermany
A Muramatsu
Institut für Theoretische Physik III
Universität Stuttgart
70550StuttgartGermany
R T Scalettar
Physics Department
University of California
95616DavisCAUSA
R R P Singh
Physics Department
University of California
95616DavisCAUSA
S Wessel
Institut für Theoretische Physik III
Universität Stuttgart
70550StuttgartGermany
Comment on "Novel Superfluidity in a Trapped Gas of Fermi Atoms with Repulsive Interaction Loaded on an Optical Lattice"
Phys. Rev. Lett
9368092004. 2003. 2004. 2004. 1992. 1993. 1990numbers: 0375Ss0530Fk7130+h7420Mn
In a recent letter [1] (referred to as I below), Machida et al. made the exciting claim that in a one-dimensional (1D) trapped gas of fermions with repulsive interactions a superfluid phase appears around the Mott-insulator (MI) at the center of the trap (COT). Their claim is based on a negative binding energy (E b ), and a large weight for a singlet formed by particles located at opposite sides of the MI. We show here that the observed effects are not related to superfluidity.After a MI forms at large U , two particles with opposite spins added to the trap prefer to sit beyond the two ends of the MI phase in order to avoid double occupancy. Hence, the large weight of the singlet [Eq. (3)] inFig. 4of I can be understood to be a simple consequence of the density distribution and the antiferromagnetic character of the MI state, i.e., it does not signal superfluidity.We then focus on the origin of the negative E b observed in I. Most of the results in I exhibit a non-zero density at the borders of the trap, i.e, they depend on the boundary conditions. We thus recalculated two cases depicted inFig. 1(a) of I, keeping the same curvature of the trap and increasing the system size to N = 20. This ensures zero density at the borders, as it should be for confined systems. We used quantum Monte Carlo (QMC) simulations [2, 3], density-matrix renormalization group (DMRG)[4], and exact diagonalization (ED).Figure 1shows that a negative E b appears for large values of U/t, at the point where the MI sets in the COT. Beyond this point adding more particles to the system causes the MI to increase in size. This is in contrast to the systems without a trap, where adding more particles to the half-filled case causes the MI to disappear. Both negative and positive E b arise in the latter doped case for different boundary conditions[5].The dashed line inFig. 1corresponds to ED results of the 1D Hubbard model without trap, at half-filling, and open boundary conditions (OBC). Here E b is calculated by adding a site when adding a particle, for OBC, in order to simulate the MI in the middle of the trap without the metallic wings. The results obtained are practically indistinguishable from the E b obtained for trapped systems after the MI appears in the COT. Therefore, the negative E b is due to the MI region, and does not signal superfluidity in the wings of the MI. Moreover, in the inset ofFig. 1we show that the negative E b in the 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 0.18 0.22 0.26 6 8 10 12 14 U/t E /t b N f E /t G * FIG. 1: (color online). QMC (points) and DMRG (continuous lines) results for E b (10) = E(5, 5) + E(6, 6) − 2E(5, 6) vs U/t for: V = 7.5t (▽, ), and V = 9.5t (♦,△) (see I), when the center of the trap is located in the middle of two lattice points (▽,♦), and in a lattice point ( ,△). The dashed line shows ED results for E b (10) in open MI systems (see text). The inset shows ED results for E * G = EG + AN f vs N f , in open MI systems with U/t = 8. Here, we choose a nonzero value of A = 0.327 to stress the even(×)-odd(+) effect, without affecting the actual value of E b < 0 it causes.MI is due to an even-odd effect. There we have plotted the ground-state energy E G for MI systems without the trap, and OBC, vs the number of particles (N f ). The even-odd effect is evident, and becomes smaller with increasing system size. Additionally, consistent with the results above, we find that: (i) Displacing the COT from the middle of two lattice points, as selected in I, leads to positive values of E b . Results for the COT on a lattice point are also shown inFig. 1. (ii)In the trap, similarly to the OBC case, the negative E b → 0 almost linearly with increasing system size, i.e, it is a finite size effect.
In a recent letter [1] (referred to as I below), Machida et al. made the exciting claim that in a one-dimensional (1D) trapped gas of fermions with repulsive interactions a superfluid phase appears around the Mott-insulator (MI) at the center of the trap (COT). Their claim is based on a negative binding energy (E b ), and a large weight for a singlet formed by particles located at opposite sides of the MI. We show here that the observed effects are not related to superfluidity.
After a MI forms at large U , two particles with opposite spins added to the trap prefer to sit beyond the two ends of the MI phase in order to avoid double occupancy. Hence, the large weight of the singlet [Eq. (3)] in Fig. 4 of I can be understood to be a simple consequence of the density distribution and the antiferromagnetic character of the MI state, i.e., it does not signal superfluidity.
We then focus on the origin of the negative E b observed in I. Most of the results in I exhibit a non-zero density at the borders of the trap, i.e, they depend on the boundary conditions. We thus recalculated two cases depicted in Fig. 1(a) of I, keeping the same curvature of the trap and increasing the system size to N = 20. This ensures zero density at the borders, as it should be for confined systems. We used quantum Monte Carlo (QMC) simulations [2,3], density-matrix renormalization group (DMRG) [4], and exact diagonalization (ED). Figure 1 shows that a negative E b appears for large values of U/t, at the point where the MI sets in the COT. Beyond this point adding more particles to the system causes the MI to increase in size. This is in contrast to the systems without a trap, where adding more particles to the half-filled case causes the MI to disappear. Both negative and positive E b arise in the latter doped case for different boundary conditions [5].
The dashed line in Fig. 1 corresponds to ED results of the 1D Hubbard model without trap, at half-filling, and open boundary conditions (OBC). Here E b is calculated by adding a site when adding a particle, for OBC, in order to simulate the MI in the middle of the trap without the metallic wings. The results obtained are practically indistinguishable from the E b obtained for trapped systems after the MI appears in the COT. Therefore, the negative E b is due to the MI region, and does not signal superfluidity in the wings of the MI. Moreover, in the inset of Fig. 1 we show that the negative E b in the
FIG. 1 :
1(color online). QMC (points) and DMRG (continuous lines) results for E b (10) = E(5, 5) + E(6, 6) − 2E(5, 6) vs U/t for: V = 7.5t (▽, ), and V = 9.5t (♦,△) (see I), when the center of the trap is located in the middle of two lattice points (▽,♦), and in a lattice point ( ,△). The dashed line shows ED results for E b (10) in open MI systems (see text). The inset shows ED results for E * G = EG + AN f vs N f , in open MI systems with U/t = 8. Here, we choose a nonzero value of A = 0.327 to stress the even(×)-odd(+) effect, without affecting the actual value of E b < 0 it causes.
PACS numbers: 03.75.Ss, 05.30.Fk, 71.30.+h, 74.20.Mn
MI is due to an even-odd effect. There we have plotted the ground-state energy E G for MI systems without the trap, and OBC, vs the number of particles (N f ). The even-odd effect is evident, and becomes smaller with increasing system size. Additionally, consistent with the results above, we find that: (i) Displacing the COT from the middle of two lattice points, as selected in I, leads to positive values of E b . Results for the COT on a lattice point are also shown inFig. 1. (ii)In the trap, similarly to the OBC case, the negative E b → 0 almost linearly with increasing system size, i.e, it is a finite size effect. This work was supported by NSF-DMR-0312261, NSF-DMR-0240918, NSF-ITR-0313390, SFB 382, HLR-Stuttgart, and NIC at FZ Jülich. We thank R. M. Noack for helpful discussions.
. M Machida, Phys. Rev. Lett. 93200402M. Machida et al., Phys. Rev. Lett. 93, 200402 (2004).
. M , Phys. Rev. Lett. 91130403M. Rigol et al., Phys. Rev. Lett. 91, 130403 (2003).
. M Rigol, A Muramatsu, Phys. Rev. A. 6953612M. Rigol and A. Muramatsu, Phys. Rev. A 69, 053612 (2004);
. Opt. Commun. 24333Opt. Commun. 243, 33 (2004).
. S R White, Phys. Rev. Lett. 692863S. R. White, Phys. Rev. Lett., 69, 2863 (1992);
. Phys. Rev. B. 4810345Phys. Rev. B 48, 10345 (1993).
. R M Fye, Phys. Rev. B. 426809R. M. Fye et al., Phys. Rev. B 42, R6809 (1990).
| {'fraction_non_alphanumeric': 0.05986261040235525, 'fraction_numerical': 0.042307272925526114, 'mean_word_length': 3.809648662821185, 'pattern_counts': {'":': 0, '<': 2, '<?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': 'In a recent letter [1] (referred to as I below), Machida et al. made the exciting claim that in a one-dimensional (1D) trapped gas of fermions with repulsive interactions a superfluid phase appears around the Mott-insulator (MI) at the center of the trap (COT). Their claim is based on a negative binding energy (E b ), and a large weight for a singlet formed by particles located at opposite sides of the MI. We show here that the observed effects are not related to superfluidity.After a MI forms at large U , two particles with opposite spins added to the trap prefer to sit beyond the two ends of the MI phase in order to avoid double occupancy. Hence, the large weight of the singlet [Eq. (3)] inFig. 4of I can be understood to be a simple consequence of the density distribution and the antiferromagnetic character of the MI state, i.e., it does not signal superfluidity.We then focus on the origin of the negative E b observed in I. Most of the results in I exhibit a non-zero density at the borders of the trap, i.e, they depend on the boundary conditions. We thus recalculated two cases depicted inFig. 1(a) of I, keeping the same curvature of the trap and increasing the system size to N = 20. This ensures zero density at the borders, as it should be for confined systems. We used quantum Monte Carlo (QMC) simulations [2, 3], density-matrix renormalization group (DMRG)[4], and exact diagonalization (ED).Figure 1shows that a negative E b appears for large values of U/t, at the point where the MI sets in the COT. Beyond this point adding more particles to the system causes the MI to increase in size. This is in contrast to the systems without a trap, where adding more particles to the half-filled case causes the MI to disappear. Both negative and positive E b arise in the latter doped case for different boundary conditions[5].The dashed line inFig. 1corresponds to ED results of the 1D Hubbard model without trap, at half-filling, and open boundary conditions (OBC). Here E b is calculated by adding a site when adding a particle, for OBC, in order to simulate the MI in the middle of the trap without the metallic wings. The results obtained are practically indistinguishable from the E b obtained for trapped systems after the MI appears in the COT. Therefore, the negative E b is due to the MI region, and does not signal superfluidity in the wings of the MI. Moreover, in the inset ofFig. 1we show that the negative E b in the 0 0.5 1 1.5 2 2.5 0 2 4 6 8 10 0.18 0.22 0.26 6 8 10 12 14 U/t E /t b N f E /t G * FIG. 1: (color online). QMC (points) and DMRG (continuous lines) results for E b (10) = E(5, 5) + E(6, 6) − 2E(5, 6) vs U/t for: V = 7.5t (▽, ), and V = 9.5t (♦,△) (see I), when the center of the trap is located in the middle of two lattice points (▽,♦), and in a lattice point ( ,△). The dashed line shows ED results for E b (10) in open MI systems (see text). The inset shows ED results for E * G = EG + AN f vs N f , in open MI systems with U/t = 8. Here, we choose a nonzero value of A = 0.327 to stress the even(×)-odd(+) effect, without affecting the actual value of E b < 0 it causes.MI is due to an even-odd effect. There we have plotted the ground-state energy E G for MI systems without the trap, and OBC, vs the number of particles (N f ). The even-odd effect is evident, and becomes smaller with increasing system size. Additionally, consistent with the results above, we find that: (i) Displacing the COT from the middle of two lattice points, as selected in I, leads to positive values of E b . Results for the COT on a lattice point are also shown inFig. 1. (ii)In the trap, similarly to the OBC case, the negative E b → 0 almost linearly with increasing system size, i.e, it is a finite size effect.', 'arxivid': 'cond-mat/0502599', 'author': ['M Rigol \nPhysics Department\nUniversity of California\n95616DavisCAUSA\n', 'S R Manmana \nInstitut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany\n\nFachbereich Physik\nPhilipps-Universität Marburg\n35032MarburgGermany\n', 'A Muramatsu \nInstitut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany\n', 'R T Scalettar \nPhysics Department\nUniversity of California\n95616DavisCAUSA\n', 'R R P Singh \nPhysics Department\nUniversity of California\n95616DavisCAUSA\n', 'S Wessel \nInstitut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany\n'], 'authoraffiliation': ['Physics Department\nUniversity of California\n95616DavisCAUSA', 'Institut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany', 'Fachbereich Physik\nPhilipps-Universität Marburg\n35032MarburgGermany', 'Institut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany', 'Physics Department\nUniversity of California\n95616DavisCAUSA', 'Physics Department\nUniversity of California\n95616DavisCAUSA', 'Institut für Theoretische Physik III\nUniversität Stuttgart\n70550StuttgartGermany'], 'corpusid': 31001814, 'doi': '10.1103/physrevlett.95.218901', 'github_urls': [], 'n_tokens_mistral': 2934, 'n_tokens_neox': 2516, 'n_words': 1613, 'pdfsha': '052da8d9a2a8a5e48d9a37ecb4d90df64270b092', 'pdfurls': ['https://export.arxiv.org/pdf/cond-mat/0502599v2.pdf'], 'title': ['Comment on "Novel Superfluidity in a Trapped Gas of Fermi Atoms with Repulsive Interaction Loaded on an Optical Lattice"', 'Comment on "Novel Superfluidity in a Trapped Gas of Fermi Atoms with Repulsive Interaction Loaded on an Optical Lattice"'], 'venue': ['Phys. Rev. Lett']} |
arxiv |
Higher Spin Gravity in AdS 3 and Folds on Fermi Surface
Suvankar Dutta [email protected]
Indian Institute of Science Education and Research Bhopal Bhopal bypass
462066BhopalIndia
Debangshu Mukherjee [email protected]
Department of Physics
Indian Institute of Technology Kanpur Kalyanpur
208016KanpurIndia
Sanhita Parihar [email protected]
Indian Institute of Science Education and Research Bhopal Bhopal bypass
462066BhopalIndia
Higher Spin Gravity in AdS 3 and Folds on Fermi Surface
In this paper, we introduce new sets of boundary conditions for higher spin gravity in AdS 3 where the boundary dynamics of spin two and other higher spin fields are governed by the interacting collective field theory Hamiltonian of Avan and Jevicki. We show that the time evolution of spin two and higher spin fields can be captured by the classical dynamics of folded fermi surfaces in the similar spirit of Lin, Lunin and Maldacena. We also construct infinite sequences of conserved charges showing the integrable structure of higher spin gravity (for spin 3) under the boundary conditions we considered. Further, we observe that there are two possible sequences of conserved charges depending on whether the underlying boundary fermions are non-relativistic or relativistic.
Introduction
The dynamics of classical gravity in three spacetime dimensions has no bulk degrees of freedom, completely fixed by the asymptotic boundary conditions. Using the Hamiltonian formalism for gravity in three-dimensional AdS (AdS 3 ) spacetime, Brown and Henneaux showed that for a specific choice of boundary conditions where the lapse and the shift functions are held constant at the asymptotic boundary, the asymptotic symmetry group is generated by two copies of Virasoro algebras [1]. Such boundary conditions are known as Brown-Henneaux boundary conditions and are considered to be the standard boundary conditions. In the language of Chern-Simons theory Brown-Henneaux boundary conditions correspond to fixing the temporal component of the gauge field (known as chemical potential) to a constant value at the asymptotic boundary 1 . However, gravity in AdS 3 admits a new family of non-standard boundary conditions leveled by a non-negative integer k [4] 2 . In such cases the chemical potentials are no longer kept fixed at the asymptotic boundary, rather they are allowed to explicitly depend on the fields (asymptotic values of 1 The condition on constant chemical potential was relaxed in [2,3] 2 Other consistent boundary conditions are also interesting and discussed in the literature. For example [5][6][7][8][9][10][11][12][13][14]. The most general asymptotically AdS 3 boundary conditions for Einstein's gravity were discussed in [15]. Also, for N = (1, 1) extended higher spin AdS 3 supergravity the boundary conditions has been explored in [16]. the angular components of the gauge fields). The classical dynamics for these boundary conditions are governed by the k-th element of the KdV hierarchy. The standard Brown-Henneaux boundary conditions correspond to the special case k = 0. For k = 1, Einstein's equations are similar to two independent copies of KdV equations and the asymptotic symmetry algebra is infinite dimensional, without any central charge. The field theoretic realisation of these infinite dimensional symmetry algebras has been discussed in [17].
Another class of boundary conditions, known as soft hairy boundary conditions were considered in [7,9] for fixed chemical potentials at boundary. The soft hairy boundary conditions were further generalised in [11,14] by choosing the chemical potentials as appropriate local functions of the fields. It was shown that for this class of boundary conditions the classical dynamics of gravity is governed by the Gardner hierarchy of nonlinear partial differential equations, known as mixed KdV-mKdV hierarchy. The story of pure gravity, discussed so far, can also be generalised to higher spin gravity (and also supergravity) in AdS 3 [2,3,10,16,[18][19][20][21][22][23][24]. In [25] a new set boundary conditions were introduced for gravity coupled with spin three field where the boundary dynamics is governed by the members of the modified Boussinesq integrable hierarchy. See also [23,24,26]. It is known that the KdV equation (also mKdV and Boussinesq equations) describes an integrable dynamical system due to the presence of an infinite number of conserved quantities (constants of motion). The above results thus show a connection between classical higher spin gravity in AdS 3 and certain special classes of integrable systems (KdV/mKdV/Boussinesq) in 1 + 1 dimensions.
Another example of simple and interesting integrable system is the one dimensional matrix model of N ×N unitary matrices with arbitrary potential. In the large N limit, the one dimensional matrix model can be described in terms of a classical cubic collective field theory in arbitrary background potential [27,28]. The large N degrees of freedom of the matrix model are captured by a real collective bosonic field, called the eigenvalue density.
A generalisation of collective field theory including the interaction between the original collective fields and an infinite set of supplementary fields was proposed in [29]. One interesting feature of the collective field theory is that it admits a free fermi phase space description in terms of two dimensional droplets due to bosonisation [30][31][32][33][34][35][36][37]. A similar phase space description also exists for interacting collective field theory. It was shown in [38,39] that the dynamics of interacting collective field theory can be described in terms of folds on fermi surfaces. It turns out that one can construct an infinite sequence of classical conserved charges (i.e. the Poisson brackets of these charges vanish) in cubic collective field theory and hence the theory admits an integrable structure [40,41] 3 . Therefore it is natural to ask whether the free as well as the interacting collective field theories have any dual gravity descriptions.
In this paper we introduce new sets of boundary conditions (similar to generalised soft hairy boundary conditions) for higher spin gravity in AdS 3 where the boundary dynamics of spin two and other higher spin fields is governed by the interacting collective field theory Hamiltonian [29]. We show that the time evolution of spin two and higher spin fields can be captured by the classical dynamics of folded fermi surfaces in the similar spirit of Lin, Lunin and Maldacena (LLM) [43]. This allows us to provide a free fermionic description of higher spin gravity in AdS 3 . For spin 3 excitations we also construct infinite sequences of conserved charges showing the integrable structure of higher spin gravity under the boundary conditions we consider. In particular, denoting the single free fermion Hamiltonian density by h(p, θ), we find that the infinite sequences of conserved charges are given by different integral powers of h(p, θ) integrated over the phase space in presence of folds. Further, we observe that there are two possible sequences depending on whether the underlying fermions are non-relativistic or relativistic. We also show that the entropy of higher spin black holes connected to BTZ black holes is proportional to the total area of the droplets with folds.
The organisation of our paper is as follows. In sec. 2 we review matrix quantum mechanics and the corresponding bosonic collective field theory. We also provide a phase space description of the theory and discuss the notion of folds. In sec. 3 we discuss about AdS 3 gravity coupled to higher spin fields and their corresponding Chern-Simons description. Asymptotic symmetries and the construction of conserved charges have also been summarised in this section. Droplet description of AdS 3 gravity coupled to higher spin is discussed in sec. 4. We construct two different sets of asymptotic conserved charges in higher spin gravity depending on whether the underlying fermions are non-relativistic or relativistic. We finally end with some discussions and future directions in sec. 5.
The collective field theory and free fermi droplets
In this section, we briefly review the basic ideas of matrix quantum mechanics and their connections with collective field theories. We also provide a phase space description of the theory and discuss the notion of folds.
We start with the partition function of a unitary matrix model in (0 + 1) dimension
Z t = [DU ] exp dt TrU 2 + W (U ) (2.1)
where U is a N × N unitary matrix, the trace is taken over fundamental representation, W (U ) is a gauge invariant function of U and D[U ] is the Haar measure over U (N ) group manifold. The matrix model (2.1) admits two equivalent descriptions -one in terms of free fermions [44] and the other in terms of collective bosonic field [27,28]. These two descriptions are related by bosonisation.
In a series of papers Jevicki and Sakita [27,28] showed that the matrix model (2.1) can be described in terms of a real collective bosonic field (the eigenvalue density)
ρ(t, θ) = 1 N N i=1 δ(θ − θ i ) (2.2)
where θ i s are eigenvalues of U and its conjugate momentum π(t, θ). The corresponding bosonic Hamiltonian is given by,
H B = dθ 1 2 ∂π(t, θ) ∂θ ρ(t, θ) ∂π(t, θ) ∂θ + π 2 ρ 3 (t, θ) 6 + W (θ)ρ(t, θ) (2.3)
where W (θ) is defined as
dθW (θ)ρ(t, θ) = W (U (t)). (2.4)
The dynamics of the collective field and its conjugate momentum is governed by Hamilton's equations,
∂ t ρ(t, θ) + ∂ θ (ρ(t, θ)v(t, θ)) = 0 ∂ t v(t, θ) + 1 2 ∂ θ v(t, θ) 2 + π 2 2 ∂ θ ρ(t, θ) 2 = −W ′ (θ),
where v(t, θ) = ∂ θ π(t, θ).
(2.5)
The above set of equations are coupled, non-linear and partial differential equations. It is possible to decouple these two equations by introducing two new variable p ± (t, θ), defined as
ρ(t, θ) = p + (t, θ) − p − (t, θ) 2π , and v(t, θ) = p + (t, θ) + p − (t, θ) 2 . (2.6)
The equations for p ± (t, θ) are given by,
∂ t p ± (t, θ) + p ± (t, θ)∂ θ p ± (t, θ) + W ′ (θ) = 0. (2.7)
The collective Hamiltonian (2.3), written in terms of these new variables p ± (t, θ), is decomposed into two disjoint sectors H + p and H − p given by
H B = H + B + H − B , where H ± B = ± 1 2π dθ p ± (t, θ) 3 6 + W (θ)p ± (t, θ) . (2.8)
The equations for p ± can be derived from this Hamiltonian for the following Poisson's structure
{p ± (t, θ), p ± (t, θ ′ )} = ∓2πδ ′ (θ − θ ′ ), {p + (t, θ), p − (t, θ ′ )} = 0. (2.9)
The droplet description
Decomposition of the Hamiltonian enables us to give a geometric description of collective field theory in terms of two-dimensional droplets [30]. The set of decoupled equations (2.7) governs the evolution of a free-fermi droplet in (p, θ) plane whose boundaries are given by p ± (t, θ). To understand this in detail, we consider a system of N non-relativistic free fermions (non-interacting) moving on S 1 under a common potential W (θ). The single particle Hamiltonian is given by
h(p, θ) = p 2 2 + W (θ). (2.10)
The Hamilton's equations obtained from the single particle Hamiltonian (2.10) are given by
dp dt = −W ′ (θ), dθ dt = p.
(2.11)
The single particle phase space of this system is given by a droplet with boundaries p ± (t, θ).
Using equations (2.11) one can check that the boundaries of a droplet p → p ± (t, θ) follow equation (2.7). Therefore the equations in (2.7) determine classical evolution of the fermi surface with time. The phase space Hamiltonian for such free fermi system is given by,
H pp = 1 2π dθ dp p 2 2 + W (θ) ϖ(p, θ) (2.12) where ϖ(p, θ) is the phase space density ϖ(p, θ) = Θ ((p + (t, θ) − p)(p − p − (t, θ))) . (2.13)
Integrating over p, it is easy to check that the phase space Hamiltonian H pp is exactly same as the bosonic Hamiltonian given by Eqn. (2.8).
There is a one to one correspondence between phase space variables and collective field theory variables. Eigenvalue density and the corresponding momentum can be obtained from phase space distribution by integrating over p ρ(t, θ) = 1 2π dp ϖ(p, θ), v(t, θ) = 1 2πρ dp p ϖ(p, θ).
(2.14)
Thus the relations (2.6) serve as a dictionary between bosonic (collective field theory) and
fermionic (phase space) variables.
Solving the field theory equations of motion (2.5) is equivalent to solving for upper and lower fermi surfaces in phase space picture. In either case, one needs to provide an initial data on a constant time slice in (t, θ) plane. After that the problem reduces to a Cauchy problem. Existence of a unique solution depends on the geometry of initial data curve 4 .
Folds
The fermi surfaces (p ± (t, θ)) we have discussed so far are single valued functions of θ at any time t. A more generic fermi surface can be multi valued in θ (see fig. 1). Such a surface is called a folded fermi surface. The folds can appear both in the upper and the lower surfaces. The folds can be connected to the main droplet or can be a separate droplet also. The classical theory of folds were discussed in [38,46] in terms of p ± (t, θ)
and an infinite set of variables w ±n (t, θ) satisfying classical w ∞ algebra [29]. In presence of folds one can parametrise different moments of p as
ϖ(p, θ) = p + − p − (2.15) p n n ϖ(p, θ) = 1 n(n + 1) p n+1 + − p n+1 − + n k=1 c n k p n−k + w +k − p n−k − w −k (2.16)
up to some constants c n k . These constants can be fixed from the integrable structure of the model. We shall fix them later. Presence of non-zero w ±n s signifies non-quadratic profile of the droplet. For example, we consider a droplet with one fold in the upper surface as shown in fig.1. For a given θ we have
{f +1 , f −1 } and {f +2 , f −2 }. From the above set of relations in (2.15) we have dp ϖ(p, θ) = f +1 − f −1 + f +2 − f −2 = p + − p − .
(2.17) Figure 1: Folds on fermi surface.
Since the fold is formed on the upper surface (in this particular example) it is natural to
parametrise p + = f +1 − f −1 + f +2 , and p − = f −2 . (2.18)
From the equations for higher order moments (2.16) one can find all the w ±n . For the fold shown in fig.1, first few of them are given by
w +1 = − 1 c 1 1 (f -1 − f +1 ) (f -1 − f +2 ) , w +2 = 1 2c 2 2 w +1 2c 2 1 f -1 + (c 1 1 − 2c 2 1 )(f +1 + f +2 ) . . . (2.19)
and all w −n = 0. From these relations we see that when fold is absent i.e. f +1 = f −1 or
f −1 = f +2 we have p + = f +2 , p − = f −2 and w ±n = 0.
In general, there can be T folds on the upper surface. Then the parameterisation is given by
p + = T i=1 f +i − T −1 i=1 f −i , p − = f −T (2.20)
and the w ±n can be found as functions of f ±i using (2.16):
T i=1 f n+1 +i − f n+1 −i n(n + 1) = 1 n(n + 1) p n+1 + − p n+1 − + n k=1 c n k p n−k + w +k − p n−k − w −k . (2.21)
The number of folds can depend on the position θ and time t.
In the context of 2D string theory w ±n represents an infinite set of discrete fields and p ± represent tachyon fields [29]. The discrete fields themselves close a classical w ∞ algebra.
Interpretation of w ±n as folds on fermi surfaces was given by [38]. Classical evolution of folded fermi surfaces is therefore governed by the dynamics of p ± and w ±n for a given Hamiltonian [38]. Later we shall see how the dynamics of higher-spin fields in AdS 3 is mapped with the evolution of folded fermi surfaces.
Higher spin gravity in AdS and Chern-Simons theory
We consider gravity in AdS 3 coupled with integral higher spin fields with spin 3 ≤ s ≤ M .
The bulk theory can be formulated in terms of Chern-Simons theory with the gauge group [19,20]. The matrix valued gauge fields can be separated in two chiral sectors and denoted by A ± (x). ′ + ′ and ′ − ′ correspond to gauge fields associated with two copies of the gauge groups. Two gauge fields A ± (x) are related to generalised vielbein e and spin connection ω and formally expressed as [19] (the index structure is suppressed)
SL(M, R) × SL(M, R)A ± = ω ± e l (3.1)
where l is the radius of the AdS 3 space. Using the relation between the generalised vielbein, spin connection and gauge fields, the Einstein-Hilbert action can be shown to be equal to CS action
I = I CS (A + ) + I CS (A − ) (3.2)
where,
I CS (A ± ) = ± k M 4π Tr A ± ∧ dA ± + 2 3 A ± 3 + B ∞ (A ± ). (3.3)
The level k M of the CS theory is related to 3 dimensional Newton's constant G and AdS radius l through In this paper we consider the principal embedding of sl(2, R) in sl(M, R) [10]. The generators of sl(M, R) are given by L ± and L 0 and W (s) m where s = 3, · · · , M and m = −(s − 1), · · · , (s − 1). Important relations between these generators, needed in this paper are given by,
k M = l 4Gϵ M where ϵ M = M (M 2 − 1) 6 . (3.4) For M = 2 k 2 = l 4G ≡ k. (3.5) B ∞ (A ± )Tr (L 0 L 0 ) = ϵ M 2 , Tr L 0 W (s) 0 = 0, Tr W (s) 0 W (s) 0 = ϵ M 2σ 2 s (3.6) where σ s = (2s − 1)!(2s − 2)! 48(s − 1)! 4 s−1 k=2 (M 2 − k 2 )
.
(3.7)
For other details see [10]. Different components of the metric can be computed from the gauge fields
g µν = l 2 2 Tr (A + − A − ) µ (A + − A − ) ν . (3.8)
In 3 dimensions gravity is locally trivial, all the dynamics is localised near the boundary and hence sensitive to the boundary conditions.
Boundary conditions
We parametrise the three dimensional manifold by r, t and θ, where θ is compact. r is the radial direction and the asymptotic region (boundary) is at r → ∞. We consider the following form of the gauge fields
A ± = b −1 ± d + a ± b ± (3.9)
where b ± are gauge group elements, they depend on the radial coordinate r only. The connections a ± depend only on the transverse coordinates t and θ. We must emphasize here that the specific radial dependence of b ± does not affect the asymptotic charges and the boundary equations of motion. However, in order to write an explicit form of the bulk metric, one is required to make a specific choice of b ± . In order to satisfy the Maxwell's equations (i.e. Einstein's equation) we do not need to specify any particular form of b ± .
We choose the connection in the following form [10] a ± (t,
θ) = (ξ ± (t, θ)dt ± p ± (t, θ)dθ) L 0 + M s=3 σ s (ζ ±s (t, θ)dt ± u ±s (t, θ)dθ) W (s) 0 (3.10)
where p ± and u ±s are the dynamical fields associated with gravity and higher spin fields respectively. ξ ± and ζ ±s are corresponding chemical potentials. The Maxwell's equations obtained from the action (3.3) are given by
dA ± + A ±2 = 0. (3.11)
Using the form of A ± and a ± one can check that the dynamical fields and the chemical potentials satisfyṗ ± (t, θ) = ±ξ ′ ± (t, θ),u ±s (t, θ) = ±ζ ′ ±s (t, θ).
(3.12)
Here · and ′ denote the partial derivatives with respect to t and θ respectively.
In order to find the boundary term B ± ∞ we demand that δI = 0 for any arbitrary variations of gauge fields. This implies
δB ± ∞ = ∓ k 4π dtdθ ξ ± δp ± + M s=3 ζ ±s δu ±s . (3.13)
Therefore to get a well-defined boundary term one needs to take the δ outside the integral.
It is possible if the chemical potential ξ ± and ζ ±s can be written as variation of a quantity H ± with respect to p ± and u ±s respectively, i.e.
ξ ± = − 4π k δH ± δp ± and ζ ±s = − 4π k δH ± δu ±s ,(3.14)
where H ± in general can be a functional of p ± , u ±s and their different θ derivatives
H ± = dθ H ± (p ± , {u ±s }). (3.15)
The boundary term, therefore, becomes
B ± ∞ = ± dtdθ H ± = ± dt H ± . (3.16)
Thus we specify the boundary conditions through the choice of the function H ± . The dynamics of the fields p ± and u ±s therefore depends on the choice of the boundary conditions H ±ṗ
± (t, θ) = ∓ 4π k ∂ ∂θ δH ± δp ± ,u ±s = ∓ 4π k ∂ ∂θ δH ± δu ±s . (3.17)
Conserved charges
The asymptotic symmetries are determined by a set of gauge transformations that preserve the asymptotic form of the gauge fields. The gauge transformation is given by
δa ± = dλ ± + [a ± , λ ± ] (3.18)
where λ ± are the gauge transformation parameters. One can check that the asymptotic form of the gauge fields (3.10) are preserved under (3.18) with the following choice of gauge transformation parameters [10]
λ ± = η ± (t, θ)L 0 + M s=3 σ s η ±s (t, θ)W (s) 0 . (3.19)
With this choice the asymptotic fields p ± , u ±s and chemical potentials ξ ± transform as δp ± = ±η ′ ± , and δξ ± =η ± (3.20)
δu ±s = ±η ′ ±s and δζ ±s =η ±s .
(3.21)
Since the chemical potentials ξ ± and ζ ±s now depend on p ± and u ±s , their variations are not zero anymore. Hence we can use (3.14) to writė
η ± (t, θ) = ∓ 4π k δ δp ± (t, θ) dθ ′ δH ± δp ± η ′ ± (t, θ ′ ) + M s=3 δH ± δu ±s η ′ ±s (t, θ ′ ) , (3.22) η ±s (t, θ) = ∓ 4π k δ δu ±s (t, θ) dθ ′ δH ± δp ± η ′ ± (t, θ ′ ) + M s=3 δH ± δu ±s η ′ ±s (t, θ ′ ) . (3.23)
The variation of the conserved charges for the local gauge symmetry is given by [47,48]
δQ ± (η ± , η ±s ) = − k M 2π dθ ′ Tr λ ± δ(a ± θ ) = ∓ k 4π dθ ′ η ± δp ± + M s=3
η ±s δu ±s . are also a conserved charges of the theory under the boundary conditions specified through H ± . Apparently the right hand side does not depend on η ± and η ±s , however it depends on the choice of these quantities.
Poisson brackets of charges satisfy [47] δ {η ± ,η ±s } Q ± n (η ± , η ±s ) = {Q ± n (η ± , η ±s ), Q ± n (η ± ,η ±s )}. (3.28) From these relations we can deduce the Poisson brackets for the field p ± and u ±s
{p ± (t, θ), p ± (t, θ ′ )} = ∓ 4π k ∂ ∂θ δ(θ − θ ′ ), (3.29) {u ±s (t, θ), u ±s ′ (t, θ ′ )} = ∓ 4π k ∂ ∂θ δ(θ − θ ′ )δ ss ′ (3.30) {p ± (t, θ), u ±s ′ (t, θ ′ )} = 0. (3.31)
Once we have the Poisson brackets of the field variables the field equations (3.12) can be written asṗ
± (t, θ) = {p ± (t, θ), H ± }, andu ±s (t, θ) = {u ±s (t, θ), H ± }. (3.32)
In [11,25], H n s are chosen to be generalised Gelfand-Dikii polynomials (or modified Gelfand-Dikii polynomials for modified Boussinesq hierarchy in presence of higher spins) and the field equations are give by left and right members of Gardner hierarchy (or modified Boussinesq hierarchy for higher spin). In the next section we shall see that a different set of H n s is possible and such choices will provide a gravity dual of integrable collective field theory.
Collective field theory and AdS 3 gravity
In this section we discuss the gravity dual of interacting collective field theory. Our construction allows us to provide a geometric description of the bulk solution in terms of the shapes of fermi surfaces. We first do the exercise for pure gravity and show how the boundary dynamics is captured by the time evolution of free fermi droplets. After that we discuss the higher spin case.
To match the classical dynamics of gravity and that of collective field theory given by eqns. (3.12) and (2.7) respectively we take the Hamiltonian H ± to be proportional to the cubic collective field theory Hamiltonian (2.8)
H ± = k 2 H ± B = ± k 4π dθdp h(p, θ)ϖ(θ, p). (4.1)
The equations of motion satisfied by p ± are given by (2.7). These are dispersion-less KdV equations with the source term 5 . Further, we can check that for this choice of Hamiltonian, η ± ∼ ∂H ± /∂p ± satisfy eqn. (3.22) and hence the Hamiltonian is a conserved quantity.
Thus we introduce the new boundary conditions for AdS 3 gravity by choosing the boundary Hamiltonian H ± in (3.14) to be proportional to the collective field theory Hamiltonian and hence the dynamics of boundary gravitons is captured by that of noninteracting non-relativistic fermions.
These boundary conditions also admit infinite sequences of conserved charges in the theory. Following [34,40,41] we can construct an infinite sequence of phase space integrals of different integer powers of the single particle Hamiltonian h(p, θ) (given by (2.10))
H 2n−1 = k 4π dθdp ϖ(p, θ) h n (p, θ) ≡ dθ H 2n−1 (θ). (4.2)
Separating these conserved charges into two chiral sectors for each n we have
H 2n−1 = H + 2n−1 + H − 2n−1 , where H ± 2n−1 = ± k 4π dθ p ± 0 dp p 2 2 + W (θ) n (4.3)
where H ± 1 ≡ H ± is the Hamiltonian. Expanding the power on the right hand side and integrating over p, one can write
H ± 2n−1 = ± k 4π dθ n k=0 n C k 2 k (2k + 1) p 2k+1 ± W (θ) n−k . (4.4)
It turns out that for these H ± 2n−1 , the gauge transformation parameters η ± given by (3.26), satisfy eqn. (3.22) for the Hamiltonian given by eqn. (4.1) for all n. Thus the phase space integrals (3.27) provide two infinite sequences (for two chiral sectors) of conserved charges of the AdS 3 gravity. Using (3.29) we see that the Poisson brackets of charges H 2n−1 for all n > 2 with the Hamiltonian vanish and hence they are constants of motion. One can also construct the conserved current associated with these charges. We define a current
density J µ (n)± = {H ± n , J θ (n)± } such that ∂ µ J µ (n)± = 0 on-shell. (4.5)
It turns out that to satisfy the on-shell conservation equation the spatial component of the current is given by,
J θ (n)± (θ) = ± k 4π(n + 1) p 2 ± 2 + W (θ) n+1 . (4.6)
Given the Poisson structure (3.29), one can show that the Poisson brackets of any two conserved charges vanishes
{Q ± n , Q ± m } = 0 ∀ m, n ≥ 1 (4.7)
implying the integrable structure of the AdS 3 gravity for the chosen boundary conditions.
One can also take the boundary Hamiltonian to be H ± 2n−1 for any fixed n > 1 to specify the boundary conditions. In that case the equations of motion becomė
p ± + nh(p ± , θ) n−1 p ± p ′ ± + W ′ = 0. (4.8)
These are the set of hierarchical equations with respect to (2.7). For this choice the other H ± 2n−1 s turn out to be the conserved charges of the motion. This shows the hierarchical nature of the AdS 3 gravity under the boundary conditions we considered.
BTZ black holes and droplets
In case of pure gravity the dynamical equations (2.7) admit time independent solutions. Forṗ ± = 0 we have ξ ± ′ = 0. This implies ξ ± are functions of time only. Since we are interested in time independent solutions we consider ξ ± = c ± (constant) and hence we get
p ± (θ) = ± √ 2 c ± − W (θ)
. The metric is given by
ds 2 = dr 2 + l 2 4 cosh 2 r l ((c + − c − ) dt + dθ (p − (θ) + p + (θ))) 2 − l 2 4 sinh 2 r l ((c − + c + ) dt + dθ (p + (θ) − p − (θ))) 2 .
(4.9)
The explicit form of the above metric is written using the maps (3.8), (3.9) and (3.10) where the gauge group element b ± is given by
b ± = exp ± r 2l (L +1 − L −1 ) . (4.10)
Such solutions are called black flower solutions [9,13,14]. Since the eigenvalue density is given by ρ(θ) = (p + (θ) − p − (θ))/2π, such solutions correspond to gapped or no-gap solution in the matrix model side depending whether p ± (θ) is defined over the whole range of θ or not. A further special case is constant solution : p ± (θ) = ± √ 2 √ κ ± . After a suitable coordinate transformation
r = l 2 log (κ − + κ + ) l 2 − 2r 2 − 4r 4 − 4 (κ − + κ + ) l 2r2 + (κ + − κ − ) 2 l 4 2 √ κ − κ + l 2 t = 2 √ 2 √ κ + κ − v l c − √ κ + − c + √ κ − , θ = ϕ + c + √ κ − + c − √ κ + l c + √ κ − − c − √ κ + v (4.11)
the metric can be written in the standard Schwarzschild coordinate,
ds 2 = −f (r)dv 2 + dr 2 f (r) +r 2 dϕ − J 2r 2 dv 2 .
(4.12)
The function f (r) is given by
f (r) =r 2 l 2 − M + J 2 4r 2 where M = (κ + + κ − ) and J = l(κ + − κ − ). (4.13)
The solution has horizons at
r ± = l √ 2 ( √ κ + ± √ κ − ) .
(4.14)
The entropy is given by
S BH = 2πr + 4G = √ 2πk ( √ κ + + √ κ − ) . (4.15)
The constant configuration p ± (θ) = ± √ 2κ ± corresponds to droplet as shown in fig.2. The area of the droplet is given by
A = 2π(p + (θ) − p − (θ)) = 2 √ 2π( √ κ + + √ κ − ) = 2 k S BH . (4.16)
The black hole mass M and angular momentum J are specified in terms of droplet data κ ± . A symmetric distribution about θ axis (i.e. κ + = κ − ) corresponds to J = 0, zero angular momentum. The extremal black hole corresponds to κ − = 0. Therefore any droplet with p ± (θ) = constant with |p + | ≥ |p − | corresponds to a BTZ black hole.
Higher spin droplets and conserved charges
Since the classical algebra (Poisson brackets) satisfied by p ± and u ±s is a disjoint unions of M − 1 Poisson brackets, a trivial generalisation of the above exercise is to construct a Hamiltonian which is a direct sum of M − 1 copies of mutually non-interacting Hamiltonians (4.1). As a result one can construct M − 1 copies of asymptotic conserved charges (4.4). The boundary dynamics, similarly, can be described by M − 1 copies of free fermi droplets. Such a generalisation is not interesting. One can, in fact, find a much more interesting boundary dynamics where higher spin excitations are coupled with spin two fields in a non-trivial way and hence conserved charges.
To specify non-trivial boundary conditions we first turn on all the integer higher spin fields and define a set of infinite number of collective variables w ±n from u ±s
w ±n (t, θ) = M =∞ s=3 u n+1 ±s (t, θ) n + 1 . (4.17)
Using (3.31) we see that w ±m satisfy the classical w ∞ algebra
{w ±m (t, θ), w ±n (t, θ ′ )} = ∓ 4π k (mw ±m+n−1 (t, θ) + nw ±m+n−1 (t, θ ′ )) ∂ ∂θ δ(θ − θ ′ ). (4.18)
In order to specify the boundary conditions in presence of higher spin excitations we again integrate the single particle Hamiltonian h(θ, p) over phase space with folds. The boundary Hamiltonian, therefore, is given by The equations of motion satisfied by p ± and different higher spin fields are given bẏ
H = k 4π dθdpϖ(θ, p)h(θ, p) = k 4π dθ p 3 + 6 − p 3 − 6 + (p + w +1 − p − w −1 ) + (w +2 − w −2 ) + W (θ) (p + − p − ) .p ± + p ± p ′ ± + ∞ s=3
u ±s u ′ ±s + W ′ = 0 u ±s + p ′ ± u ±s + p ± u ′ ±s + 2u ±s u ′ ±s = 0.
(4.21)
In terms of of collective excitations w ±n these equations can be written as, p ± + p ± p ′ ± + w ′ ±1 + W ′ = 0 andẇ ±n + (n + 1)p ′ ± w ±n + p ± w ′ ±n + 2w ′ ±(n+1) = 0. (4.22) Considering p ± and w ±n to be independent fields, these equations can also be obtained using the Poisson brackets of w ±n (4.18) for the Hamiltonian (4.19). One can check that the eqns. (3.22) and (3.23) are satisfied for the choice : η ± ∼ ∂H ∂p ± and η s± ∼ ∂H ∂u s± . Therefore the Hamiltonian (4.19) provides consistent boundary conditions for higher spin gravity. In presence of arbitrary higher spins the above sets of equations (4.22), in general, neither admit any integrable structure nor a free fermi description.
Integrable structure
It turns out that if we turn on a single higher spin (spin s = 3) i.e.
H 2n−1 = k 4π dθdph n (θ, p) = H + 2n−1 + H − 2n−1 , n ≥ 1. (4.24)
The integrals are determined up to the constants c n m , defined in (2.15) and (2.16). It turns out that one we can fix this constants (for a given choice of c 2 1 and c 2 2 in eqn. (4.20)) by taking η ± ∼ ∂H ± 2n−1 ∂p ± and η ±s ∼ ∂H ± 2n−1 ∂u ±s such that eqns. (3.22) and (3.23) are satisfied. We fix the coefficients c 2n 1 , · · · , c 2n 2n to find the conserved charge H ± 2n−1 . Thus H ± 2n−1 generate two infinite sequences of conserved charges of the higher spin (spin 3) gravity in AdS 3
H ± H ± 3 = ± k 4π dθ p 4 ± 4 + 3p 2 ± w ±1 + 6p ± w ±2 + 5w ±3 + 6W p 3 ± 6 + p ± w ±1 + w ±2 +3W 2 p 2 ± 2 + w +1 + W 3 p ± , (4.32) H ± 4 = ± k 4π dθ p 5 ± 5 + 4p 3 ± w ±1 + 12p 2 ± w ±2 + 20p ± w ±3 + 12w ±4 + 12W p 4 ± 12 + p 2 ± w ±1 + 2p ± w ±2 + 5 3 w ±3 + 12W 2 p 3 ± 6 + p ± w ±1 + w +2 + 4W 3 p 2 ± 2 + w ±1 + W 4 p ± .
(4.33)
One interesting point to note here is that for constant (or zero) potential both the sectors merge together and provide a larger class of conserved charges.
Free fermi description
There are two possible cases. First we consider that only a single higher spin (i.e. spin 3)
is turned on such that the integrable structure is preserved. In the second case we give up the integrable structure and turn on all possible higher spins. In both the cases it is possible to give a free fermi description of the AdS 3 solution.
In presence of single higher spin the equations of motion are given by (4.21). Suppose the dynamics is governed by folded fermi surfaces with boundaries f ±i . The fold boundaries f ±i satisfyḟ
±i + f ±i f ′ ±i + W ′ (θ) = 0 (4.34)
for non-relativistic fermions. These equations are obtained from the single particle Hamiltonian (2.10).
The phase space Hamiltonian in presence of folds is given by
H = k 4π dθφ(θ, p)h(θ, p) = k 4π i 1 6 f 3 +i − f 3 −i + W ′ (θ) (f +i − f −i ) . (4.35)
Equating this with the collective field theory Hamiltonian we get,
i (f +i − f −i ) = p + − p − i 1 6 f 3 +i − f 3 −i = 1 6 p 3 + − p 3 − + u 2 +s 2 p + − u 2 −s 2 p − + 1 3 u 3 +s − u 3 −s .
(4.36)
However, in order to make the fold equations (4.34) consistent with the field equations (4.21), fold variables f ±i satisfy some further constraints given by eqn. (2.21) with the same c n k , obtained for the calculations of conserved charges. This is little surprising that the c n k were determined by choosing H n such that the gauge transformation parameters satisfy the corresponding equations. Such conditions have a priori no connection with the consistency between the fold equations and the higher spin field equations. The dynamics of higher spin gravity is captured by the evolution of folded fermi surfaces. Therefore different geometries or shapes of droplets (with folds) correspond to different solutions of higher spin gravity. However, for a given higher spin solution the droplets/folds are highly constrained because of (2.21).
If we give up the integrable structure and turn on all the higher spins, then it is possible to parametrise eqns (2.21) in a different way such that the fold equations (4.34) and equations for w ±n are consistent. However, this imposes an infinite sequence of restrictions on w ±n depending on the choice of parametrisation.
Summary and Discussion
Summary : We introduce new sets of boundary conditions (following the work on generalised soft hairy boundary conditions [11,14]) for higher spin gravity in AdS 3 where the boundary dynamics of spin two and other higher spin fields are governed by the interacting collective field theory Hamiltonian [29]. However there is a difference between this Hamiltonian and the Hamiltonian introduced by Avan and Jevicki. In our case the collective field theory Hamiltonian is defined over a cylinder (i.e. the spatial direction is a circle) and hence can be obtained from a unitary matrix quantum mechanics in presence of an arbitrary potential W (θ). The interaction of this free Hamiltonian with the w ∞ excitations was introduced in [29]. We consider this interacting Hamiltonian to impose the boundary conditions on spin 2 and other higher spin fields. We show that the classical evolution of metric and other higher spin fields are governed by the dynamics collective fields and other supplementary fields. Since the dynamics of collective and other supplementary fields has a geometric interpretation in terms of evolution of free fermi droplets in presence of folds [38,46], our construction therefore provides a phase space description of higher spin gravity in AdS 3 in the spirit of LLM [43]. We also show that for spin 3 gravity one can construct an infinite number of gauge transformations (for a given Hamiltonian) that preserve the asymptotic structure of the bulk spacetime and hence render infinite sequences of conserved charges whose mutual Poisson brackets are zero and thus exhibit an integrable structure.
Discussion : From the relation between p ± , w ±n and folds f ±i (eqns. (2.20, 2.21)) we see that if folds are developed on the upper (for example) fermi surface then all w +n s are non-zero in general and are given in terms of f ±i and vice versa.
For the static solutions of the field equations (3.12) we have ξ ± = constant and ζ ±s = constant ∀ s. We consider spin 3 black holes whose dynamics is governed by the Hamiltonian (4.19).
For such black holes these conditions are given by From the second condition (5.4) we see that one trivial solution is u ±3 = 0, i.e. no higher spin modes are turned on. The non-trivial solution is u ±3 = −p ± . To find the droplet geometry one has to solve the equations in (4.36) to find possible values of f ±i . The entropy of the black hole, which is connected to the BTZ black hole, is proportional to 2π(p + − p − ) [2,3,10,25]. Therefore, from eqn. (4.36) we see that the entropy of such black holes is equal to the total area covered by the droplets. However for other black hole solutions (not connected to BTZ) the entropy is no longer equal to the area of the droplets. This is worth mentioning that any arbitrary droplet geometry may not satisfy the regularity conditions at the horizon.
p 2 ± 2 + w +1 + V = 2π,(5.
Following the Hamiltonian reduction method developed in [14,17,25] In general this is a difficult problem to address [49]. However, we can try to compute the partition function for a given classical solution. In absence of higher spins, one can consider the classical solution to be a BTZ black hole which is given by a constant droplet as discussed in sec. 4.1. Thus excitations about this classical solution correspond to different deformations of the droplet and one has to sum over all such deformations.
Expanding p ± in Fourier modes we can classify all possible deformations as quantum states in the Hilbert space. The problem was discussed in [37]. It turns out that the partition function is equal to 2D Yang-Mills partition function on torus. It would be interesting to find the partition function in presence of different higher spin fields in the bulk. In classical theory one needs an infinite number of fields (w ±n ) to describe folds on the fermi surface. However, the situation is different in quantum theory. It turns out that w ±n are not additional degrees of freedom in quantum theory; rather they represent O(1)
quantum dispersions of the collective fields [39].
is a boundary term added to the bulk action to make δI CS (A ± ) = 0. The gauge fields A ± are matrix valued and the trace in (3.3) acts on the generators of the algebra sl(M, R) in the fundamental representation.
can check that if η ± = − 4π k ∂H ± ∂p ± and η ±s = − 4π k ∂H ± ∂u ±s then eqns.(3.22) and(3.23) are satisfied and henceQ ± (η ± , η ±s ) = ± dθH ± (3.25)are conserved charges, which are the Hamiltonians for the two chiral sectors. Moreover, one can also find some other H ± n = dθH ± n such that .(3.22) and (3.23) for a given choice of Hamiltonians H ± . In that case Q ± n (η ± , η ±s ) = ± dθ H ± n (3.27)
Figure 2 :
2Droplet for BTZ black hole.
is possible to construct an infinite sequence of conserved charges for each chiral sector. The conserved charges are obtained by integrating h n over phase space in presence of folds.
to maintain the regularity of the Euclidean black hole solution it was shown in [2,3,10,25] that the chemical potentials ξ ± and ζ ±s depend linearly on M − 1 arbitrary integers when M spin fields are turned one. Black holes which are connected to BTZ black holes the above conditions are given by, ξ ± = 2π, ζ ±s = 0. (5.2)
one can show that the total action is governed by left and right moving chiral bosons for a specific choice of boundary term in (3.16). The CS action (3.3) receives contributions only from the boundary degrees of freedom and is given by The Euclidean AdS 3 partition function for specific boundary conditions, given by the choice of boundary action, can be written as Z =I(ϕ ± , ψ ±s ) = dt
k
4π
dθ ϕ ′
±φ ± +
s
ψ ′
±sψ ±s
∓ H ± .
(5.5)
classical solutions
[DΦ]e −βH .
(5.6)
The classical integrability was generalised to quantum integrability in[42].
See[45], for an example.
Our boundary conditions are related to the standard boundary conditions[4] by a gauge transformation on-shell[7].
Some of the c n+1 m that remain unfixed after satisfying (3.22) and (3.23) have been fixed by demanding {H n , H m } = 0.
with the boundary conditions specified through the Hamiltonian given in(4.19). The first few conserved charges (for n = 2, n = 3 and n = 4) are given by (for w ±n given by eqn.(4.23)) We can also find a different set of conserved charges if we consider that the boundary dynamics of gravity and higher spin fields is governed by a system of relativistic fermions with single particle Hamiltonian h = p + V (θ). The total Hamiltonian is therefore given(4.28)Here again we have chosen the constant c 1 1 = 1. The equation of motion for this Hamiltonian is given byṗFor the relativistic Hamiltonian the infinite set of conserved charges are defined bỹAs before we fix the set of coefficients c n+1 1 , · · · c n+1 n+1 by demanding that eqns. (3.22) and (3.23) are satisfied for η ± ∼ ∂H ± n ∂p ± and η ±s ∼ ∂H ± n ∂u ±s . The first few such charges are given by 6
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Quantum Gravity Partition Functions in Three Dimensions. A Maloney, E Witten, JHEP. 02155A. Maloney and E. Witten, "Quantum Gravity Partition Functions in Three Dimen- sions," JHEP, vol. 02, p. 029, 2010, 0712.0155.
| {'fraction_non_alphanumeric': 0.08247784991583892, 'fraction_numerical': 0.046279340769102714, 'mean_word_length': 3.6390938733482066, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 2, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 32, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In this paper, we introduce new sets of boundary conditions for higher spin gravity in AdS 3 where the boundary dynamics of spin two and other higher spin fields are governed by the interacting collective field theory Hamiltonian of Avan and Jevicki. We show that the time evolution of spin two and higher spin fields can be captured by the classical dynamics of folded fermi surfaces in the similar spirit of Lin, Lunin and Maldacena. We also construct infinite sequences of conserved charges showing the integrable structure of higher spin gravity (for spin 3) under the boundary conditions we considered. Further, we observe that there are two possible sequences of conserved charges depending on whether the underlying boundary fermions are non-relativistic or relativistic.', 'arxivid': '2302.08471', 'author': ['Suvankar Dutta [email protected] \nIndian Institute of Science Education and Research Bhopal Bhopal bypass\n462066BhopalIndia\n', 'Debangshu Mukherjee [email protected] \nDepartment of Physics\nIndian Institute of Technology Kanpur Kalyanpur\n208016KanpurIndia\n', 'Sanhita Parihar [email protected] \nIndian Institute of Science Education and Research Bhopal Bhopal bypass\n462066BhopalIndia\n'], 'authoraffiliation': ['Indian Institute of Science Education and Research Bhopal Bhopal bypass\n462066BhopalIndia', 'Department of Physics\nIndian Institute of Technology Kanpur Kalyanpur\n208016KanpurIndia', 'Indian Institute of Science Education and Research Bhopal Bhopal bypass\n462066BhopalIndia'], 'corpusid': 256901003, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 19013, 'n_tokens_neox': 16267, 'n_words': 9712, 'pdfsha': '6a13c2559ac2cd697234dc4985ccd0f0d2225c73', 'pdfurls': ['https://export.arxiv.org/pdf/2302.08471v3.pdf'], 'title': ['Higher Spin Gravity in AdS 3 and Folds on Fermi Surface', 'Higher Spin Gravity in AdS 3 and Folds on Fermi Surface'], 'venue': []} |
arxiv |
17 Apr 2013 (April 17, 2013) 17 April 2013.
W-Y Pauchy Hwang [email protected]
Center for Cosmology and Particle Astrophysics
Institute of Astrophysics
Center for Theoretical Sciences
Department of Physics
Asia Pacific Organization
National Taiwan University
106TaipeiTaiwan
17 Apr 2013 (April 17, 2013) 17 April 2013.An Extended Standard Model 1 Correspondence Author;
Following the thinking underlying the minimal Standard Model, we propose an extended Standard Model, based on the gauge groupThe extension, which is rather unique, derives from the family concept that there are three generations of quarks and of leptons. It yields neutrino oscillations in a natural manner. It also predicts a variety of lepton-flavor-violating rare decays. PACS Indices: 12.60.-i (Models beyond the standard model); 98.80.Bp (Origin and formation of the Universe); 12.10.-g (Unified field theories and models).
Nowadays it is firmly established that there are three generations of quarks and of leptons, at the level of the so-called "point-like Dirac particles". According to another newly established belief in Cosmology, the content of the current Universe would be 25 % in the dark matter while only 5% in the ordinary matter, the latter described by the "minimal Standard Model" (mSM). The dark-matter particles are supposed to be described by an extended Standard Model. Thus, there is certain urgent need for an extended Standard Model, which accommodates those phenomena which are "beyond the Standard Model".
In the attempt to write down another extended Standard Model, we would try to write it as a whole -strong interactions, electroweak interactions, and others as a whole, and not separately. In that way, the "basic units", which are based on the right-handed Dirac components or left-handed Dirac components, would be more appropriate than just "the building blocks of matter". Each basic unit derives from one kinetic-energy term, −Ψ R γ µ ∂ µ Ψ R or −Ψ L γ µ ∂ µ Ψ L , and from only one such term. The gauge principle tells us that ∂ µ is to be replaced by some "gauge-invariant" derivative D µ . This would be a way to write down a theory in the globally consistent manner.
We may split the ordinary-matter world into the "quark" world and the "lepton" world. As we know so far, the dark-matter world does not involve directly strong and electromagnetic interactions. We could decide which multiplet of the dark-matter group SU f (3) each basic unit should belong -for instance, we may rule out the direct coupling between the quark world and the dark-matter world, altogether.
How about the couplings between the family group SU f (3) and the lepton world? We know that the right-handed neutrinos do not appear in the minimal Standard Modelso, (ν τ R , ν µR , ν eR ) would make a perfect triplet under SU f (3). In a recent note [1], we proposed to put ((ν τ , τ ) L , (ν µ , µ) L , (ν e , e) L ) (columns) (≡ Ψ L (3, 2)) as the SU f (3) triplet and SU L (2) doublet. The next question is how to assign τ R , µ R , and e R under SU f (3). We could write down the mass term for the charged leptons -if they are singlets or (τ R , µ R , e R ) is an SU f (3) triplet? The singlet assignment can be ruled out since there are undesirable crossed terms in, e.g., Ψ L (3, 2)e R Φ (3,2). So, three of them would form a triplet -Ψ C R (3, 1). In fact, this would imply that three charged leptons would have equal masses, to the zeroth order. Or, we go back to the ground-zero: the SU f (3) is completely decoupled. Nevertheless, it might not be so bad to have the beginning point that we have to look for why the three charged leptons have different masses. This would complete the list of "the basic units".
So, there are only three members in the lepton world -two righted-handed triplets and one "composite" left-handed triplet under SU f (3). It is rather simple if we wish to play with SU f (3).
Since we wish to propose the SU f (3) family gauge theory as a way to understand why there are three generations, it requires all additional particles, i.e., (eight) gauge bosons and (four) residual family Higgs, very massive. In the proposal of Hwang and Yan [1], we would have one Standard-Model Higgs field Φ(1, 2), one complex family Higgs triplet Φ(3, 1), and another family triplet-doublet complex scalar fields Φ (3,2). Amazingly enough [2], two neutral complex triplets, Φ(3, 1) and Φ 0 (3, 2) would indeed undergo the desired Higgs mechanism and the three charged scalar fields would remain massive, i.e. there is no spontaneous symmetry breaking (SSB) in the charged Higgs sector.
So far, we have decided on the basic units -those left-handed and right-handed quarks and leptons; the gauge group is chosen to be SU c (3)×SU L (2)×U (1)×SU f (3). For notations, we use Wu and Hwang [3].
In the quark world, we have, for the up-type right-handed quarks u R , c R , and t R ,
D µ = ∂ µ − ig c λ a 2 G a µ − i 2 3 g ′ B µ ,(1)
and, for the rotated down-type right-handed quarks d ′ R , s ′ R , and b ′ R ,
D µ = ∂ µ − ig c λ a 2 G a µ − i(− 1 3 )g ′ B µ .(2)
On the other hand, we have, for the SU L (2) quark doublets,
D µ = ∂ µ − ig c λ a 2 G a µ − ig τ 2 · A µ − i 1 6 g ′ B µ .(3)
In the lepton world, we introduce the family triplet,
(ν R τ , ν R µ , , ν R e ) (column) (≡ Ψ R (3, 1)), under SU f (3)
. Since the minimal Standard Model does not see the right-handed neutrinos, it would be a natural way to make an extension of the minimal Standard Model. Or, we have, for the triplet (ν R τ , ν R µ , ν R e ),
D µ = ∂ µ − iκλ a 2 F a µ .(4)
and, for the left-handed SU f (3)-triplet and SU L (2)-doublet ((ν L τ , τ L ), (ν L µ , µ L ), (ν L e , e L )) (all columns) (≡ Ψ L (3, 2)),
D µ = ∂ µ − iκλ a 2 F a µ − ig τ 2 · A µ + i 1 2 g ′ B µ .(5)
As discussed earlier, it follows as the only option that three of the right-handed charged leptons have to form an SU f (3) triplet Ψ C R (3, 1). The generation of the various quark masses is through the Standard-Model way. For the charged leptons, we have to chooseΨ L (3, 2)Ψ C R (3, 1)Φ(1, 2). Note that, in the U-gauge, the SM Higgs looks like (0, 1 √ 2 (v + η(x))), only one degree of freedom [3]. Thus, the neutral neutrino triplet Ψ R (3, 1) cannot be used in the above coupling, because of the charge conservation.
For neutrinos in the lepton world, the neutrino mass term becomes [1,2]
i η 2Ψ L (3, 2) × Ψ R (3, 1) · Φ(3, 2) + h.c..(6)
The cross (curl) product is somewhat new [4], referring to the singlet combination of three triplets in SU (3) -suitable for SU (3); not an ordinary matrix operation. In this note, we are careful enough to distinguish the anti-triplet from the triplet and to realize that there are a lot of "conjugates", such as "complex conjugate", "Dirac adjoint", and "anti-triplet" (though sometime the same).
The last expression serves as the operator describing neutrino oscillations in general. For point-like Dirac particles, this way of describing oscillations is rather natural and unique.
To summarize on the origin of masses, the quark masses are determined by the vacuum expectation values (VEV's) of the SSB of the Standard-Model Higgs Φ(1, 2) and of the adjoint, the charged lepton mass is determined cooperatively by Φ (1, 2), while the neutrino masses are determined by the purely and mixed family Higgs Φ(3, 1) and Φ 0 (3, 2) (neutral). In the leading order, the equality of the three charged leptons remains to be explainedpresumably by higher-order loop diagrams.
To be more precise, our gauge group is (
SU c (3) × SU L (2) × U (1)) × SU f (3), rather than SU c (3) × SU L (2) × U (1) × SU f (3)
. That is why it is so difficult to find.
As pointed out in an early paper [2], we may imagine that, in the U-gauge, the standardmodel Higgs Φ(1, 2) looks like (0, (v + η(x))/ √ 2) (column) and Φ † (3, 2)Φ(1, 2) would pick out the neutral sector naturally. In fact, the term (Φ † (3, 2)Φ(1, 2))(Φ † (1, 2)Φ(3, 2)) with a suitable sign, would modify a massive Φ(3, 2) field such that the neutral sector has SSB while the charged sector remains massive. This "project-out" Higgs mechanism is what we need.
We may [2] write down the terms for potentials among the three Higgs fields, subject to (1) that they are renormalizable, and (2) that symmetries are only broken spontaneously (via the Higgs or induced Higgs mechanism). We write [2]
V = V SM + V 1 + V 2 + V 3 ,(7)V 1 = M 2 2 Φ † (3, 2)Φ(3, 2) + λ 1 4 (Φ † (3, 2)Φ(3, 2)) 2 +ǫ 1 (Φ † (3, 2)Φ(3, 2))(Φ † (1, 2)Φ(1, 2)) + η 1 (Φ † (3, 2)Φ(1, 2))(Φ † (1, 2)Φ(3, 2)) +ǫ 2 (Φ † (3, 2)Φ(3, 2))(Φ † (3, 1)Φ(3, 1)) + η 2 (Φ † (3, 2)Φ(3, 1))(Φ † (3, 1)Φ(3, 2)) +(δ 1 iΦ † (3, 2) × Φ(3, 2) · Φ † (3, 1) + h.c.),(8)V 2 = µ 2 2 2 Φ † (3, 1)Φ(3, 1) + λ 2 4 (Φ † (3, 1)Φ(3, 1)) 2 + (δ 2 iΦ † (3, 1) · Φ(3, 1) × Φ(3, 1) + h.c.) +λ ′ 2 Φ † (3, 1)Φ(3, 1)Φ † (1, 2)Φ(1, 2),(9)V 3 = (δ 3 iΦ † (3, 2) · Φ(3, 2) × (Φ † (1, 2)Φ(3, 2)) + h.c.) +(δ 4 i(Φ † (3, 2)Φ(1, 2)) · Φ † (3, 1) × Φ(3, 1) + h.c.) +η 3 (Φ † (3, 2)Φ(1, 2)Φ(3, 1) + c.c.).(10)
We maintain that every terms are naively renormalizable since they are of power four or less. Note that the terms in δ i i involve the so-called "SU (3) operations", as before.
In the simplest case, we could assume
M 2 > 0, λ 1 > 0, η 1 = 0, η 2 = 0, µ 2 2 < 0, λ 2 > 0, all other couplings vanish.(11)
In the present case, we require M 2 + η 1 v 2 /2 < 0 to ensure that SSB takes place in the neutral sector of Φ (3,2). The cross term in η 2 might be optional (i.e. with other options) since we follow the early paper [5] on the colored Higgs mechanism to absorb the four DOF's from one complex triplet. So, we see that, at the Lagrangian level, the SU c (3) × SU L (2) × U (1) × SU f (3) gauge symmetry is protected but such symmetry is violated via spontaneous symmetry breaking (via the SM Higgs mechanism and the project-out Higgs mechanism).
The scenario for the masses of the family particles might be as follows: For those eight familons (or family gauge bosons), we could assume a few T eV or slightly more. For those four family Higgs (those participating Higgs mechanisms), maybe even slightly more heavier. We don't have definitive expectations for the charged scalar particles, except that they could be much heavier.
We note that the terms allowed by the renormalizability, in V 1 , V 2 , and V 3 given above, are rather rich. This is the nature of the scalar fields and their vacuum expectation values.
As shown earlier [4,5], two triplets of complex scalar fields would make the eight family gauge bosons and four residual family Higgs particles all massive, presumably heavier that a few T eV . In the SU f (3) family gauge theory treated alone, there are many ways of accomplishing such goal. In the present complicated case, the equivalence between two triplets is lost, but presumably in a minor way.
Neutrinos have tiny masses far smaller than the masses of the quarks or of charged leptons. Neutrinos oscillate among themselves, giving rise to the lepton-flavor violation (LFV). There are other oscillation stories, such as the oscillation in the K 0 −K 0 system, but there is a fundamental "intrinsic" difference here -the K 0 −K 0 system is composite while neutrinos are regarded as "point-like" Dirac particles. We could draw Feymann diagrams for the oscillating K 0 −K 0 system; but none so far on neutrino oscillations. It is fair to say that neutrino masses and neutrino oscillations may be regarded as one of the most important experimental facts over the last thirty years [6].
In fact, certain LFV processes such as µ → e+γ [6] and µ+A → A * +e are closely related to the most cited picture of neutrino oscillations so far [6]. In recent publications [7], it was pointed out that the cross-generation or off-diagonal neutrino-Higgs interaction may serve as the detailed mechanism of neutrino oscillations, with some vacuum expectation value(s) of the new Higgs fields, Φ(3, 1) and Φ 0 (3,2). So, even though we haven't seen, directly, the family gauge bosons and family Higgs particles, we already see the manifestations of their vacuum expectation values.
In the other words, Eq. (6), the SU (3)-generalized curl product, can be used as the basis to analyze the various lepton-flavor-violating decays and reactions, in addition to its implications on the tiny neutrino masses and on neutrino oscillations.
To close this note, We would like to speculate what the dark-matter world look like, if the extended Standard Model discussed is true.
In a slightly different context [8], it was proposed that we could work with two working rules: "Dirac similarity principle", based on eighty years of experience, and "minimum Higgs hypothesis", from the last forty years of experience. Using these two working rules, the extended model mentioned above becomes rather unique -so, it is so much easier to check it against the experiments. These two working rules merely assert that our world is rather simple.
We would be curious about how the dark-matter world looks like, though it is difficult to verify experimentally. The first question would be: The dark-matter world, 25 % of the current Universe (in comparison, only 5 % in the ordinary matter), would clusterize to form the dark-matter galaxies, maybe even before the ordinary-matter galaxies. The dark-matter galaxies would then play the hosts of (visible) ordinary-matter galaxies, like our own galaxy, the Milky Way. Note that a dark-matter galaxy is by our definition a galaxy that does not possess any ordinary strong and electromagnetic interactions (with our visible ordinary-matter world). This fundamental question deserves some thoughts, for the structural formation of our Universe.
Of course, we should remind ourselves that, in our ordinary-matter world, those quarks can aggregate in no time, to hadrons, including nuclei, and the electrons serve to neutralize the charges also in no time. Then atoms, molecules, complex molecules, and so on. These serve as the seeds for the clusters, and then stars, and then galaxies, maybe in a time span of 1 Gyr (i.e., the age of our young Universe). The aggregation caused by strong and electromagnetic forces is fast enough to help giving rise to galaxies in a time span of 1 Gyr. On the other hand, the seeded clusterings might proceed with abundance of extra-heavy dark-matter particles such as familons and family Higgs, all greater than a few T eV and with relatively long lifetimes (owing to very limited decay channels). So, further simulations on galactic formation and evolution may yield clues on our problem. This work is supported in part by the National Science Council (NSC99-2112-M-002-009-MY3).
. W-Y. Pauchy Hwang, Tung-Mow Yan, arXiv:1212.4944hep-phW-Y. Pauchy Hwang and Tung-Mow Yan, arXiv:1212.4944 [hep-ph] 20 Dec 2012.
. W-Y. Pauchy Hwang, arXiv:1301.6464v3hep-phW-Y. Pauchy Hwang, arXiv:1301.6464v3 [hep-ph] 10 April 2013.
Ta-You Wu, W-Y. Pauchy Hwang, Relatistic Quantum Mechanics and Quantum Fields. World ScientificTa-You Wu and W-Y. Pauchy Hwang, "Relatistic Quantum Mechanics and Quantum Fields" (World Scientific 1991).
. W-Y. Pauchy Hwang, 978-0-7354-0687-2/09Intern. J. Mod. Phys. Conf. Series. 844American Institute of PhysicsInternational J. Mod. Phys.. ibid.W-Y. Pauchy Hwang, Nucl. Phys. A844, 40c (2010); ibid., International J. Mod. Phys. A24, 3366 (2009); ibid., Intern. J. Mod. Phys. Conf. Series 1, 5 (2011); ibid., American Institute of Physics 978-0-7354-0687-2/09, pp. 25-30 (2009).
. W-Y P Hwang, Phys. Rev. 32824W-Y. P. Hwang, Phys. Rev. D32, 824 (1985).
Review of Particle Physics. J. Phys. G: Nucl. Part. Phys. 371Particle Data Groupand its biennual publicationsParticle Data Group, "Review of Particle Physics", J. Phys. G: Nucl. Part. Phys. 37, 1 (2010); and its biennual publications.
. W-Y. Pauchy Hwang, arXiv:1207.6443v1arXiv:1209.5488v1Hypefine Interactions. 215105hep-ph. ibid.. ibid.. hep-phW-Y. Pauchy Hwang, arXiv:1207.6443v1 [hep-ph] 27 Jul 2012; ibid., Hypefine Interac- tions 215, 105 (2013); ibid., arXiv:1209.5488v1 [hep-ph] 25 Sep 2012.
W-Y P Hwang, arXiv:11070156v1Plenary talk given at the 10th International Conference on Low Energy Antiproton Physics. Vancouver, Canadahep-phW-Y. P. Hwang, arXiv:11070156v1 (hep-ph, 1 Jul 2011), Plenary talk given at the 10th International Conference on Low Energy Antiproton Physics (Vancouver, Canada, April 27 -May 1, 2011).
| {'fraction_non_alphanumeric': 0.080907926047959, 'fraction_numerical': 0.037830251998291535, 'mean_word_length': 3.85773562537048, 'pattern_counts': {'":': 0, '<': 2, '<?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': 12, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Following the thinking underlying the minimal Standard Model, we propose an extended Standard Model, based on the gauge groupThe extension, which is rather unique, derives from the family concept that there are three generations of quarks and of leptons. It yields neutrino oscillations in a natural manner. It also predicts a variety of lepton-flavor-violating rare decays. PACS Indices: 12.60.-i (Models beyond the standard model); 98.80.Bp (Origin and formation of the Universe); 12.10.-g (Unified field theories and models).', 'arxivid': '1304.4705', 'author': ['W-Y Pauchy Hwang [email protected] \nCenter for Cosmology and Particle Astrophysics\nInstitute of Astrophysics\nCenter for Theoretical Sciences\nDepartment of Physics\nAsia Pacific Organization\nNational Taiwan University\n106TaipeiTaiwan\n', 'W-Y Pauchy Hwang [email protected] \nCenter for Cosmology and Particle Astrophysics\nInstitute of Astrophysics\nCenter for Theoretical Sciences\nDepartment of Physics\nAsia Pacific Organization\nNational Taiwan University\n106TaipeiTaiwan\n'], 'authoraffiliation': ['Center for Cosmology and Particle Astrophysics\nInstitute of Astrophysics\nCenter for Theoretical Sciences\nDepartment of Physics\nAsia Pacific Organization\nNational Taiwan University\n106TaipeiTaiwan', 'Center for Cosmology and Particle Astrophysics\nInstitute of Astrophysics\nCenter for Theoretical Sciences\nDepartment of Physics\nAsia Pacific Organization\nNational Taiwan University\n106TaipeiTaiwan'], 'corpusid': 118530884, 'doi': '10.1036/1097-8542.650950', 'github_urls': [], 'n_tokens_mistral': 5469, 'n_tokens_neox': 4725, 'n_words': 2913, 'pdfsha': '1c5b17bc8141f297b3a2378f7b1260529db5bd62', 'pdfurls': ['https://arxiv.org/pdf/1304.4705v3.pdf'], 'title': [], 'venue': []} |
arxiv |
An engineered planar plasmonic reflector for polaritonic mode confinement
Shima Rajabali [email protected]
Josefine Enkner
Erika Cortese
Mattias Beck
Simone De Liberato
Jérôme Faist
Giacomo Scalari [email protected]
of Quantum Electronics
‡School of Physics and Astronomy
ETH Zürich
8093ZürichSwitzerland
University of Southampton
SO17 1BJSouthamptonUnited Kingdom
An engineered planar plasmonic reflector for polaritonic mode confinement
It was recently demonstrated that, in deep subwavelength gap resonators coupled to two-dimensional electron gases, coupling to propagating plasmons can lead to energy leakage and prevent the formation of polaritonic resonances. This process, akin to Landau damping, limits the achievable field confinement and thus the value of light-matter coupling strength. In this work, we show how plasmonic subwavelength reflectors can be used to create an artificial energy stopband in the plasmon dispersion, confining them and enabling the recovery of the polaritonic resonances. Using this approach we demonstrate a normalized light-matter coupling ratio of Ω ω = 0.35 employing a single quantum well with a gap size of λ/2400 in vacuum.
Introduction
The terahertz (THz) range is especially suited for the study of ultrastrong light-matter coupling phenomena 1,2 since it combines material systems with large optical dipoles with extremely subwavelength resonant cavities obtained by exploiting metals in a lumped-circuit approach. 3,4 The Landau polariton platform 5,6 has proven especially successful in obtaining high values of the normalized coupling constant Ω ω > 1, 7 extremely high cooperativity, 8,9 as well as providing a testbed for investigating the influence of enhanced vacuum fields on the DC magnetotransport both in the linear 10 and the integer Quantum Hall regimes. 11 Several demonstrations of few-electron systems have been also realized in Landau polariton platforms. 12,13 Reducing the modal volume of the electromagnetic resonator allows both to increase the strength of the light-matter coupling and to reduce the number of involved electrons, 14 thus approaching the non-linear polaritonic regime. In standard nanophotonic platforms, both metallic 15 and dielectric, 16 this strategy is known to break down for resonator features small enough to excite propagative electronic excitation. In this case the standard local description of light-matter coupling fails and more accurate non-local approaches have to be used. 17,18 A recent work 19 demonstrated the existence of a related effect in Landau polaritons due to the non-local excitation of propagative plasmons, limiting the possibility of arbitrarily increasing electromagnetic confinement by reducing the cavity modal volume. In the aforementioned Landau-polariton paper, the magnetoplasmons which are collective inter-Landau level excitations are (ultra)strongly coupled to the near-field of metamaterial complementary split-ring resonators (cSRR). The authors observed the progressive broadening and amplitude reduction of the upper polariton (UP) mode and partial disappearance of the lower polariton (LP) branch by reducing the gap of the coupled cSRRs below a critical length.
According to our theoretical analysis, strongly subwavelength fields can excite a continuum of high-momenta propagative magnetoplasmons. These propagating modes act as loss channels and reduce the field confinement ultimately limiting the achievable field enhancement.
As a result, certain polaritonic modes can broaden/disappear and the system enters a new regime of discrete-to-continuum strong coupling. 20,21 In this work, we show how, by defining an engineered plasmonic mirror around the gap of the resonator we can again confine these propagating waves and retrieve well-defined polariton branches.
Results and Discussion
The reflector design In order to confine the broadened polaritonic mode whose energy is leaking out as a result of polaritonic non-local effects, we can introduce a plasmonic bandgap structure: a planar reflector in the magnetoplasmon propagation path, similar to plasmonic Bragg reflectors. 22,23 To build such a planar reflector for the magnetoplasmon waves, we need to introduce a modulation of the complex refractive index, i.e., of the dielectric function. One simple method is to introduce a modulation in the carrier density of the two-dimensional electron gas (2DEG) by structuring the 90 nm-thick gallium arsenide (GaAs) cap layer on top of the quantum well channel, as illustrated in figure 1.
Landau polaritons in a plasmonic reflector
After a set of finite element simulation to find the right dimensions for reconfining the UP mode, a sample with a trench depth of h = 20 nm, was fabricated (more details can be found in Methods section). Figure 1b of an external out-of-plane magnetic field swept between 0 and 4 T. In the THz-TDS setup, a pair of off-axis parabolic mirrors first are used to collimate and focus the incident THz beam from a photoconductive switch 24 on the sample. Then, through another pair of offaxis parabolic mirrors, the transmitted signal from the sample is collected, collimated, and focused on a zinc telluride (ZnTe) crystal. Ultimately, the THz signal is detected using an electro-optic detection scheme. 25 Detailed information about the THz-TDS setup is available in reference. 26 The transmission measurement without and with the reflector in figure 2a shows the confinement of the UP and also the appearance of the LP mode at finite values of the magnetic respectively. The y-z plane cuts (figure 3c and 3f) clearly displays that the high-quality factor mode does not couple to the resonator as there is no field confinement below the gap in the GaAs substrate below the 2DEG. On the contrary, the field distribution of the UP mode shows the field confinement below the resonator gap as a result of the coupling of the resonator to this mode. This high-quality factor mode which only appears in the simulation seems to have a wave vector comparable to the free electron wave vector. Hence, it easily dissipates and cannot appear in the measurement due to Landau damping. 27 Another key point in the simulated electric field distribution is the electric field confinement in y-direction. Even though the presence of the reflector structure around the resonator gap can reconfine the UP mode, the electric field cannot be confined inside the gap of the resonator (d = 250 nm) and it is only confined to the central defect width (w = 1.5 µm).
Thus, we can retrieve the ultrastrong coupling of our Landau polaritonic system but the highest achievable strength of the coupling will remain limited and the electromagnetic field
Conclusion
In this work, an engineered reflector design is proposed to retrieve the broadened polaritonic mode in systems with sub-micron size cavities coupled to inter-Landau level excitations in 2DEG. The initial broadening of the polaritonic mode originates from polaritonic nonlocality effects, discussed in our previous work. 19 The proposed design is based on periodic one-dimensional structures on both sides of the resonator gap to reflect the propagative magnetoplasmon waves and confine them in the gap of the resonator. After implementing such a structure around a coupled cSRR's gap with 250 nm width, in which polaritonic non-locality effect was observed, the polaritonic modes are retrieved. The recovered branches show a normalized coupling strength of Ω R /ω 0 = 35% which is slightly lower than the expected coupling strength without considering the nonlocal effects. This is because the reconfined electromagnetic field is confined to the central width (w = 1.5 µm) of the proposed structure and not to the resonator's gap (d = 250 nm). The measurements are also verified by our finite element simulation. Our result can be further improved by optimizing the reflector structure and can be also used for the direct resonators. This method can offer a solution to overcome the polaritonic nonlocal effects that introduce loss channels into the coupled systems and limit the enhancement of the coupling, however, the maximum achievable coupling strength saturates as the electromagnetic field confinement remains limited to the central width of the reflector structure.
Methods
To fabricate the plasmonic reflector, first, the trenches were defined using a direct laser
Figure 1 :
1The scheme of the plasmonic reflector design for reconfinement of the broadened UP mode (a) The 3D image, taken by atomic force microscopy, of the cSRR with a gap size of d = 250 nm on top a plasmonic reflector (the periodic trenches on the 2DEG). The parameters d, w 1 , w 2 , and w are 0.25, 1, 1, and 1.5 µm, respectively.(b) A scanning electron microscopy image of the same structure shown in panel (a).
This structure, made by etching shallow trenches on both sides of the resonator gap, can define a stop-band in the transmission spectrum of the wave propagating across the 2DEG plane.22,23 As a result, specifying such a periodic structure with a central defect below the sub-micron gap of the cSRR should allow retrieving the contrast in the amplitude and enhance the lifetime of the UP, effectively decoupling it from the lossy continuum of plasmonic modes. The three dimensional image of the fabricated resonator processed by atomic force microscopy is also exhibited infigure 1aindicating the important design parameters such as the resonator's gap (d), the central defect's width (w), and the width of the periodic structure of the reflector (w 1 and w 2 ).
shows a scanning electron microscopy image of the etched trenches in the 2DEG with a cSRR on top. The resonators were aligned and written on the etched substrate using electron-beam lithography.
Figure 2a
2ashows the transmission measurement for the cSRR with 250 nm gap size on a plasmonic reflector with 20 nm-deep trenches. The measurements are conducted in a THz time-domain spectroscopy (TDS) setup at cryogenic temperature T = 2.7 K as a function
Figure 2 :
2The cSRR with a gap size of 250 nm and the plasmonic reflector: experiment vs. simulation (a) The measured transmission of the coupled cSRR to inter-Landau level transitions in a 2DEG without and with the plasmonic reflector (with h = 20 nm etching depth). The one without the reflector illustrates the nonlocal effects (broadening of the modes) in its polaritonic modes while the one with the reflector has confined modes. The polariton branches from the measurement of the sample with reflector can be fitted by Hopfield model (black solid lines). The fitting indicates a normalized coupling of Ω R ω 0 = 35%. The red arrows mark the linear dispersions at multiple of the cyclotron frequency discussed in the main text.(b) Finite element simulation of the same resonator with the reflector in which the trenches are etched as deep as the entire cap layer thickness (h = 90 nm). A high-quality factor mode starting at 405 GHz at zero magnetic field appears only in the simulated colormap. The normalized coupling for the simulated spectra is Ω R ω 0 = 42%. (c) Sections of experimental data in panel (a) at three different values of the magnetic field, B = 0, 1 (anti-crossing), and 4 T with (solid) and without (dashed) the plasmonic reflector, confirm the retrieval of the broadened mode with the reflector. field for the coupled cSRR on the reflector structure. There is no sign of magnetoplasmon dispersion in the measurement as the fitted LP (solid black line) converges to zero energy at zero magnetic field. For a better comparison between the polaritonic modes of the measurement without and with the reflector, section of the colormaps at three different values of the magnetic field, B = 0, 1 (anti-crossing), and 4 T are displayed in figure 2c. Interestingly, the linear dispersions corresponding to optical transitions at multiples of the cyclotron frequency (indicated with red arrows in figure 2a) are appearing in the transmission spectra of the sample with the reflector structure at the same place in the spectra of the sample without reflector which was also reported in our previous work. 19 These additional absorption lines are possible signatures of the breaking of the dipole approximation which results in the relaxation of the optical selection rules between the Landau levels. We have also simulated the transmission spectrum of the coupled system with the reflector structure as a function of the magnetic field using the finite element method (figure 2b). Given the difficulty of precisely modeling the modulation of the 2DEG's carrier due to the etching of the cap layer, in the simulations the trenches are etched as deep as the entire cap layer thickness, h = 90 nm. The computed normalized coupling for the measured (figure 2a) and the simulated (figure 2b) spectrum of this coupled sample combined with the plasmonic reflector are Ω R ω 0 = 35% and Ω R ω 0 = 42%, respectively. Besides the UP and LP modes, an additional high-quality factor mode starting from 405 GHz at zero magnetic field appears in the simulated transmission spectra, which is not visible in the experimental data. To assess the origin of this mode, the field distribution of the high-quality factor mode at f = 405 GHz and of the UP mode at f = 620 GHz are investigated at ∼zero magnetic field, shown in figure 3. By comparing the tangential electric field in x-y plane in figure 3a, 3b and 3d, 3e, the high-quality factor mode at f = 405 GHz seems to be a higher order mode. The full width half maximum (FWHM) of the central mode at 405 GHz and 620 GHz are 190 and 660 nm,
confinement can be only reduced to the width of the central defect area. This explains why the achieved normalized coupling strength for the recovered polaritonic modes (figure 2a) was a few percent lower than the expected normalized coupling strength for a coupled cSRR with a gap of d = 250 nm.
Figure 3 :
3Field distribution of the high-quality factor mode and the UP mode (a), (b), and (c): The tangential electric field distribution for the UP mode at f = 620 GHz at zero magnetic field. Panel (b) shows the x-y plane at z = z 2DEG . Panel (a) is a cross section of (b) (the real part) at x = 0 marked by dashed white line. Panel (c) illustrates the y-z plane at x = 0. The 2DEG is marked by dotted white line. The dashed black line indicates the place of the cross-section x-y plane for panel (b). (d), (e), and (f): similar to (a), (b), and (c) but for the high-quality factor mode at f = 405 GHz at zero magnetic field.
writing lithography with Heidelberg DWL66+. The trenches were etched by a highly diluted etching solution (slow etchant solution: H 2 O, 1 : 3). The slow etchant solution itself was made of (H 2 SO 4 : H 2 O 2 : H 2 O, 1 : 8 : 60) with an etch rate of ∼ 9 nm/s for semi-insulating GaAs. The diluted solution had an etch rate of ∼ 3 nm/s for semi-insulating GaAs. Reducing the etch rate was done to increase the etching time and the accuracy of the etching to be able to etch only a few tens of nanometers. The etch rates were evaluated by a Dektak surface step profiler (with a 5 µm radius tip) and a scanning electron microscopy. After an aligning step, the cSRRs were fabricated on top of the etched trenches with electron beam lithography using a bilayer resist process with 450 nm 495K-PMMA-A4 and 90 nm Dow Corning XR-1541-006 electron beam negative resist. The lithography step was followed by deposition of 4 nm titanium and 200 nm of gold and a lift-off process.
Ultrastrong coupling regimes of light-matter interaction. P Forn-Díaz, L Lamata, E Rico, J Kono, E Solano, Reviews of Modern Physics. 91Forn-Díaz, P.; Lamata, L.; Rico, E.; Kono, J.; Solano, E. Ultrastrong coupling regimes of light-matter interaction. Reviews of Modern Physics 2019, 91
Ultrastrong coupling between light and matter. A Frisk Kockum, A Miranowicz, S De Liberato, S Savasta, F Nori, Nature Rev. Phys. 1Frisk Kockum, A.; Miranowicz, A.; De Liberato, S.; Savasta, S.; Nori, F. Ultrastrong coupling between light and matter. Nature Rev. Phys. 2019, 1, 19-40
Absorption Engineering in an Ultrasubwavelength Quantum System. M Jeannin, T Bonazzi, D Gacemi, A Vasanelli, L Li, A G Davies, E Linfield, C Sirtori, Y Todorov, Nano Lett. 20Jeannin, M.; Bonazzi, T.; Gacemi, D.; Vasanelli, A.; Li, L.; Davies, A. G.; Linfield, E.; Sirtori, C.; Todorov, Y. Absorption Engineering in an Ultrasubwavelength Quantum System. Nano Lett. 2020, 20, 4430-4436
Ultrastrong Coupling Regime and Plasmon Polaritons in Parabolic Semiconductor Quantum Wells. M Geiser, F Castellano, G Scalari, M Beck, L Nevou, J Faist, Phys. Rev. Lett. 108106402Geiser, M.; Castellano, F.; Scalari, G.; Beck, M.; Nevou, L.; Faist, J. Ultrastrong Coupling Regime and Plasmon Polaritons in Parabolic Semiconductor Quantum Wells. Phys. Rev. Lett. 2012, 108, 106402
Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor. D Hagenmüller, S De Liberato, C Ciuti, Phys. Rev. B. 235303Hagenmüller, D.; De Liberato, S.; Ciuti, C. Ultrastrong coupling between a cavity resonator and the cyclotron transition of a two-dimensional electron gas in the case of an integer filling factor. Phys. Rev. B 2010, 81, 235303
Ultrastrong Coupling of the Cyclotron Transition of a 2D Electron Gas to a THz Metamaterial. G Scalari, C Maissen, D Turčinková, D Hagenmüller, S De Liberato, C Ciuti, C Reichl, D Schuh, W Wegscheider, M Beck, J Faist, Science. 335Scalari, G.; Maissen, C.; Turčinková, D.; Hagenmüller, D.; De Liberato, S.; Ciuti, C.; Reichl, C.; Schuh, D.; Wegscheider, W.; Beck, M.; Faist, J. Ultrastrong Coupling of the Cyclotron Transition of a 2D Electron Gas to a THz Metamaterial. Science 2012, 335, 1323-1326
Terahertz Light-Matter Interaction beyond Unity Coupling Strength. A Bayer, M Pozimski, S Schambeck, D Schuh, R Huber, D Bougeard, C Lange, Nano Lett. 17Bayer, A.; Pozimski, M.; Schambeck, S.; Schuh, D.; Huber, R.; Bougeard, D.; Lange, C. Terahertz Light-Matter Interaction beyond Unity Coupling Strength. Nano Lett. 2017, 17, 6340-6344
Vacuum Bloch-Siegert shift in Landau polaritons with ultra-high cooperativity. X Li, M Bamba, Q Zhang, S Fallahi, G C Gardner, W Gao, M Lou, K Yoshioka, M J Manfra, J Kono, Nature Photonics. 12Li, X.; Bamba, M.; Zhang, Q.; Fallahi, S.; Gardner, G. C.; Gao, W.; Lou, M.; Yosh- ioka, K.; Manfra, M. J.; Kono, J. Vacuum Bloch-Siegert shift in Landau polaritons with ultra-high cooperativity. Nature Photonics 2018, 12, 324-329
Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons. Q Zhang, M Lou, X Li, J L Reno, W Pan, J D Watson, M J Manfra, J Kono, Nature Physics. 12Zhang, Q.; Lou, M.; Li, X.; Reno, J. L.; Pan, W.; Watson, J. D.; Manfra, M. J.; Kono, J. Collective non-perturbative coupling of 2D electrons with high-quality-factor terahertz cavity photons. Nature Physics 2016, 12, 1005-1011
Magneto-transport controlled by Landau polariton states. G L Paravicini-Bagliani, F Appugliese, E Richter, F Valmorra, J Keller, M Beck, N Bartolo, C Rössler, T Ihn, K Ensslin, C Ciuti, G Scalari, J Faist, Nature Physics. 15Paravicini-Bagliani, G. L.; Appugliese, F.; Richter, E.; Valmorra, F.; Keller, J.; Beck, M.; Bartolo, N.; Rössler, C.; Ihn, T.; Ensslin, K.; Ciuti, C.; Scalari, G.; Faist, J. Magneto-transport controlled by Landau polariton states. Nature Physics 2018, 15, 186-190
Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect. F Appugliese, J Enkner, G L Paravicini-Bagliani, M Beck, C Reichl, W Wegscheider, G Scalari, C Ciuti, J Faist, Science. 2022Appugliese, F.; Enkner, J.; Paravicini-Bagliani, G. L.; Beck, M.; Reichl, C.; Wegschei- der, W.; Scalari, G.; Ciuti, C.; Faist, J. Breakdown of topological protection by cavity vacuum fields in the integer quantum Hall effect. Science 2022, 375, 1030-1034
An ultrastrongly coupled single terahertz meta-atom. S Rajabali, S Markmann, E Jöchl, M Beck, C A Lehner, W Wegscheider, J Faist, G Scalari, Nature Communications. 20222528Rajabali, S.; Markmann, S.; Jöchl, E.; Beck, M.; Lehner, C. A.; Wegscheider, W.; Faist, J.; Scalari, G. An ultrastrongly coupled single terahertz meta-atom. Nature Com- munications 2022, 13, 2528
Few-Electron ultrastrong light-matter coupling at 300 GHz with nanogap hybrid LC microcavities. J Keller, G Scalari, S Cibella, C Maissen, F Appugliese, E Giovine, R Leoni, M Beck, J Faist, 17Keller, J.; Scalari, G.; Cibella, S.; Maissen, C.; Appugliese, F.; Giovine, E.; Leoni, R.; Beck, M.; Faist, J. Few-Electron ultrastrong light-matter coupling at 300 GHz with nanogap hybrid LC microcavities. Nano letters 2017, 17, 7410-7415
Polaritonics: from microcavities to sub-wavelength confinement. D Ballarini, S De Liberato, Nanophotonics. 8Ballarini, D.; De Liberato, S. Polaritonics: from microcavities to sub-wavelength con- finement. Nanophotonics 2019, 8, 641-654
Probing the Ultimate Limits of Plasmonic Enhancement. C Ciracì, R T Hill, J J Mock, Y Urzhumov, A I Fernández-Domínguez, S A Maier, J B Pendry, A Chilkoti, D R Smith, Science. 337Ciracì, C.; Hill, R. T.; Mock, J. J.; Urzhumov, Y.; Fernández-Domínguez, A. I.; Maier, S. A.; Pendry, J. B.; Chilkoti, A.; Smith, D. R. Probing the Ultimate Lim- its of Plasmonic Enhancement. Science 2012, 337, 1072-1074
Optical Nonlocality in Polar Dielectrics. C R Gubbin, S De Liberato, Phys. Rev. 202021027Gubbin, C. R.; De Liberato, S. Optical Nonlocality in Polar Dielectrics. Phys. Rev. X 2020, 10, 021027
Transformation-Optics Description of Nonlocal Effects in Plasmonic Nanostructures. A I Fernández-Domínguez, A Wiener, F J García-Vidal, S A Maier, J B Pendry, Phys. Rev. Lett. 108106802Fernández-Domínguez, A. I.; Wiener, A.; García-Vidal, F. J.; Maier, S. A.; Pendry, J. B. Transformation-Optics Description of Nonlocal Effects in Plasmonic Nanostructures. Phys. Rev. Lett. 2012, 108, 106802
Impact of phonon nonlocality on nanogap and nanolayer polar resonators. C R Gubbin, S De Liberato, Phys. Rev. 2020Gubbin, C. R.; De Liberato, S. Impact of phonon nonlocality on nanogap and nanolayer polar resonators. Phys. Rev. B 2020, 102, 201302
Polaritonic nonlocality in light-matter interaction. S Rajabali, E Cortese, M Beck, S De Liberato, J Faist, G Scalari, Nature Phot. 15Rajabali, S.; Cortese, E.; Beck, M.; De Liberato, S.; Faist, J.; Scalari, G. Polaritonic nonlocality in light-matter interaction. Nature Phot. 2021, 15, 690-695
Strong coupling of ionizing transitions. E Cortese, I Carusotto, R Colombelli, S De Liberato, Optica. 6Cortese, E.; Carusotto, I.; Colombelli, R.; De Liberato, S. Strong coupling of ionizing transitions. Optica 2019, 6, 354-361
Exact solution of polaritonic systems with arbitrary light and matter frequency-dependent losses. E Cortese, S De Liberato, The Journal of Chemical Physics. 156Cortese, E.; De Liberato, S. Exact solution of polaritonic systems with arbitrary light and matter frequency-dependent losses. The Journal of Chemical Physics 2022, 156,
Detuned plasmonic Bragg grating sensor based on a defect metal-insulator-metal waveguide. S Qu, C Song, X Xia, X Liang, B Tang, Z.-D Hu, J Wang, Sensors. 16784Qu, S.; Song, C.; Xia, X.; Liang, X.; Tang, B.; Hu, Z.-D.; Wang, J. Detuned plasmonic Bragg grating sensor based on a defect metal-insulator-metal waveguide. Sensors 2016, 16, 784
Characteristics of plasmonic Bragg reflectors with graphene-based silicon grating. C Song, X Xia, Z.-D Hu, Y Liang, J Wang, Nanoscale research lett. 11Song, C.; Xia, X.; Hu, Z.-D.; Liang, Y.; Wang, J. Characteristics of plasmonic Bragg reflectors with graphene-based silicon grating. Nanoscale research lett. 2016, 11, 1-8
Frequency tunable terahertz interdigitated photoconductive antennas. J Madeo, N Jukam, D Oustinov, M Rosticher, R Rungsawang, J Tignon, S S Dhillon, Elec. Lett. 46Madeo, J.; Jukam, N.; Oustinov, D.; Rosticher, M.; Rungsawang, R.; Tignon, J.; Dhillon, S. S. Frequency tunable terahertz interdigitated photoconductive antennas. Elec. Lett. 2010, 46, 611-U25
Electro-optic detection of terahertz radiation. G Gallot, D Grischkowsky, J. Opt. Soc. Am. B. 16Gallot, G.; Grischkowsky, D. Electro-optic detection of terahertz radiation. J. Opt. Soc. Am. B 1999, 16, 1204-1212
THz ultrastrong light-matter coupling. G Scalari, C Maissen, S Cibella, R Leoni, C Reichl, W Wegscheider, M Beck, J Faist, 31II Nuovo SaggiatoreScalari, G.; Maissen, C.; Cibella, S.; Leoni, R.; Reichl, C.; Wegscheider, W.; Beck, M.; Faist, J. THz ultrastrong light-matter coupling. II Nuovo Saggiatore 2015, 31, 3-4, 4-14
Landau Damping and Limit to Field Confinement and Enhancement in Plasmonic Dimers. J Khurgin, W.-Y Tsai, D P Tsai, G Sun, ACS Photonics. 4Khurgin, J.; Tsai, W.-Y.; Tsai, D. P.; Sun, G. Landau Damping and Limit to Field Confinement and Enhancement in Plasmonic Dimers. ACS Photonics 2017, 4, 2871- 2880
| {'fraction_non_alphanumeric': 0.05487731657527119, 'fraction_numerical': 0.026818236400752513, 'mean_word_length': 4.443137254901961, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, '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': 'It was recently demonstrated that, in deep subwavelength gap resonators coupled to two-dimensional electron gases, coupling to propagating plasmons can lead to energy leakage and prevent the formation of polaritonic resonances. This process, akin to Landau damping, limits the achievable field confinement and thus the value of light-matter coupling strength. In this work, we show how plasmonic subwavelength reflectors can be used to create an artificial energy stopband in the plasmon dispersion, confining them and enabling the recovery of the polaritonic resonances. Using this approach we demonstrate a normalized light-matter coupling ratio of Ω ω = 0.35 employing a single quantum well with a gap size of λ/2400 in vacuum.', 'arxivid': '2212.13482', 'author': ['Shima Rajabali [email protected] ', 'Josefine Enkner ', 'Erika Cortese ', 'Mattias Beck ', 'Simone De Liberato ', 'Jérôme Faist ', 'Giacomo Scalari [email protected] ', '\nof Quantum Electronics\n‡School of Physics and Astronomy\nETH Zürich\n8093ZürichSwitzerland\n', '\nUniversity of Southampton\nSO17 1BJSouthamptonUnited Kingdom\n'], 'authoraffiliation': ['of Quantum Electronics\n‡School of Physics and Astronomy\nETH Zürich\n8093ZürichSwitzerland', 'University of Southampton\nSO17 1BJSouthamptonUnited Kingdom'], 'corpusid': 255186379, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 7793, 'n_tokens_neox': 6601, 'n_words': 3907, 'pdfsha': '6c97fbb8f9b1dfa923f096bb72e92038549f452d', 'pdfurls': ['https://export.arxiv.org/pdf/2212.13482v1.pdf'], 'title': ['An engineered planar plasmonic reflector for polaritonic mode confinement', 'An engineered planar plasmonic reflector for polaritonic mode confinement'], 'venue': []} |
arxiv |
THE SUM OF THE r'TH ROOTS OF FIRST n NATURAL NUMBERS AND NEW FORMULA FOR FACTORIAL
13 Feb 2013
Snehal Shekatkar
THE SUM OF THE r'TH ROOTS OF FIRST n NATURAL NUMBERS AND NEW FORMULA FOR FACTORIAL
13 Feb 2013
Using the simple properties of Riemman integrable functions, Ramanujan's formula for sum of the square roots of first n natural numbers has been generalized to include r ′ th roots where r is any real number greater than 1. As an application we derive formula that gives factorial of positive integer n similar to Stirling's formula.
The formula for the sum of the square roots of first n natural numbers has been given by Srinivas Ramanujan ([Ra15]). Here we extend his result to the case of r ′ th roots, where r is a real number greater than 1. Statement of result:
Theorem 0.1. Let r be a real number with r ≥ 1 and n be a positive integer. Then
(1) x=n x=1 x 1 r = r r + 1 (n + 1) 1+r r − 1 2 (n + 1) 1 r − φ n (r)
where φ n is a function of r with n as a parameter. This function is bounded between 0 and 1 2 .
Proof. For a closed interval [a,b] we define partition of this interval as a set of points x 0 = a, x 1 , ..., x n−1 = b where x i < x j whenever i < j . Now consider the closed interval [0,n] and consider a partition P of this interval,where P is a set {0, 1, 2, ..., n}.
Consider a function defined as f (x)=x 1 r . We have,
I = n 0 f (x)dx = lim ∆x i →0 n−1 i=0 f (x i )∆x i where ∆x i = x i+1 − x i
We define lower sum for partition P as:
L = n−1 i=0 f (i)∆x i = n−1 i=0 i 1 r Similarly,upper sum for P is U = n i=1 f (i)∆x i = n−1 i=0 (i + 1) 1 r
We write value of integral I as average of L and U with some correction term.
2I = L + U + φ ∴ 2 n 0 x 1 r dx = n−1 i=0 i 1 r + (i + 1) 1 r + φ ∴ 2r r + 1 x 1+r r n 0 = 0 1 r + 2 n−1 i=1 i 1 r + n 1 r + φ ∴ n−1 i=1 i 1 r = r r + 1 n 1+r r − n 1 r 2 − φ
where the term of 1 2 has been absorbed into φ.
(2) n i=1 i 1 r = r r + 1 (n + 1) This gives φ n (1) = 0
Since the difference between first and second term can easily be shown to be monotonic, we see that φ is bounded between between 0 and 1 2 for 1 ≤ r < ∞ As an application of the formula derived above, we derive formula to derive factorial of positive integer. We begin by taking derivative of (2) with respect to r. After rearranging the terms, we get,
(3) i=n i=1 i 1 r log i = r r + 1 (n + 1) − 1 2 (n+1) 1 r log(n+1)− r 2 (r + 1) 2 (n+1)
1+r r +r 2 dφ dr
After taking limit of this equation as r → ∞, we get following equation:
(4) n i=1
log i = (n + 1 2 ) log(n + 1) − (n + 1) + lim r→∞ r 2 dφ dr
In the above expression, L.H.S is just log(n!). Let us assume that limit in the last term of the above equation exists and is finite and say that it is ξ. Then we can rewrite above equation as follows:
(5) n! = (n + 1) n+ 1 2 e −n−1 e ξ Numerically it turns out that the quantity e ξ indeed converges to finite value, the value being close to √ 2π. This formula is similar to precise version of Stirling's formula ( [St1]).
Equation (3) allows us to find one more interesting formula. After putting r = 1 in (3) and after little rearrangement, we get following beautiful formula:
(6) log 1 1 .2 2 ...n n = n(n + 1) 2 log(n + 1) − 1 4 (n + 1) 2 + dφ dr | r=1
Numerically it turns out that quantity dφ dr | r=1 is very small and can be neglected.
Taking limit of (2) as r → ∞,L.H.S. → n and R.H.S. → (n + 1 2 − φ), so that in the limit φ →
On the sum of the square roots of the first n natural numbers. S Ramanujan, J. Indian Math. Soc. Ramanujan S., On the sum of the square roots of the first n natural numbers., J. Indian Math. Soc., V II, (1915), 173-175.
M Abramowitz, I Stegun, Handbook of Mathematical Functions. Abramowitz, M. and Stegun, I. (2002), Handbook of Mathematical Functions.
Indian institute of science education and research. Dept, Physics. Pin:411021 E-mail address: [email protected] Physics, Indian institute of science education and research, Pune, India, Pin:411021 E-mail address: [email protected]
| {'fraction_non_alphanumeric': 0.08422894276430892, 'fraction_numerical': 0.042989252686828294, 'mean_word_length': 3.1046153846153848, 'pattern_counts': {'":': 0, '<': 3, '<?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': 8, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Using the simple properties of Riemman integrable functions, Ramanujan's formula for sum of the square roots of first n natural numbers has been generalized to include r ′ th roots where r is any real number greater than 1. As an application we derive formula that gives factorial of positive integer n similar to Stirling's formula.", 'arxivid': '1204.0877', 'author': ['Snehal Shekatkar '], 'authoraffiliation': [], 'corpusid': 118268840, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 1516, 'n_tokens_neox': 1327, 'n_words': 828, 'pdfsha': '691aab63184d99f87e100da5d4a299dd1bfbd270', 'pdfurls': ['https://arxiv.org/pdf/1204.0877v2.pdf'], 'title': ["THE SUM OF THE r'TH ROOTS OF FIRST n NATURAL NUMBERS AND NEW FORMULA FOR FACTORIAL", "THE SUM OF THE r'TH ROOTS OF FIRST n NATURAL NUMBERS AND NEW FORMULA FOR FACTORIAL"], 'venue': []} |
arxiv |
Quadrupole-quadrupole gravitational waves
7 Oct 1997
Luc Blanchet
Département d'Astrophysique Relativiste et de Cosmologie (UPR 176 du CNRS)
Observatoire de Paris
92195Meudon CedexFrance
Quadrupole-quadrupole gravitational waves
7 Oct 1997Submitted to: Classical Quantum Grav. Date: 24 March 2022 1Short title: Quadrupole-quadrupole gravitational waves PACS number(s): 04.25.Nx, 04.30.Db
This paper investigates the non-linear self-interaction of quadrupole gravitational waves generated by an isolated system. The vacuum Einstein field equations are integrated in the region exterior to the system by means of a post-Minkowskian algorithm. Specializing in the quadrupole-quadrupole interaction (at the quadratic non-linear order), we recover the known results concerning the nonlocal modification of the ADM mass-energy of the system accounting for the emission of quadrupole waves, and the non-local memory effect due to the re-radiation by the stress-energy distribution of linear waves. Then we compute all the local (instantaneous) terms which are associated in the quadrupole-quadrupole metric with the latter non-local effects. Expanding the metric at large distances from the system, we obtain the corresponding radiation-field observables, including all non-local and transient contributions. This permits notably the completion of the observable quadrupole moment at the 5/2 post-Newtonian order.
Introduction
In general relativity, the multipole moments of any finite distribution of energy and momentum interact with each other in vacuum, through the non-linearities of the field equations. In particular, the multipole moments which describe the gravitational waves emitted by an isolated system do not evolve independently, but rather couple together (including with themselves), giving rise to non-linear physical effects.
The simplest multipole interaction which contributes to the radiation field is that between the (mass-type) quadrupole moment M ij and the mass monopole M . The latter moment is the constant mass-energy of the source as measured at spatial infinity (ADM mass). Associated with the multipole interaction M ij × M is the non-linear effect of tails. This effect is due to the backscatter of linear waves (described by M ij ) onto the space-time curvature generated by the mass-energy M . The tails can be computed within the theory of gravitational perturbations of the Schwarzschild background (see e.g. [1][2][3]). A consequence of the existence of tails is the non-locality in time, as the tails are in the form of integrals depending on the history of the source from −∞ in the past to the retarded time t − r/c [4][5][6][7]. It is known that the tails appear both in the radiation field and in the radiation reaction forces at the 3/2 post-Newtonian order (1.5PN, or order c −3 when c → ∞) relatively to the quadrupole radiation [8,9].
Next in complexity is the interaction of the quadrupole moment with itself, or quadrupole self-interaction M ij × M kl . Two closely related non-local (or hereditary) effects are known with this particular interaction. As shown by Bonnor and collaborators [10,11,4,6], the total mass M gets modified by a non-local integral accounting for the energy which is radiated by the quadrupole waves (there is agreement with the Einstein quadrupole formula). On the other hand the radiation field involves a non-local contribution whose physical origin is the re-emission of waves by the linear waves [12][13][14][15][16]9]. This contribution can be easily computed by using as the source of waves in the right side of the Einstein field equations the effective stress-energy tensor of gravitational waves (averaged over several wavelengths). As shown by Christodoulou [14] and Thorne [15] this implies a permanent change in the wave amplitude from before to afterward a burst of gravitational waves, which can be interpreted as the contribution of gravitons in the known formulas for the linear memory [17,18]. The latter non-linear memory integral appears at the 2.5PN order in the radiation field (1PN order relatively to the tail integral).
Now the metric which corresponds to the quadrupole-quadrupole interaction involves also, besides the non-local contributions, many terms which by contrast depend on the multipole moments at the sole retarded instant t − r/c. In the following we shall often qualify these local terms as instantaneous (following the terminology of [8]). The instantaneous terms in the radiation field (to order 1/r) are transient in the sense that they return to zero after the passage of a burst of gravitational waves.
On physical grounds, it can be argued that the instantaneous terms do not play a very important role. However, these terms do exist and form an integral part of the field generated by very relativistic sources like inspiraling compact binaries (see e.g. [19]). In fact, the complete quadrupole-quadrupole metric including all the hereditary and instantaneous contributions will be needed in the construction of very accurate theoretical wave forms to be used by the future detectors LIGO and VIRGO.
The present paper is devoted to the computation of the quadrupole-quadrupole metric, using the so-called multipolar-post-Minkowskian method proposed by Blanchet and Damour [20,21] after previous work by Bonnor [10] (his double-series approximation method) and Thorne [22] (how to start the post-Minkowskian iteration using STF multipole moments). The instantaneous quadrupole-quadrupole terms have never been computed within the multipolar-post-Minkowskian method. However, they have been computed by Hunter and Rotenberg [6] within the double-series method, in the case where the source is axi-symmetric. The non-linear interaction between quadrupoles has also received attention more recently within the double-series method [23].
The main result of this paper (having in mind the application to astrophysical sources to be detected by VIRGO and LIGO) is the completion of the observable quadrupole moment of a general isolated source at the 2.5PN order. To reach this result we also take into account previous results concerning the tails at the 1.5PN order [9], and the multipole moments given by explicit integrals over the source to 2.5PN order [24,25]. (Note that the 2.5PN approximation in the observable quadrupole moment gives no contribution to the phase evolution of inspiraling compact binaries [25], however it will be required when we compute the 2.5PN wave form.)
In a following paper [26], that we shall refer to as paper II, we shall investigate the monopole-monopole-quadrupole interaction (at the cubic non-linear order) which enters the radiation field at the 3PN approximation. The present paper and paper II are part of the program of computing the field generated by inspiraling compact binaries with 3PN and even 3.5PN accuracy (see [27][28][29][30] for why such a very high accuracy is necessary).
The plan of the paper is as follows. In Section 2 we summarize from [20,21] the method for computing the field non-linearities. In Section 3 we investigate (following [8]) the general structure of the quadratic non-linearities. Section 4 deals with the explicit computation of the quadrupole-quadrupole metric. The results are presented in the form of tables of numerical coefficients. Finally, in Section 5, we expand the metric at infinity and obtain the observable moments in the radiation field (essentially the quadrupole). The technical formulas for integrating the wave equation are relegated to Appendix A, and some needed results concerning the dipolequadrupole interaction are derived in Appendix B. Henceforth we pose c = 1, except when we discuss the post-Newtonian order at the end of Section 5.
Non-linearities in the external field
We summarize the method set up in [20] for the computation of the quadratic and higher non-linearities in the field generated by an isolated system. The computation is performed in the exterior (vacuum) weak-field region of the system, where the components of the gravitational field h µν are numerically small as compared to one.
Here h µν denotes the metric deviation h µν = √ −gg µν − η µν , with g µν the inverse and g the determinant of the metric g µν , and η µν the Minkowski metric diag(−1, 1, 1, 1).
The field h µν admits in the exterior weak-field region a post-Minkowskian expansion,
h µν = Gh µν 1 + G 2 h µν 2 + ... + G n h µν n + ... ,(2.⊓ ⊔h µν 1 = 0 , (2.2a) ∂ ν h µν 1 = 0 . (2.2b)
In the first equation ⊓ ⊔ denotes the flat space-time wave (d'Alembertian) operator.
The second equation (divergenceless of the field) is the harmonic gauge condition. The equations (2.2), supplemented by the condition of retarded potentials, are solved by means of a multipolar expansion. It is known that only two sets of multipole moments, the mass-type moments M L and current-type moments S L (both depending on the retarded time t − r), are sufficient to parametrize the general multipole expansion (see e.g. [22]). In terms of these moments the general solution reads [22] h 00
1 = − 4 ℓ≥0 (−) ℓ ℓ! ∂ L r −1 M L (t − r) , (2.3a) h 0i 1 =4 ℓ≥1 (−) ℓ ℓ! ∂ L−1 r −1 M (1) iL−1 (t − r) + 4 ℓ≥1 (−) ℓ ℓ (ℓ + 1)! ε iab ∂ aL−1 r −1 S bL−1 (t − r) , (2.3b) h ij 1 = − 4 ℓ≥2 (−) ℓ ℓ! ∂ L−2 r −1 M (2) ijL−2 (t − r) − 8 ℓ≥2 (−) ℓ ℓ (ℓ + 1)! ∂ aL−2 r −1 ε ab(i S (1) j)bL−2 (t − r) . (2.3c)
Here the superscript (n) denotes n time derivatives. The index L is a shorthand for a multi-index composed of ℓ indices,
L = i 1 i 2 ...i ℓ (similarly, L − 1 = i 1 i 2 ...i ℓ−1 ,
aL − 2 = ai 1 ...i ℓ−2 , and so on), and ∂ L denotes a product of ℓ space derivatives,
∂ L = ∂ i 1 ∂ i 2 ...∂ i ℓ .
The multipole moments M L and S L are symmetric and tracefree (STF) with respect to all their indices (see [31] for a recapitulation of our notation and conventions).
A priori the multipole moments M L (t) and S L (t) are arbitrary functions of time.
They describe potentially the physics of a general isolated source as seen in its exterior field [22,20]. The only physical restriction is that the mass monopole M (total massenergy or ADM mass), the mass dipole M i (position of the center of mass times the mass), and the current dipole S i (total angular momentum) are constant. Technically speaking, this is a consequence of the harmonic gauge condition (2.2b) (see e.g. [22]). In this paper we shall set the mass dipole M i to zero by shifting the origin of coordinates to the center of mass. In order to describe an isolated system, we must implement a condition of no-incoming radiation, ensuring that the radiation field is entirely generated by the system. We assume that the field is stationary in the remote past, i.e. that the moments M L (t) and S L (t) are constant before some finite instant in the past, say when t ≤ −T (hence M i cannot be a linear function of time and is necessarily constant or zero). This assumption may seem to be somewhat restrictive, but we can check a posteriori that the formulas derived in this paper and paper II admit a well-defined limit when −T → −∞ in more general physical situations, such as the formation of the system by initial gravitational scattering. The multipole moments M L (t) and S L (t), subject to the previous restrictions, play the role of "seed" moments for the construction of the exterior field (2.1). In particular we shall express the results of this paper in terms of products of M ij with itself. We shall not use the expressions of the multipole moments M L and S L as explicit integrals over the source.
However, these expressions are known in the post-Newtonian approximation [24,25], and should be used in applications.
The coefficient of G 2 in (2.1) is the quadratically non-linear metric, whose precise definition we recall. The field equations for this coefficient read, still using harmonic coordinates,
⊓ ⊔h µν 2 = N µν 2 , (2.4a) ∂ ν h µν 2 = 0 . (2.4b)
The d'Alembertian equation involves a quadratic source N µν 2 = N µν (h 1 , h 1 ) generated by the linearized gravitational field (2.3), where
N µν (h, h) = − h ρσ ∂ ρ ∂ σ h µν + 1 2 ∂ µ h ρσ ∂ ν h ρσ − 1 4 ∂ µ h∂ ν h − 2∂ (µ h ρσ ∂ ρ h ν)σ + ∂ σ h µρ (∂ σ h ν ρ + ∂ ρ h νσ ) + η µν − 1 4 ∂ λ h ρσ ∂ λ h ρσ + 1 8 ∂ ρ h∂ ρ h + 1 2 ∂ ρ h σλ ∂ σ h ρλ . (2.5) From (2.4b) we deduce ∂ ν N µν 2 = 0 . (2.6)
Then the quadratic metric h µν 2 , solving (2.4) and the condition of stationarity in the past, is obtained as the sum of two distinct contributions,
h µν 2 = u µν 2 + v µν 2 . (2.7)
Basically the first contribution u µν 2 is the retarded integral of the source N µν 2 . However, in our case the source is in the form of a multipole expansion (valid only in the exterior of the system and singular at r = 0), so we cannot apply directly the usual retarded integral operator, whose range of integration intersects the system at retarded time. A way out of this problem, proposed in [20], consists of multiplying the actual source term N µν [Actually we consider separately each multipolar pieces, with given multipolarities ℓ, so that the maximal power of the singularities is finite, and B can indeed be chosen in such a way.] Applying the retarded integral on each multipolar pieces of the product (r/r 0 ) B N µν 2 results in a function of B whose definition can be analytically continued to a neighbourhood of B = 0, at which value it admits a Laurent expansion. The finite part at B = 0 (in short FP B=0 ) of the latter expansion is our looked-for solution, as it satisfies the correct wave equation (⊓ ⊔u µν 2 = N µν 2 ), and is like N µν 2 in the form of a multipole expansion. Note that the latter process represents simply a convenient mean to find a solution of the wave equation whose source is in the form of a multipole expansion. Other processes could be used as well, but this one is particularly powerful as it yields many explicit formulas to be used in practical computations (see Appendix A below and Appendix A of paper II). Hence the first contribution in (2.7) reads
u µν 2 = FP B=0 ⊓ ⊔ −1 R r r 0 B N µν 2 , (2.8) where ⊓ ⊔ −1 R denotes the usual retarded integral (⊓ ⊔ −1 R f )(x, t) = − 1 4π d 3 x ′ |x − x ′ | f (x ′ , t − |x − x ′ |) . (2.9)
When dealing with the metric at quadratic order, it can be proved that the Bdependent retarded integral in (2.8) is actually finite when B → 0 (the finite part is not followed by any pole). So u µν 2 is simply given by the value at B = 0 of the retarded integral (see [8] and Section 4). But this is due to the special structure of the quadratic source N µν 2 , and does not remain true at cubic and higher non-linear approximations (see for instance paper II).
The first contribution u µν 2 solves (2.4a), but not the harmonic gauge condition (2.4b). The divergence of u µν 2 , say w µ 2 = ∂ ν u µν 2 , is a priori different from zero. Using (2.6) we find
w µ 2 = FP B=0 ⊓ ⊔ −1 R B r r 0 B n i r N µi 2 .
(2.10)
The explicit factor B comes from the differentiation of r B in (2.8) (we denote has a structure which is different from N µν 2 , and unlike in (2.8) the integral admits in general a (simple) pole at B = 0.] The second contribution v µν 2 in (2.7) is then defined in such a way as to compensate exactly the (a priori) non-zero divergence w µ 2 of u µν 2 , while being a homogeneous solution of (2.4a). This is possible because w µ 2 is a particular retarded solution of ⊓ ⊔w µ 2 = 0 (in the exterior region). As such, it admits a unique multipolar decomposition in terms of four sets of STF tensors A L , B L , C L ,
n i = ∂ i r = x i /r).D L , namely w 0 2 = ℓ≥0 ∂ L r −1 A L (t − r) , (2.11a) w i 2 = ℓ≥0 ∂ iL r −1 B L (t − r) + ℓ≥1 ∂ L−1 r −1 C iL−1 (t − r) + ε iab ∂ aL−1 r −1 D bL−1 (t − r) . (2.11b)
These tensors can be computed straigthforwardly from the known expression (2.10), and the contribution v µν 2 is defined in terms of these tensors. A particular definition was proposed in the equations (4.13) of [20], where it was denoted q µν 2 . Here we shall define this second contribution slightly differently, and accordingly we use the different notation v µν 2 . The various components of v µν 2 are given by v 00
2 = − r −1 A + ∂ a r −1 − A a + C a − 3B a , (2.12a) v 0i 2 = r −1 − C i + 3B (1) i − ε iab ∂ a r −1 D b − ℓ≥2 ∂ L−1 r −1 A iL−1 , (2.12b) v ij 2 = − δ ij r −1 B + ℓ≥2 2δ ij ∂ L−1 r −1 B L−1 − 6∂ L−2(i r −1 B j)L−2 + ∂ L−2 r −1 (A (1) ijL−2 + 3B (2) ijL−2 − C ijL−2 ) − 2∂ aL−2 r −1 ε ab(i D j)bL−2 . (2.12c)
Like in (2.11) all the tensors are evaluated at the retarded time t − r. We note that the formulas (2.12) are non-instantaneous, as they depend on the moments M L and S L at any time less than t − r through the first and second time anti-derivatives [20] for discussions). [To quadratic order the tensors A L , ..., D L are given by some instantaneous functionals of the moments M L and S L .] The main property of v µν 2 is ∂ ν v µν 2 = −w µ 2 , which is easily checked on the expressions (2.12). Furthermore ⊓ ⊔v µν 2 = 0, so the quadratic metric (2.7) is, indeed, a solution of both the wave equation (2.4a) and the gauge condition (2.4b). Note that the spatial trace v ii
of A, A a , C a , denoted e.g. by A = t−r −∞ A(t ′ )dt ′ and C a = t−r −∞ C a (t ′ )dt ′ (see2 = δ ij v ij 2 is especially simple, v ii 2 = −3r −1 B . (2.12d)
The choice of definition (2.12) adopted here, which differs from the choice adopted in [20], is for convenience in future work. Of course we are free to adopt one definition or another because such a choice is equivalent to a choice of gauge. However, the definition (2.12) is slightly preferable to the definition proposed in [20] when we want to express the multipole moments M L and S L as integrals over the source. Thus we take the present opportunity to redefine the construction of the exterior metric using To the third (cubic) and higher non-linear iterations the construction of the external metric proceeds exactly along the same line, namely
h µν n = u µν n + v µν n ,(2.13)
where the first term u µν n is the finite part of the retarded integral of the source to the nth post-Minkowskian order, u µν
n = FP B=0 ⊓ ⊔ −1 R [(r/r 0 ) B N µν n (h 1 , .
.., h n−1 )], and where the second term v µν n is defined from the divergence w µ n = ∂ ν u µν n by the same formulas (2.11)-(2.12). (See [20] for the proof that the construction of the metric can be implemented to any post-Minkowskian order.)
Structure of the quadratically non-linear field
In this section we investigate the structure of the quadratic metric h µν 2 defined by (2.7)-(2.12). For simplicity we omit most of the numerical coefficients and indices, so as to focus our attention on the basic structure of the metric. A precise computation of the numerical coefficients is dealt with in Section 4.
The structure of the linearized metric (2.3) is that of a sum of retarded multipolar waves, consisting of p spatial derivatives (say) acting on monopolar waves r −1 X(t−r),
h 1 ≈ ∂ P [r −1 X(t − r)] .
(3.1)
Our notation ≈ refers to the structure of the expression. By expanding the derivatives ∂ P (which act both on the pre-factor r −1 and on the retardation t − r) we get
h 1 ≈ j≥1n Q r −j Z(t − r) , (3.2)
where the powers of 1/r are j ≥ 1, and where we have expressed the angular dependence of each term using STF products of unit vectorsn Q = n <i i n i 2 ...n i q > = STF part of n Q (see [31]). In practice, one may compute (3.2) from (3.1) by decomposing ∂ P on the basis of STF spatial derivatives∂ Q and using (A.15) in Appendix A. After insertion of (3.2) into the quadratic source term N µν 2 one finds
N 2 ≈ k≥2n L r −k F (t − r) , (3.3)
where the powers of 1/r start with k = 2 (as is clear from the fact that N 2 is quadratic in h 1 which is of order 1/r). The functions F are composed of sums of quadratic products of derivatives of the functions Z in (3.2).
The main problem is to compute u 2 defined by (2.8). In view of the structure (3.3),
it is a priori required to compute the finite part FP B=0 of the retarded integral of any
termn L r −k F (t − r)
with multipolarity ℓ and radial dependence with k ≥ 2. All the required formulas are listed in Appendix A (which summarizes results obtained mostly in previous works [20,8,9]). Notably, we know from Appendix A that the (finite part of the) retarded integral in the case k = 2 is irreducibly non-local or non-instantaneous. . This is probably the most efficient way to obtain u 2 . However, in doing so we would discover that the only non-local integrals left in u 2 come from the source terms having a radial dependence with k = 2, in other words all the non-local integrals coming from terms having k ≥ ℓ + 3 actually cancel out. Practically speaking the source terms having k ≥ ℓ + 3 turn out to combine into combinations such as (A.16) yielding purely instantaneous contributions. This fact (proved in [8]) is special to the quadratic non-linearities, and does not stay true at higher orders, e.g. at the cubic order as seen in paper II.
See (A.3)-(A.7) in Appendix A. When the power k satisfies 3 ≤ k ≤ ℓ + 2, where
The proof of the latter assertion uses specifically the quadratic structure of the source N 2 . Instead of inserting in (2.5) the linearized metric in expanded form (3.2) and then working out all derivatives to arrive at (3.3), we keep the structure of the source as it basically is, i.e. a sum of quadratic products of multipolar waves,
N 2 ≈ ∂ P r −1 X(t − r) ∂ Q r −1 Y (t − r) , (3.4)
involving spatial multi-derivatives with P = i 1 ...i p and Q = j 1 ...j q and some functions of time X and Y . Then we perform on each term of (3.4) a sequence of operations by
parts [i.e. ∂ i A∂ j B = ∂ i (A∂ j B) − A∂ i ∂ j B]
, by which the spatial derivatives acting on the wave in the left (say) are shifted in front and to the right. This leads to
N 2 ≈ ∂ P r −1 X(t − r)∂ R [r −1 Y (t − r)] . (3.5)
Only at this stage does one expand the space derivatives ∂ R (inside the curly brackets), while leaving the derivatives ∂ P in front un-expanded. The result reads after multiplication by the factor r −1 on the left. The next operation is to single out the terms with pure radial dependence k = 2. All these terms can be obtained by applying ∂ P on the terms inside the brackets having k = 2. In this way one generates besides all the terms k = 2 many other terms with k ≥ 3, but the latter terms can be recombined into terms of the same form as in (3.6) (and thus having 3 ≤ k ≤ ℓ + 2).
N 2 ≈ 2≤k≤ℓ+2 ∂ P n L r −k H(t − r) ,(3.
See [8] for the proof. Thus (3.6) can be rewritten as
N 2 ≈ r −2 Q(n, t − r) + 3≤k≤ℓ+2 ∂ P n L r −k F (t − r) ,(3.7)
where Q(n, t−r) denotes the coefficient of r −2 in the (finite) expansion of the quadratic source when r → ∞ with t − r = const. The definition of Q(n, t − r) is
N µν 2 = 1 r 2 Q µν (n, t − r) + O 1 r 3 . (3.8)
Next, in anticipation of applying the finite part of the retarded integral, we multiply (3.7) by a factor r B , and introduce r B inside the brackets using again a series of operations by parts. In this way we get many new terms, but which all involve at least one factor B coming from the differentiation of r B during the latter operations.
Thus
r B N 2 ≈ r B−2 Q(n, t − r) + 3≤k≤ℓ+2 ∂ P n L r B−k F (t − r) + O(B) . (3.9)
Applying the retarded integral on both sides of (3.9), commuting ⊓ ⊔ −1 R with ∂ P , and taking the finite part, we are left with (the finite part of) retarded integrals of three types of terms: (i) the first term in (3.9) which has radial dependence r −2 , (ii) the terms in the brackets of (3.9) having radial dependence such that 3 ≤ k ≤ ℓ the retarded integrals are also instantaneous. Therefore we can state the following result [8]: the only non-local integrals in u 2 = FP⊓ ⊔ −1 R N 2 come from Case (i), namely from the source terms whose radial dependence is r −2 , and which are denoted by
u 2 ≈ t 2 + k≥1n L r −k G(t − r) , (3.10)
where the functions G(t − r) depend instantaneously on the multipole moments M L and S L (i.e. at the retarded time t − r only), and where the first term is non-local and given by
t µν 2 = ⊓ ⊔ −1 R [r −2 Q µν (n, t − r)] . (3.11)
On the other hand the second contribution v 2 , defined by (2.11)-(2.12), involves some time anti-derivatives of quadratic products of moments. We denote these time antiderivatives by s 2 . Thus the quadratic-order metric can be written as
h 2 ≈ t 2 + s 2 + k≥1n L r −k P (t − r) , (3.12)
where t 2 is the non-local integral (3.11), where s 2 are some anti-derivatives [given by (4.12) below in the case of the quadrupole-quadrupole interaction], and where we have many instantaneous terms. See (4.13) and Table 2 below for the complete expression of h 2 in the case of the quadrupole-quadrupole interaction.
The quadrupole-quadrupole metric
We specialize the previous investigation in the case of the interaction between two quadrupole moments M ab and M cd . Thus we keep in the linearized metric (2.3) only the terms corresponding to M ab ,
h 00 1 = −2∂ ab r −1 M ab , (4.1a) h 0i 1 = 2∂ a r −1 M (1) ai , (4.1b) h ij 1 = −2r −1 M(2)
ij .
(4.1c)
(Henceforth we use the same notation for the metric constructed out of the quadrupole moment as for the complete metric involving all multipolar contributions.)
Inserting (4.1) into the quadratic source (2.5), we can work out explicitly all the terms composing the source either in the all-expanded form (3.3) or in the more elaborate form (3.7). Notably, we find that the terms with radial dependence r −2 take the classic form of the stress-energy tensor of a massless field,
Q µν (n, t − r) = k µ k ν Π(n, t − r) ,(4.2)
where k µ denotes the Minkowskian null vector k µ = (1, n i ), and where Π is given by
Π = n abcd M (3) ab M (3) cd − 4n ab M (3) ac M (3) bc + 2M (3) ab M(3)
ab .
t µν 2 = ⊓ ⊔ −1 R k µ k ν r 2 Π = − +∞ r ds dΩ ′ 4π
k ′µ k ′ν s − rn.n ′ Π(n ′ , t − s) .
kl ∂ µν ijkl {6} − 4M (3) ai M(3)aj ∂ µν ij {4} + 2M (3) ab M (3) ab ∂ µν {2} , (4.7a)(3)
where the moments in the integrand are evaluated at the time t − s, where the multiderivative operators mean for instance ∂ µν ij = ∂ µ ∂ ν ∂ i ∂ j with ∂ µ = (−∂/∂s, ∂ i ), and where we use the special notation
{p} = (s − r) p ln(s − r) − (s + r) p ln(s + r) p!r (4.7b) [see (A.4) in Appendix A].
Having the term t µν 2 , we undertake the computation of all the instantaneous terms M ab × M cd in u µν 2 [second terms in (3.10)]. The computation is straigthforward but tedious. As said above, when doing practical computations (notably by computer), the best method is the somewhat brute force method consisting of obtaining the source in the all-expanded form (3.3), and applying the (finite part of the) retarded integral on each term of (3.3) with k ≥ 3 using the formulas (A.11) and (A.13). This is simpler than working out the source in the more elaborate form (3.9), and using the manifestly instantaneous formulas (A.11) and (A.18). The brute force method has also the advantage that one can check that all the non-local integrals but those coming from the r −2 term cancel out (as well as the associated logarithms of r). The
+ w k mna(i M (6−k−m) j)b M (m) ab + x k m n ab M (6−k−m) ij M (m) ab + y k mn ab M (6−k−m) a(i M (m) j)b + z k m M (6−k−m) a(i M (m) j)a , (4.8c)
where all moments are evaluated at time t − r and where a k m , b k m , ..., z k m are purely numerical coefficients. Using the algebraic computer program Mathematica [32] we have obtained the numerical coefficients listed in Table 1.
Next we follow the second part of the construction of the metric, and compute the divergence w µ 2 = ∂ ν u µν 2 [see (2.10)-(2.12)]. To this end we need the divergence of the integral t µν 2 , which is easily evaluated by noticing that ∂ ν t µν 2 can be written, analogously to (2.10), as some residue at B = 0 of a retarded integral (because of the explicit factor B), and furthermore that the radial dependence of the integrand is merely r −3 (because it comes from the differentiation of the source term r −2 ). From (A.18), we know that when k = 3 the residue is non-zero only when the multipolarity is ℓ = 0. This yields immediately
∂ ν t µν 2 = FP B=0 ⊓ ⊔ −1 R B r r 0 B k µ r 3 Π = − 1 r dΩ 4π k µ Π(n, t − r) . (4.9)
The µ = 0 component of (4.9) is proportional to the angular average of Π, already computed in (4.5a). One can check that the µ = i component is zero. Thus,
∂ ν t 0ν 2 = − 4 5 r −1 M (3) ab M(3)
ab , (4.10a)
∂ ν t iν 2 = 0 . (4.10b)
Knowing (4.10) we can obtain w µ 2 by direct differentiation of the expressions (4.8), using the coefficients in Table 1. Again the computation is quite lengthy, but it provides us with an important check. Indeed, from (2.11) one must find that the divergence w µ 2 is a solution of the source-free d'Alembertian equation, namely ⊓ ⊔w µ 2 = 0. This test is very stringent, as a single erroneous coefficient in Table 1 would signify almost certainly its failure. Thus we determine all the tensors A L , ..., D L in (2.11). The non-zero ones are given by ab , (4.11c)
A = − 2 75 M(C ij = 4 25 M (5) a<i M j>a − 4 5 M (4) a<i M (1) j>a + 8 5 M (3) a<i M (2) j>a , (4.11d) D i = ε iab − 2 5 M (5) ac M bc − 2M (4) ac M (1) bc − 4 5 M (3) ac M (2) bc . (4.11e)
By inserting the latter values into (2.12), we obtain the components of the second part v µν 2 . In particular, we find some time anti-derivatives which define s µν 2 [see (3.12)] as
s 00 2 = 4 5 r −1 t−r −∞ duM (3) ab M (3) ab (u) , (4.12a) s 0i 2 = − 4 5 ε iab ∂ a r −1 ε bcd t−r −∞ duM (3) ce M (2) de (u) , (4.12b) s ij 2 = 0 . (4.12c)
The physical interpretation of these anti-derivatives is clear. radiation. Now the integral s 00 2 given by (4.12a) represents a small modification, due to the emission of radiation, of the initial mass M . This is clear from the comparison of (4.12a) and (2.3a), showing that there is exact agreement with the energy loss by radiation as given by the standard Einstein quadrupole formula. This result is originally due to Bonnor [10], and Bonnor and Rotenberg [4]. Similarly the integral s 0i 2 given by (4.12b) represents a modification of the total angular momentum in agreement with the quadrupole formula for the angular momentum loss. [There is no loss of total linear momentum at the level of the quadrupole-quadrupole interaction (one needs to consider also the mass octupole M abc and/or the current quadrupole
S ab ).]
The quadratic metric h µν 2 can now be completed. We add up the two contributions u µν 2 [given by (4.8) and where the non-local integrals t µν 2 and s µν 2 are given by (4.7) and (4.12), and where all the coefficients a ′k m , ..., z ′k m are listed in Table 2.
The quadrupole-quadrupole metric in the far zone
We investigate the behaviour of the quadrupole-quadrupole field h µν 2 in the far zone, near future null infinity (i.e. at large distances when we recede from the source at the speed of light). The degrees of freedom of the radiation field, at leading order in the inverse of the distance, are contained in the transverse and tracefree (TT) projection of the spatial components ij of the metric (say g TT ij ). These are the so-called observable (or radiative) multipole moments, which are "measured" in an experiment located far away from the system. The TT projection g TT ij to first order in 1/r reads
g TT ij = 4 r P ijab ∞ ℓ=2 1 ℓ! n L−2 U ijL−2 − 2ℓ ℓ + 1 n aL−2 ε ab(i V j)bL−2 + O 1 r 2 (5.1) (with G = c = 1),
where U L and V L denote the mass-type and current-type observable moments (both are functions of t − r), and where the TT projection operator is
P ijab (n) = (δ ia − n i n a )(δ jb − n j n b ) − 1 2 (δ ij − n i n j )(δ ab − n a n b ) . (5.2)
The ℓ-dependent coefficients in (5.1) are chosen so that U L and V L agree at the Let us consider first the non-local integral t µν 2 [see (4.6)-(4.7)]. As we know from (A.8), the asymptotic expansion when r → ∞, t − r = const of the retarded integral of a source with radial dependence r −2 is composed of terms 1/r n and ln r/r n . As such, t µν 2 behaves like ln r/r when r → ∞, t − r = const. The logarithm is due to the deviation of the flat cones t − r = const in harmonic coordinates from the true space-time null cones. The metric in harmonic coordinates is not of the normal Bondi-type at future null infinity [33,34]. Removal of the logarithm is done using radiative coordinates, so defined that the associated flat cones agree, asymptotically when r → +∞, with the true null cones (see e.g. [21]). This method, adopted in [9], permits to compute the observable moments in the non-local term t µν 2 . Here we follow another method, found by Thorne [15] and Wiseman and Will [16], which consists of applying first the TT projection operator (5.2) on t µν 2 . Because the TT projection kills any (linear) gauge term in the 1/r part of the metric, this method shortcuts the need of a transformation to radiative coordinates. However, one must be cautious in taking the limit r → ∞, t − r = const using (4.6). It is not allowed, for instance, to work out a leading 1/r term from the second expression in (4.6) because this term would involve a divergent integral (in accordance with the fact that the leading term is actually ln r/r). But, as pointed out in [15,16], the divergent parts of the integral cancel out after application of the TT projection, and at the end one recovers the correct result. Thus, we compute
(t ij 2 ) TT = −P ijab (n) +∞ r ds dΩ ′ 4π
n ′a n ′b s − rn.n ′ Π(n ′ , t − s) .
(t ij 2 ) TT = −P ijab (n) ℓ≥0 t−r −∞ du Π L (u) dΩ ′ 4π n ′ a n ′ b n ′ L t − u − rn.n ′ . (5.4)
We decompose the product of unit vectors n ′ a n ′ b n ′ L on the basis of STF tensors [31], we drop the terms having zero TT-projection, and we express the remaining terms using the Legendre function of the second kind Q ℓ [see (A.6c)]. Restoring the traces on the STF tensors, and droping further terms having zero TT-projection, we obtain
(t ij 2 ) TT = − 1 r P ijab ℓ≥2 ℓ(ℓ − 1) (2ℓ + 3)(2ℓ + 1)(2ℓ − 1) n L−2 t−r −∞ du Π abL−2 (u) × (2ℓ − 1)Q ℓ+2 t − u r − 2(2ℓ + 1)Q ℓ t − u r + (2ℓ + 3)Q ℓ−2 t − u r .
(5.5)
The limit at future null infinity can now be applied. Indeed it suffices to insert the expansion of Q ℓ (x) when x → 1 + as given by (A.8). As expected the limit is finite, because the terms ln(x − 1)/2 in the expansions of the Q ℓ 's cancel out. This yields [13,15,16,9] (t ij
2 ) TT = − 2 r P ijab ℓ≥2 n L−2 (ℓ + 1)(ℓ + 2) t−r −∞ du Π abL−2 (u) + O ln r r 2 . (5.6)
In the case of the quadrupole-quadrupole interaction we find [using (4.5)]
(t ij 2 ) TT = 1 r P ijab t−r −∞ du 4 7 M (3) c<a M (3) b>c − 1 15 n cd M (3) <ab M (3) cd> + O ln r r 2 (5.7)
(where the brackets <> denote the STF projection). The non-local integral (5.7)
represents the quadrupole-quadrupole contribution to the non-linear memory [14][15][16]9].
We now turn our attention to the instantaneous part of the metric. All the terms have been obtained in Section 4. We need only to apply the TT projection on the 1/r part of h ij 2 as given by (4.13) and the coefficients listed in Table 2. After several transformations using STF techniques we obtain (2) cd> + n dg ε acg ε ef <b 1 10
(h ij 2 − t ij 2 ) TT = 1 r P ijab − 2 7 M(M (5) ce M d>f − 1 2 M (4) ce M (1) d>f + O 1 r 2 (5.8)
(where the overbar on e means that the index e is to be excluded from the STF projection).
With (5.7) and (5.8) we deduce the observable moments by comparison with (5.1). The quadrupole-quadrupole interaction contributes only to the observable mass quadrupole moment U ij , mass 2 4 -pole moment U ijkl , and current octupole moment V ijk . We find
δU ij = − 2 7 t−r −∞ M (3) a<i M (3) j>a + 1 7 M (5) a<i M j>a − 5 7 M (4) a<i M (1) j>a − 2 7 M (3) a<i M (2) j>a , (5.9a) δU ijkl = 2 5 t−r −∞ M (3) <ij M (3) kl> − 21 5 M (5) <ij M kl> − 63 5 M (4) <ij M (1) kl> − 102 5 M (3) <ij M(2)
kl> ,
(5.9b) δV ijk = ε ab<i 1 10 M (5) ja M k>b − 1 2 M (4) ja M (1) k>b . (5.9c)
Note that the non-local integrals are present in U ij and U ijkl , but not in the current
moment V ijk .
Finally we add back in (5.9) the factor G and the powers of 1/c which are required in order to have the correct dimensionality. When this is done we find that δU ij is of order 1/c 5 in the post-Newtonian expansion (2.5PN order), while both δU ijkl and δV ijk are of order 1/c 3 or 1.5PN. This permits writing the complete expressions of U ij to 2.5PN order, and U ijkl , V ijk to 1.5PN order. In the case of U ij the reasoning has been done in [25], which shows that besides the quadrupole-quadrupole terms [25],
U ij (t − r/c) = M (2) ij + 2GM c 3 t−r/c −∞ du ln t − r/c − u 2b + 11 12 M (4) ij (u) + G c 5 − 2 7 t−r/c −∞ duM (3) a<i M (3) j>a (u) + 1 7 M (5) a<i M j>a − 5 7 M (4) a<i M (1) j>a − 2 7 M (3) a<i M (2) j>a + 1 3 ε ab<i M (4) j>a S b + O 1 c 6 . (5.10)
The tail integral involves a constant 11/12 computed in Appendix B of [9]. (See also [9] for the definition of the constant b.) The coefficient in front of the term M ab × S c is computed in Appendix B below. In a similar way, we find that U ijkl and V ijk involve the same types of multipole interactions, but that the terms M ab × M cd are of the same 1.5PN order as the tail terms. Thus,
U ijkl (t − r/c) = M (4) ijkl + G c 3 2M t−r/c −∞ du ln t − r/c − u 2b + 59 30 M (6) ijkl (u) + 2 5 t−r/c −∞ duM (3) <ij M (3) kl> (u) − 21 5 M (5) <ij M kl> − 63 5 M (4) <ij M (1) kl> − 102 5 M (3) <ij M (2) kl> + O 1 c 4 , (5.11) V ijk (t − r/c) = S (3) ijk + G c 3 2M t−r/c −∞ du ln t − r/c − u 2b + 5 3 S (5) ijk (u) + ε ab<i 1 10 M (5) ja M k>b − 1 2 M (4) ja M (1) k>b − 2S <i M (4) jk> + O 1 c 4 . (5.12)
The constants 59/30 and 4/3 are obtained from Appendix C in [24]. The coefficient of the term S <i M (4) jk> is computed in Appendix B below. The obtention in (5.10) of the observable quadrupole moment U ij up to the 2.5PN order yields the total power contained in the radiation field (or energy flux) complete up to the same 2.5PN order. Indeed it suffices to insert into (5.10) the intermediate quadrupole moment M ij determined in [25] as an explicit integral extending over the source at the 2.5PN order (the other needed moments being the octupole M ijk and current quadrupole S ij at 1PN order, and M ijkl and S ijk at Newtonian order, which are all known). Note that in order to obtain the wave form itself (and not only the energy flux it contains) one needs M ijk and S ij at the higher 2PN order.
Appendix 1. Formulas to compute the quadratic non-linearities
This appendix presents a unified compendium of formulas, many of them issued from previous works [20,8,9], which permit the computation of the quadratic nonlinearities (involving any interaction between two multipole moments). The source of the quadratic non-linearities takes the structure (3.3), so we present the formulas for the finite part [as defined in (2.8)] of the retarded integral of any termn L r −k F (t − r) with multipolarity ℓ and radial dependence r −k where k is an integer ≥ 2. [For practical computations it is convenient to use the source in expanded form (3.3) rather than in the more precise form (3.9).] The problem was solved in [20] which obtained a basic formula for the B-dependent retarded integral
⊓ ⊔ −1 R r r 0 Bn L r k F (t − r) = 2 k−3 (2r 0 ) B (B − k + 2)(B − k + 1) · · · (B − k − ℓ + 2) × +∞ r ds F (t − s)∂ L (s − r) B−k+ℓ+2 − (s + r) B−k+ℓ+2 r . (A.1)
This formula is valid (by analytic continuation) for all values of B in the complex plane, except possibly at integer values of B where there is a simple pole. Note that to the STF product of unit vectorsn L in the left side corresponds a STF product of spatial derivatives∂ L in the right side [31].
Our first case of interest is that of a source having k = 2. In this case (A.1)
becomes
⊓ ⊔ −1 R r r 0 Bn L r 2 F (t − r) = 1 2(2r 0 ) B B(B − 1) · · · (B − ℓ) × +∞ r ds F (t − s)∂ L (s − r) B+ℓ − (s + r) B+ℓ r , (A.2)
of which we compute the finite part in the Laurent expansion when B → 0. We repeat briefly the reasoning of [8]: the coefficient in front of the integral admits a simple pole at B = 0, but at the same time the integral vanishes at B = 0 thanks to the identity (A36) in [20] (see also (4.20a) in [8]). As a result the right side of (A.2) is finite at B = 0, in agreement with the fact that the retarded integral in its usual form (2.9) is convergent, with value
⊓ ⊔ −1 R n L r 2 F (t − r) = (−) ℓ 2 +∞ r ds F (t − s) ×∂ L (s − r) ℓ ln(s − r) − (s + r) ℓ ln(s + r) ℓ!r , (A.3)
where we have removed the reference to taking the finite part at B = 0. Note that the length scale r 0 drops out in the result (this is thanks to (4.20a) in [8]). The formula (A.3) can be generalized to the case where the angular dependence is contained in any (non-tracefree) product k α 1 ...k α ℓ of ℓ Minkowskian null vectors k α = (−1, n),
⊓ ⊔ −1 R k α 1 ...k α ℓ r 2 F (t − r) = (−) ℓ 2 +∞ r ds F (t − s) × ∂ α 1 ... ∂ α ℓ (s − r) ℓ ln(s − r) − (s + r) ℓ ln(s + r) ℓ!r , (A.4)
where the space-time derivatives in the right side are defined by ∂ α = (∂/∂s, ∂ i ). A useful alternative form of (A.3), proved e.g. in Appendix A of [9], reads
⊓ ⊔ −1 R n L r 2 F (t − r) = −n L r +∞ r dsF (t − s)Q ℓ s r , (A.5)
where Q ℓ (x) denotes the ℓth-order Legendre function of the second kind (with branch cut from −∞ to 1, so x > 1). The Legendre function is given by
Q ℓ (x) = 1 2 1 −1 P ℓ (y) dy x − y (A.6a) = 1 2 P ℓ (x)ln x + 1 x − 1 − ℓ j=1 1 j P ℓ−j (x)P j−1 (x) , (A.6b)
where P ℓ is the Legendre polynomial (see e.g. [35]). Note that by combining (A.6a) and the expansion of 1/(x − n.n ′ ) in terms of Legendre polynomials (see e.g. (A26)
in [20]), one hasn
L Q ℓ (x) = dΩ ′ 4πn ′ L x − n.n ′ , (A.6c)
where the angular integration dΩ ′ is associated with the unit direction n ′i . Combining (A.5) and (A.6c) we obtain another alternative formula,
⊓ ⊔ −1 R 1 r 2 F (n, t − r) = − +∞ r ds dΩ ′ 4π F (n ′ , t − s) s − rn.n ′ . (A.7)
This formula is valid for any function F (n, u) [not only for a function having a definite multipolarity ℓ like in (A. 5)]. It can also be recovered from the retarded integral in its usual form (2.9).
The leading terms in the expansion when r → ∞ (with t − r = const) follow from the expansion of the Legendre function when x → 1 + . The expression (A.6b) yields
Q ℓ (x) = − 1 2 ln x − 1 2 − ℓ j=1 1 j + O[(x − 1) ln(x − 1)] , (A.8a)
and with (A.5) this implies [9]
⊓ ⊔ −1 R n L r 2 F (t − r) =n L 2r +∞ 0 dλF (t − r − λ) ln λ 2r + ℓ j=1 2 j + O ln r r 2 . (A.8b)
This formula gives the leading term ln r/r and the sub-dominant term 1/r. If we try to find the leading term using (A.7) instead of (A.5) [i.e. by changing the variable s = r + λ in (A.7) and expanding the integrand when r → ∞, t − r = const], we get formally a 1/r term but in factor of a divergent integral, as expected since the leading term is actually ln r/r.
In the case of a source term corresponding to k ≥ 3, the relevant formula is obtained by performing k − 2 integrations by parts of (A.1). We get in this way
⊓ ⊔ −1 R r r 0 Bn L r k F (t − r) = r r 0 B k−3 i=0 α i (B)n L F (i) (t − r) r k−i−2 + β(B)⊓ ⊔ −1 R r r 0 Bn L r 2 F (k−2) (t − r) , (A.9)
where the second term is a retarded integral of the type studied before, and where the coefficients are
α i (B) = 2 i (B − k + 2 + i)..(B − k + 3) (B − k + 2 − ℓ + i)..(B − k + 2 − ℓ)(B − k + 3 + ℓ + i)..(B − k + 3 + ℓ)
,
(A.10a) β(B) = 2 k−2 B(B − 1)..(B − ℓ) (B − k + 2)..(B − k − ℓ + 2)(B + ℓ)..(B − k + ℓ + 3)
.
(A.10b)
The factors symbolized by dots decrease by steps of one unit from left to right. Before taking the finite part one must study the occurence of poles at B = 0 in the coefficients (A.10). Two cases must be distinguished. In the case 3 ≤ k ≤ ℓ + 2, none of the denominators in (A.10) vanish at B = 0. This implies that the second term in (A.9)
is zero at B = 0, owing to the explicit factor B in the numerator of β(B). Thus we obtain (in the case 3 ≤ k ≤ l + 2) a finite result when B → 0, which is local and independent of r 0 ,
⊓ ⊔ −1 R r r 0 Bn L r k F (t − r) | B=0 = − 2 k−3 (k − 3)!(ℓ + 2 − k)! (ℓ + k − 2)!n L × k−3 j=0 (ℓ + j)! 2 j j!(ℓ − j)! F (k−3−j) (t − r) r j+1 . (A.11)
This formula can be checked by applying the d'Alembertian operator on both sides.
When r → ∞, t − r = const, it gives
⊓ ⊔ −1 R r r 0 Bn L r k F (t − r) | B=0 = − 2 k−3 (k − 3)!(ℓ + 2 − k)! (ℓ + k − 2)!n L F (k−3) (t − r) r + O 1 r 2 . (A.12)
Next, in the case k ≥ ℓ + 3, the denominators of both α(B) and β(B) vanish at B = 0, so α(B) involves a (simple) pole, while β(B) is finite (because the pole is compensated by the factor B in the numerator). In this case (k ≥ ℓ + 3) the finite part reads
FP B=0 ⊓ ⊔ −1 R r r 0 Bn L r k F (t − r) = k−3 i=0α inL F (i) (t − r) r k−i−2 + (−) k+ℓ 2 k−2 (k − 3)! (k + ℓ − 2)!(k − ℓ − 3)! × (−) ℓ 2 ln r r 0 ∂ L F (k−ℓ−3) (t − r) r + ⊓ ⊔ −1 R n L r 2 F (k−2) (t − r) . (A.13)
The coefficients are given, when 0 ≤ i ≤ k − ℓ − 4, bỹ
α i = (−2) i (k − 3)! (k + ℓ − 2)!(k − ℓ − 3)! (k + ℓ − 3 − i)!(k − ℓ − 4 − i)! (k − 3 − i)! , (A.14a) and, when k − ℓ − 3 ≤ i ≤ k − 3, bỹ α i = 2 i (−) k+ℓ (k − 3)! (k + ℓ − 2)!(k − ℓ − 3)! (k + ℓ − 3 − i)! (k − 3 − i)!(ℓ − k + i + 3)! × k−ℓ−3 j=1 1 j − ℓ+3+i−k j=1 1 j − k−3 j=k−2−i 1 j + k+ℓ−2 j=k+ℓ−2−i 1 j . (A.14b)
[One can express (A.14) with the help of the Euler Γ-function and its logarithmic derivative ψ.] Note that (A.13) has a genuine dependence on the length scale r 0 through the logarithm of r/r 0 , which is in factor of a homogeneous solution, saŷ
∂ L G(t − r) r = (−) ℓn L ℓ j=0 (ℓ + j)! 2 j j!(ℓ − j)! G (ℓ−j) (t − r) r j+1 . (A.15)
The last term in (A.13) involves a non-local integral known from (A.3)-(A.5) or (A.7).
There exists a special combination of retarded integrals (A.13) which is purely local, finite at B = 0, logarithm-free and independent of r 0 . This combination, particularly useful in practical computations, reads
⊓ ⊔ −1 R r r 0 Bn L 2(k − 2) F (1) (t − r) r k + [(k − 1)(k − 2) − ℓ(ℓ + 1)] F (t − r) r k+1 | B=0 =n L F (t − r) r k−1 + γ∂ L F (k−ℓ−2) (t − r) r , (A.16a) where γ = (−) k 2 k−2 (k − 3)! (k + ℓ − 1)!(k − ℓ − 2)! (k + ℓ − 1)(k − ℓ − 2) − (2k − 3)(k − 2) . (A.16b)
The formula is valid when k ≥ ℓ+2; when k = ℓ+2 we recover (A.11). The second term
⊓ ⊔ −1 R B r r 0 Bn L r k F (t − r) | B=0 = (−) k 2 k−3 (k − 3)! (k + ℓ − 2)!(k − ℓ − 3)!∂ L F (k−ℓ−3) (t − r)
r .
(A. 18) As there are only simple poles, the result is zero when the power of B is strictly larger than one.
Appendix 2. The quadrupole-(current-)dipole interaction
We start from the linearized metric (2.3) written for the mass quadrupole M ab and the (static) current dipole S a ,
h 00 1 = − 2∂ ab (r −1 M ab ), (B.1a) h 0i 1 = − 2ε iab ∂ a (r −1 )S b + 2∂ a (r −1 M (1) ai ), (B.1b) h ij 1 = − 2r −1 M(2)
ij , (B.1c) and replace it into (2.5), keeping only the terms M ab × S c .
Once the source N µν 2 for this interaction is known in all-expanded form (3.3), we can apply on each of the terms the formulas of Appendix A. However, as we are only interested in the two coefficients of the terms M ab × S c in (5.10) and (5.12), it is sufficient to obtain the 1/r part of the metric h µν 2 when r → ∞, t − r = const. Thus we proceed like in Appendix C of [24], which computed the 1/r part of h µν 2 in the cases of quadrupole-monopole interactions. We recall the necessary formulas derived in [24],
FP B=0 ⊓ ⊔ −1 R r B−1∂ L r −1 F (t − r) = (−) ℓn L 2r ∞ 0 dλF (ℓ) (t − r − λ) × ln λ 2r + ℓ j=1 1 j + O ln r r 2 , (B.2a) FP B=0 ⊓ ⊔ −1 R r B ∂ i (r −1 )∂ L r −1 F (t − r) = (−) ℓ 2(ℓ + 1) (n inL − δ i<a ℓ n L−1> ) F (ℓ) (t − r) r + O 1 r 2 , (B.2b) FP B=0 ⊓ ⊔ −1 R r B ∂ ij (r −1 )∂ L r −1 F (t − r) = (−) ℓ+1 2(ℓ + 1)(ℓ + 2) − 4 3 δ ij δ ℓ0 + (n ij + δ ij )n L − 2[δ i<a ℓ n L−1> n j + δ j<a ℓ n L−1> n i ] + 2δ i<a ℓ δ ja ℓ−1 n L−2> F (ℓ+1) (t − r) r + O 1 r 2 . (B.2c)
We have added in (B.2c) the term − 4 3 δ ij δ ℓ0 (where δ ij and δ ℓ0 are Kronecker symbols) with respect to the formula (C3) given in Appendix C of [24]. Indeed this term is mistakenly missing in the formula (C3) of [24] (but it does not change any of the results derived in [24]).
The formulas (B.2) yield straightforwardly the 1/r term in the first part u µν 2 of the metric [see (2.8)]. Then we compute the (1/r term of the) divergence w µ 2 = ∂ ν u µν 2 and deduce from (2.11)-(2.12) the second part v µν 2 of the metric. We find that v µν 2 is non zero in the case M ab × S c , contrarily to the cases M × M L and M × S L studied in [24] for which the v µν 2 's are zero. The result of the computation is We thus obtain the coefficients quoted in (5.10) and (5.12) of the text.
2 by a factor (r/r 0 ) B , where B is a complex number and r 0 denotes a certain constant having the dimension of a length. When the real part of B is large enough all the power-like (except for logarithms) singularities of the multipole expansion at the spatial origin of the coordinates r = 0 are cancelled.
Owing to this factor the finite part in (2.10) is in fact a residue at B = 0, or coefficient of B −1 in the Laurent expansion. [The source term in (2.10)
this new definition. [Actually it can be checked that all intermediate and final results of this paper and paper II are independent of the choice of definition.]
ℓ
is the multipolarity (this excludes the monopolar case ℓ = 0), we know that the corresponding retarded integral is instantaneous, and given by (A.11)-(A.12). Finally, when k ≥ ℓ + 3, we have again a non-local expression, given by (A.13)-(A.14), except in special combinations like (A.16) for which the non-local integrals cancel out. The computation of u 2 = FP⊓ ⊔ −1 R N 2 can be implemented by an algebraic computer program, following the successive steps (3.1)-(3.3) and applying on each terms composing (3.3) the formulas (A.5), (A.11) and (A.13)-(A.14)
functions H are sums of products of X and time-derivatives of Y , and where we have projected the angular dependence of the terms inside the brackets on STF tensorsn L . The point is that the radial dependence of the terms inside the brackets is related to the multipolarity ℓ by 2 ≤ k ≤ ℓ + 2. This can easily be seen from (3.5), as the expansion of the multipolar wave ∂ R [r −1 Y ] is composed of a sum of terms∂ L [r −1 Y ′ ] which have 1 ≤ k ≤ ℓ + 1 [see (A.1)], yielding 2 ≤ k ≤ ℓ + 2
iii) the terms O(B) which have the structure (3.3) with any radial dependence k but carry at least one factor B. In Case (ii), the retarded integrals are given by the instantaneous expressions (A.11). In Case (iii), the retarded integrals are given by (A.18) when the power of B is one, and are zero for higher powers. So in Case (iii)
r − 2
2Q(n, t − r) in(3.8). These non-local integrals are given by (A.3)-(A.7). This result holds true only in the case of the quadratic non-linearity. In cubic and higher non-linearities, some hereditary integrals are generated by source terms with radial dependence such that k ≥ ℓ + 3.Incidentally, note that the decomposition (3.9) of the source shows that the B-dependent retarded integral ⊓ ⊔ −1 R [r B N 2 ] is finite at B =0, i.e. does not involve any pole when B → 0. Thus the finite part at B = 0 is simply equal to the value of ⊓ ⊔ −1 R [r B N 2 ] at B = 0. This can be checked from the formulas (A.3), (A.11) and (A.18) in Appendix A, which are all finite at B = 0. Here again this is a peculiarity of the quadratic approximation.We are now in the position to write down the structure of the first contribution
u 2 .
2From (3.9) and (A.3), (A.11) and (A.18) we have
Π is proportional to the power (per unit of steradian) carried by the linearized waves. In the general case, Q µν involves besides the quadrupole-quadrupole terms all the interacting terms between multipole moments with ℓ ≥ 2 and the mass monopole M . We introduce the STF multipole decomposition of Π, Π(n, u) = ℓ≥0 n L Π L (u) .
0 denotes the coefficient with multipolarity ℓ = 0 or spherical average.) With this definition, and with the help of (A.7) in Appendix A, we can write the non-local integral t 2 as
+
q k mn ijab M (6−k−m) ac M (m) bc + r k m δ ijnabcd M (6−k−m) ab M (m) cd + s k mn ij M (6−k−m) ab M (m) ab + t k m δ ijnab M (6−k−m)
Indeed the linearized metric (2.3) depends in particular on the mass monopole M and current dipole S i of the source (both are constant). Physically M and S i represent the total massenergy and angular momentum of the system before the emission of gravitational
linearized order with the ℓth time derivatives of the moments M L and S L [compare with (2.3c)].
[
Note that the TT projection as defined in (5.2) is purely algebraic. Strictly speaking it agrees with the true TT projection only when acting on the leading 1/r term.] With the multipole decomposition (4.4) we have
M
ab × M cd there is an interaction between M ab and the (static) current dipole S c also at 2.5PN order, and there is the standard contribution of tails (computed in [9]) at 1.5PN order. [In principle there is also an interaction between the mass octupole M abc and the mass dipole M d at 2.5PN order, but we have chosen M d = 0.] Thus, from (5.9a) and the equation (5.7) in
FP
in (A.16a) is an homogeneous solution fixed by our particular way of integrating the wave equation. Thus the first term by itself is a solution of the required equation, but the homogeneous solution must absolutly be taken into account when doing practical computations with this method. To leading-order when r → ∞ with t − r = const, (A.13) reads B=0 the case k = 2 given by (A.8b), the leading 1/r term is non-local. However, in contrast with (A.8b), the expansion is free of logarithms of r (this is true to all orders in 1/r), and depends irreducibly on the length scale r 0 . Finally we give the expression of the pole part when B → 0 of the retarded integral in the case k ≥ ℓ + 3. The result, easily obtained from (A.9)-(A.10), is
iab n acd M(4) cd S b + O(r −2 ) , (B.3b) h ij 2 =r −1 1 3 ε abc n ijad M (4) bd S c − 8 3 ε abc n a(i M(4)j)b S c + δ ij ε abc n ad M B.3c) we obtain the TT projection and then deduce the associated observable moments. Only the mass quadrupole and current octupole moments receive
1 )
1where G is the Newtonian gravitational constant, which plays here the role of book-keeping parameter in the non-linearity expansion. The first term in (2.1) satisfies the
vacuum Einstein equations linearized around the Minkowski metric. In harmonic (or
De Donder) coordinates this means two equations,
Table 1 ]
1and v µν
2 [given by (4.11) and (2.12)]. The local terms
are written in the same form as in (4.8). Thus,
h µν
2 = t µν
2 + s µν
2 +
6
k=1
1
r k
6−k
m=0
same expressions as in (4.8abc) but
Table 1 :
1The numerical values of coefficients entering the expression of u µν 2 as defined in(4.8).83/12
−1/7
−2/15
−7/8
−5/9
1/6
89/18
−2/5
−49/40
70/33
−205/132
−7/18
86/63
0
−113/33
4
25/6
−12/7
0
−1/8
−13/18
23/210
16/9
−8/35
−7/40
10/33
−25/33
−5/252
103/126
0
−2/33
0
69/4
−18/7
−1/5
−7/8
−65/36
0
187/36
−6/5
−7/4
15/11
−75/22
−5/252
103/126
0
291/22
69/4
−18/7
−1/5
−7/8
−65/36
0
40/9
0
−7/4
15/11
−75/22
−5/252
103/126
0
−3/11
0
27
−39/7
−9/10
−9/4
−13/4
−23/70
−49/4
−102/35
−8
45/11
−153/44
5/63
37/63
−11/210
279/22
−63/4
−9
−21/10
0
0
0
0
0
−75/4
90/11
−9/44
25/84
−29/42
−11/70
−18/11
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Our notation is the following: signature − + ++. greek indices =0,1,2,3Our notation is the following: signature − + ++; greek indices =0,1,2,3;
3; g = det (g µν ); η µν = η µν = flat metric = diag. 1,1,1latin indices =1,2,latin indices =1,2,3; g = det (g µν ); η µν = η µν = flat metric = diag (- 1,1,1,1);
L−1 , etc. . . ; n L and∂ L are the (symmetric) and trace-free (STF) parts of n L and ∂ L , also denoted by n <L>. ∂ <L> ; the superscript (n) denotes n time derivativesr = |x| = (x 2 L−1 , etc. . . ; n L and∂ L are the (symmetric) and trace-free (STF) parts of n L and ∂ L , also denoted by n <L> , ∂ <L> ; the superscript (n) denotes n time derivatives;
T (αβ) = 1 2 (T αβ + T βα ) and T (ij) = 1 2 (T ij + T ji ). T (αβ) = 1 2 (T αβ + T βα ) and T (ij) = 1 2 (T ij + T ji )
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E T Whittaker, G N Watson, A course of Modern Analysis. Cambridge Univ. Pressreprinted editionWhittaker E T and Watson G N 1990 A course of Modern Analysis reprinted edition (Cambridge Univ. Press)
| {'fraction_non_alphanumeric': 0.08848305254045415, 'fraction_numerical': 0.0442332023706466, 'mean_word_length': 3.2674765558397274, 'pattern_counts': {'":': 0, '<': 37, '<?xml version=': 0, '>': 42, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 20, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'This paper investigates the non-linear self-interaction of quadrupole gravitational waves generated by an isolated system. The vacuum Einstein field equations are integrated in the region exterior to the system by means of a post-Minkowskian algorithm. Specializing in the quadrupole-quadrupole interaction (at the quadratic non-linear order), we recover the known results concerning the nonlocal modification of the ADM mass-energy of the system accounting for the emission of quadrupole waves, and the non-local memory effect due to the re-radiation by the stress-energy distribution of linear waves. Then we compute all the local (instantaneous) terms which are associated in the quadrupole-quadrupole metric with the latter non-local effects. Expanding the metric at large distances from the system, we obtain the corresponding radiation-field observables, including all non-local and transient contributions. This permits notably the completion of the observable quadrupole moment at the 5/2 post-Newtonian order.', 'arxivid': 'gr-qc/9710037', 'author': ["Luc Blanchet \nDépartement d'Astrophysique Relativiste et de Cosmologie (UPR 176 du CNRS)\nObservatoire de Paris\n92195Meudon CedexFrance\n"], 'authoraffiliation': ["Département d'Astrophysique Relativiste et de Cosmologie (UPR 176 du CNRS)\nObservatoire de Paris\n92195Meudon CedexFrance"], 'corpusid': 119332401, 'doi': '10.1088/0264-9381/15/1/008', 'github_urls': [], 'n_tokens_mistral': 22175, 'n_tokens_neox': 19884, 'n_words': 11925, 'pdfsha': '071b87a14060c246865933bd23bb32ddf33d61a4', 'pdfurls': ['https://export.arxiv.org/pdf/gr-qc/9710037v1.pdf'], 'title': ['Quadrupole-quadrupole gravitational waves', 'Quadrupole-quadrupole gravitational waves'], 'venue': []} |
arxiv |
Transverse momentum broadening and gauge invariance
20 Sep 2012
Antonio Vairo
Physik-Department
Technische Universität München
James-Franck-Str. 185748GarchingGermany
Transverse momentum broadening and gauge invariance
20 Sep 2012Jet Quenching, SCET
In the framework of the soft-collinear effective theory, we present a gauge invariant definition of the transverse momentum broadening probability of a highly-energetic collinear quark in a medium and consequently of the jet quenching parameterq.
INTRODUCTION
Jet quenching occurs when in a heavy-ion collision an energetic parton propagating in one light-cone direction loses sufficient energy that few high momentum hadrons are seen in the final state, where in the vacuum there would be a jet. In this context, a parton is considered highly energetic when its momentum Q is much larger than any other energy scale, including those characterizing the medium. Jet quenching, which has been observed at RHIC [1] and at LHC [2], manifests itself in many ways. In particular, the hard partons produced in the collision lose energy and change direction of their momenta. This last phenomenon goes under the name of transverse momentum broadening. A way to describe the transverse momentum broadening is by means of the probability, P(k ⊥ ), that after propagating through the medium for a distance L (→ ∞) the hard parton acquires a transverse momentum k ⊥ (see Fig. 1
): d 2 k ⊥ (2π) 2 P(k ⊥ ) = 1.
A related quantity is the jet quenching parameter,q, which is the mean square transverse momen-tum picked up by the hard parton per unit distance traveled:
q = 1 L d 2 k ⊥ (2π) 2 k 2 ⊥ P(k ⊥ ) .(1)
In the following, we will review the derivation of a gauge invariant expression for P(k ⊥ ) in the case of the propagation of a highly energetic quark. The original detailed derivation can be found in [3].
SCALES AND EFFECTIVE FIELD THEORY
We consider a highly energetic quark of momentum Q propagating along one light-cone directionn = (1, 0, 0, −1)/ √ 2. The light-cone momentum coordinates are q + =n · q, q − = n · q, with n = (1, 0, 0, 1)/ √ 2, and q ⊥ , which is the momentum component that is transverse with respect to the light-cone directions n andn, see Fig. 2. If the quark propagates in a medium whose energy scales are much smaller than Q, then we can define a parameter λ ≪ 1, which is the ratio of the energy scale characterizing the medium and Q. This small parameter may serve to classify the different modes of the propagating quark and interacting gluons. We assume that the quark, after traveling along the medium, undergoes a transverse momentum broadening of order Qλ . If the virtuality of the quark is small, i.e. of order Q 2 λ 2 , then the parton has momentum q ∼ Q(λ 2 , 1, λ ) and is called collinear. We set up to describe the propagation of a single collinear quark in the medium. A collinear quark may scatter in the medium with ultrasoft gluons, whose momenta scale like Q(λ 2 , λ 2 , λ 2 ), with Glauber gluons , whose momenta scale like Q(λ 2 , λ , λ ) or Q(λ 2 , λ 2 , λ ) or with soft gluons scaling like Q(λ , λ , λ ) through the emission of virtual hard-collinear quarks scaling like Q(λ , 1, λ ). The relevant degrees of freedom are shown in Fig. 3.
The effective field theory that describes the propagation of a collinear quark in then light-cone direction is the soft-collinear effective theory (SCET) [4] coupled to Glauber gluons [5]. After rescaling the quark field by ξn → e −iQx + ξn, the Lagrangian may be organized as an expansion in λ : where
Ln =ξn in /n · D ξn +ξn D 2 ⊥ 2Q n / ξn +ξn i gF µν ⊥ 4Q γ µ γ ν n / ξn + . . . ,(2)iD µ = i∂ µ + gA µ and F µν ⊥ = i[D µ ⊥ , D ν ⊥ ]/g is the gluon field strength.
The fragmentation of the collinear quark into collinear partons is not taken into account by the above Lagrangian; a preliminary study of this effect can be found in [6].
MOMENTUM BROADENING IN COVARIANT GAUGES
Collinear and hard-collinear quark fields, ξn(x), scale in the same way. The operators n · ∂ and ∇ ⊥ scale like Qλ 2 and Qλ respectively when acting on a collinear field ξn(x), and both scale like Qλ when acting on a hard-collinear field ξn(x). Soft gluon fields scale like Qλ and ultrasoft gluon fields scale like Qλ 2 , for they are homogeneous in the soft and ultrasoft scale respectively. In contrast, the power counting of Glauber gluons depends on the gauge. The equations of motion require A + (x) to scale like Qλ 2 . In a covariant gauge, if the gluon field is coupled to a homogeneous soft source, this also implies that A ⊥ (x) ∼ Qλ 2 . The leading order Lagrangian in λ is then
Ln =ξn in /n · D ξn +ξn ∇ 2 ⊥ 2Q n / ξn .(3)
Because ultrasoft gluons decouple at lowest order from collinear quarks trough the field
redefinition ξn(x) → P exp ig x − −∞ dyn · A us (x + , y, x ⊥ ) ξn(x),
where P stands for the path ordering operator, only one relevant vertex involving either Glauber or soft gluons has to be taken into account:
A + = igT anµ n /.
The transverse momentum broadening probability is then given by the imaginary part of the differential scattering amplitude q =Q(0,1,0) 0 q =Q(0,1,0) 0 (k ,k ,k )
+ _ = P(k ⊥ ) ,
taken for k ⊥ = 0 and normalized by the number of collinear quarks in the medium. The scattering amplitude has the form (evaluated on a background of gluon fields)
∏ i d 4 q i (2π) 4 · · · iQ 2Qq + 2 − q 2 2⊥ + iεn / A + (q 2 − q 1 )n / iQ 2Qq + 1 − q 2 1⊥ + iεn / A + (q 1 − q 0 )n / ξn(q 0 ),
where the Dirac spinor ξn(q 0 ) satisfiesn / ξn(q 0 ) = 0 and is normalized as ξ † n (q 0 ) ξn(q 0 ) = √ 2Q. For Glauber gluons, the free propagator may be approximated by (e.g. in Feynman gauge)
D µν (k) = D(k 2 )g µν ≈ D(k 2 ⊥ )g µν ,(4)
which implies that the scattering amplitude in coordinate space is at leading order
dy + d 2 y ⊥ ∏ i dy − i · · · θ (y − 3 − y − 2 )A + (y + , y − 2 , y ⊥ ) θ (y − 2 − y − 1 )A + (y + , y − 1 , y ⊥ )ξn(q 0 ) .(5)
The same result also holds when considering the case of (hard-)collinear quarks interacting with soft gluons.
x − = −∞ x − = ∞ 0⊥ x⊥P(k ⊥ ) = d 2 x ⊥ e ik ⊥ ·x ⊥ 1 N c Tr W † [0, x ⊥ ]W [0, 0] ,(6)
where . . . denotes a field average. The Wilson lines of (6) are shown in Fig. 4. The relation between jet quenching and Wilson lines oriented along one of the light-cone directions was derived within different approaches in [7] and within SCET in [8]. Clearly the above expression is, in general, not gauge invariant (e.g. in the light-cone gauge A + = 0, one would have W = 1).
MOMENTUM BROADENING IN LIGHT-CONE GAUGE
In the light-cone gauge A + = 0, the free gluon propagator reads
D µν (k) = D(k 2 ) g µν − k µnν + k νnµ [k + ] .(7)
For Glauber gluons k ⊥ /[k + ] ∼ 1/λ , which leads to on enhancement of order λ in the singular part of the propagator. Moreover, because of the k ⊥ /[k + ] singularity, one can write [9]
A ⊥ (x + , x − , x ⊥ ) = A cov ⊥ (x + , x − , x ⊥ ) + A sin ⊥ (x + , x − , x ⊥ ), where A cov ⊥ (x)
contributes to the non-singular part of the propagator and vanishes at
x − = ±∞, while A sin ⊥ (x + , x − , x ⊥ ) = θ (x − )A ⊥ (x + , ∞, x ⊥ ) + θ (−x − )A ⊥ (x + , −∞, x ⊥ ) with A ⊥ (x + , ±∞, x ⊥ ) = −∇ ⊥ φ ± (x + , x ⊥ ).
The field A ⊥ does not vanish at infinity where it becomes pure gauge, for the field tensor does (the energy of the gauge field is finite).
In the A + = 0 gauge, the scaling of the Glauber fields appearing in the Lagrangian changes to A cov ⊥ ∼ Qλ 2 and A sin ⊥ ∼ Qλ . The leading order Lagrangian in λ is then
Ln =ξn in /n · ∂ ξn +ξn (∇ ⊥ + igA sin ⊥ ) 2 2Q n / ξn ,(8)
where gluons are just Glauber gluons. The relevant vertices are now two
A q q' = −ig q ′ ⊥ · A sin ⊥ (q ′ − q) + A sin ⊥ (q ′ − q) · q ⊥ 2Q n /, A A q q'' = − ig 2 2Q d 4 q ′ (2π) 4 A sin i ⊥ (q ′′ − q ′ )A sin i ⊥ (q ′ − q)n /.
From the vertices one constructs the scattering amplitude (on the left of the cut)
q =Q(0,1,0) 0 1 2 3 4 5 n−1 n (k /2Q,k ,k ) _ 2 = G n (k − , k ⊥ ) .
The function G n is a convolution of G + n− j , which involves only fields at x − = ∞ and G − j , which involves only fields at x − = −∞:
G n (k − , k ⊥ ) = n ∑ j=0 d 4 q (2π) 4 G + n− j (k − , k ⊥ , q) iQn / 2Qq + − q 2 ⊥ + iε G − j (q) .(9)
The computation is done by solving recursively the equation (analogously for G + n (q))
G − n (q) = d 4 q ′ (2π) 4 G − n−1 (q ′ ) q' q + d 4 q ′′ (2π) 4 G − n−2 (q ′′ ) q'' q ,(10)
writing the differential amplitude as
1 L 3 √ 2Q dk + 2π dk − 2π 2π Q δ (2Qk + − k 2 ⊥ )ξn(q 0 ) G † m (k − , k ⊥ )n / G n (k − , k ⊥ ) ξn(q 0 ) ,
and eventually summing over all m and n. The expression of the transverse momentum broadening probability in light-cone gauge then reads
P(k ⊥ ) = d 2 x ⊥ e ik ⊥ ·x ⊥ 1 N c Tr T † [0, −∞, x ⊥ ]T [0, ∞, x ⊥ ]T † [0, ∞, 0]T [0, −∞, 0] ,(11)
where
T [0, ±∞, x ⊥ ] = P exp −ig 0 −∞ ds l ⊥ · A ⊥ (0, ±∞, x ⊥ + sl ⊥ ) (for the definition of
T see also [10]). The transverse vector l ⊥ is arbitrary. The Wilson lines of (11) are shown in Fig. 5.
x⊥ 0⊥ −∞⊥ x − = −∞ x − = ∞ FIGURE 5.
Wilson lines contributing to P(k ⊥ ) in light-cone gauge. We have chosen l ⊥ x ⊥ .
GAUGE INVARIANT MOMENTUM BROADENING
Combining the results in covariant and light-cone gauge for L → ∞, we obtain a gauge invariant expression for P(k ⊥ ), which reads
P(k ⊥ ) = d 2 x ⊥ e ik ⊥ ·x ⊥ 1 N c Tr T † [0, −∞, x ⊥ ]W † [0, x ⊥ ]T [0, ∞, x ⊥ ] × T † [0, ∞, 0]W [0, 0]T [0, −∞, 0] .(12)
The Wilson lines of (12) are shown in Fig. 6. Note that the fields are path ordered but not time ordered as in usual Wilson loops [11]. This difference should not be surprising x⊥ 0⊥
−∞⊥ since it reflects the fact that P(k ⊥ ) describes the propagation of a single particle, while usual Wilson loops describe the propagation of a particle-antiparticle pair. The expression of P(k ⊥ ) may be simplified into
x − = −∞ x − = ∞P(k ⊥ ) = d 2 x ⊥ e ik ⊥ ·x ⊥ 1 N c Tr [0, x ⊥ ] − W † [0, x ⊥ ] [x ⊥ , 0] + W [0, 0] ,(13)
where
[x ⊥ , y ⊥ ] ± = P exp −ig 0 1 ds (y ⊥ − x ⊥ ) · A ⊥ (0, ±∞, x ⊥ + s(y ⊥ − x ⊥ )) , because
contiguous adjoint lines cancel, fields separated by space-like intervals commute and because of the cyclicity of the trace. The Wilson lines of (13) are shown in Fig. 7.
x⊥
0⊥
−∞⊥
x − = −∞ x − = ∞
FIGURE 1 .
1transverse momentum with respect to the original hard parton direction Kinematics of transverse momentum broadening.
FIGURE 2 .
2Parton momentum in light-cone coordinates.
FIGURE 3 .
3Relevant degrees of freedom.
FIGURE 4 .dy
4Wilson lines contributing to P(k ⊥ ) in a covariant gauge.Because(5)is just a term in the expansion of the Wilson line W [y + , y ⊥ ] = − A + (y + , y − , y ⊥ ) for L → ∞, the transverse momentum broadening probability of a quark in covariant gauges is given by
FIGURE 6 .
6Wilson lines contributing to the gauge invariant expression of P(k ⊥ ) given in(12). We have chosen l ⊥ x ⊥ . The fields at x − = ∞ are contiguous while those at x − = −∞ are not.
FIGURE 7 .
7Wilson lines contributing to the gauge invariant expression of P(k ⊥ ) given in (13). The fields in (0, −∞, 0) are not contiguous.The obtained expression for P(k ⊥ ) does not depend on l ⊥ . It is also gauge invariant. In fact, under a gauge transformation Ω,Tr{T † [0, −∞, x ⊥ ]W † [0, x ⊥ ] · · · T [0, −∞, 0]} transforms to Tr {Ω[0, −∞, −∞ l ⊥ ]T † [0, −∞, x ⊥ ]W † [0, x ⊥ ] · · · T [0, −∞, 0]Ω † [0, −∞, −∞ l ⊥ ]},which is equal to the original expression, Tr{T † [0, −∞, x ⊥ ]W † [0 + , x ⊥ ] · · · T [0, −∞, 0]}, after noticing that the fields in Ω[0, −∞, −∞ l ⊥ ] commute with all the others (because of space-like separations) and after using the cyclicity of the trace.
Having derived the transverse momentum broadening probability, P(k ⊥ ), we are in the position to write the jet quenching parameterq in a manifestly gauge invariant fashion:where the fields F +iacting on the Wilson lines, and U. We recall that the above expression holds when the integral over k ⊥ has an ultraviolet cutoff of order Qλ , which is the size of the transverse momentum broadening that we have been considering.ACKNOWLEDGMENTSI thank Michael Benzke, Nora Brambilla and Miguel Angel Escobedo for collaboration on the work presented here. I acknowledge financial support from the DFG cluster of excellence "Origin and structure of the universe" (http://www.universe-cluster.de).
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| {'fraction_non_alphanumeric': 0.10939342329359361, 'fraction_numerical': 0.045788025677977205, 'mean_word_length': 3.119535887749595, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, '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': 'In the framework of the soft-collinear effective theory, we present a gauge invariant definition of the transverse momentum broadening probability of a highly-energetic collinear quark in a medium and consequently of the jet quenching parameterq.', 'arxivid': '1209.4486', 'author': ['Antonio Vairo \nPhysik-Department\nTechnische Universität München\nJames-Franck-Str. 185748GarchingGermany\n'], 'authoraffiliation': ['Physik-Department\nTechnische Universität München\nJames-Franck-Str. 185748GarchingGermany'], 'corpusid': 119221168, 'doi': '10.1063/1.4763541', 'github_urls': [], 'n_tokens_mistral': 6461, 'n_tokens_neox': 5497, 'n_words': 2959, 'pdfsha': 'f00f6bf07e7693e8fcc3faf51c162121aafc912c', 'pdfurls': ['https://arxiv.org/pdf/1209.4486v1.pdf'], 'title': ['Transverse momentum broadening and gauge invariance', 'Transverse momentum broadening and gauge invariance'], 'venue': []} |
arxiv |
A new measurement method of electrode gains for orthogonal symmetric type beam position monitor*
Jun-Ying Zou
University of Science and Technology of China
230029HefeiP. R. China
W U Fang-Fang
University of Science and Technology of China
230029HefeiP. R. China
Yong-Liang Yang
University of Science and Technology of China
230029HefeiP. R. China
Bao-Gen Sun
University of Science and Technology of China
230029HefeiP. R. China
Zhou Ze
University of Science and Technology of China
230029HefeiP. R. China
A new measurement method of electrode gains for orthogonal symmetric type beam position monitor*
Submitted to 'Chinese Physics C' _ * Supported by the Natural Science Foundation of China (11175173, 11375178, 11105141) 1) PACS: 29.20.Ej, 29.90.+rbeam position monitorelectrode gaincalibrationorthogonal symmetric
The new beam position monitor (BPM) system of the injector at the upgrade project of Hefei Light Source (HLS II) has 19 stripline beam position monitors. Most consist of four orthogonal symmetric stripline electrodes. The differences in electronic gain and mismachining tolerance can cause the change of the beam response of the BPM electrodes. This variation will couple the two measured horizontal positions in order to bring the measuring error. To alleviate this effect, a new technique to measure the relative response of the four electrodes has been developed. It is irrelevant to the beam charge and the related coefficient can be theoretical calculated. The effect of electrodes coupling on this technique is analyzed. The calibration data is used to fit the gain for all 19 injector beam position monitors. The results show the standard deviation of the distribution of measured gains is about 5%.
Introduction
Recently, Hefei Light Source (HLS) is being upgraded to HLS II. The injector beam position monitoring (BPM) system is composed of 19 beam position monitors, mostly are regular stripline type BPM.
They are precisely calibrated and carefully installed in place [1]. We have developed a new technique that provides a measure of the relative gain of the four stripline electrodes.
The method we developed is similar to the technique of D.L. Rubin [2]
Derivation of new expression
As Fig. 1
I V Z Z Z Z b I V Z Z Z Z b I V Z Z Z Z b I V Z Z Z Z b (1)
Which, I beam is the beam charge, is the electrodes opening angle, b is the distance from center of the beam position monitor to the electrodes. Z 1x , Z 1y , Z 2 , Z 3x , Z 3y and x y
x y x y Z Z b b x y Z b x y x Z b b x y y Z b b x y x y Z b (2)
Which, x 0 and y 0 are the positions of the beam.
When the beam is near the center of the beam pipe, x 0 and y 0 are small compared to b. In this case, the third order and up can be ignored, so the electrode signals can be approximated as a quadratic polynomial expansion 1 2
1 2 1 2 1 2 (1 ) 2 (1 ) 2 (1 ) 2 (1 ) 2 beam R x beam T y beam L x beam B y I V Z Z b I V Z Z b I V Z Z b I V Z Z b (3)
Taking sums and differences of Eq. (3) gives
2 1 1 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 R L T B mn R L T B R L T B m R L T B x y x y R L T B n R L T B x y x y V V V V Z V V V V V V V V V V V V Z Z Z Z Z Z Z Z V V V V V V V V Z Z Z Z Z Z Z Z (4)
Also, ignore the third order and up we can simply get
2 1 1 1 1 mn m x y n x y Z Z Z Z Z (5)
Combining Eq. (2) and Eq. (5) to eliminate x 0 and y 0 gives an expression that simply relates the electrode
signals 4 tan / 2 mn mm m n mm k k (6)
In this case, k mn is a constant only determined by the electrodes opening angle of BPM. To the regular injector stripline BPM of HLS II, is 45 degree, so we can simply calculate that k mn is 0.474. Eq. (6) not only shows that Σ mn is proportional to the product Σ m Σ n , more importantly, the equation is irrelevant to the beam charge, which is useful when fit the gain errors using real beam.
Simulation
To simulate the connection between Σ mn and Σ m Σ n , we used a finite element code to create a map of each electrode response as a function of beam position [4].
The simulated beam was moved in a 5 mm 5 mm square area with a step of 0.5 mm. Σ mn and Σ m Σ n was calculated with the exact response of electrodes at every beam positions. The product Σ m Σ n is plotted versus Σ mn in Fig. 2. In Fig.2, the points deviation from the straight line only slightly appears at large amplitudes, shows the extent to which the higher than second order terms can be ignored.
We see that our quadratic term approximation is good, the product Σ m Σ n approximated to Σ mn , which fits the form of Eq. (5) with slightly deviation at large amplitudes.
R R L T B L R L T B T R L T B T R L T B V V K V K V K V V K V V K V K V V K V K V V KV V K V K V KV V (7)
In this case, we calculate Eq.
2 R L T B R L T B R L T B m R L T B R L T B n R L T B mn x y x y V V V V K K Z K K V V V V K Z Z V V V V K K V V V V K Z Z V V V V K K V V V V (8)
Electrode gain fit with new expression
We assume the deviations from Eq. (9) are determined by the gain variations between different electrodes. We use a nonlinear least squares fit to get the electrode gains (g R , g L , g T and g B ). The merit function to be minimized is
2 2 1 R R L L T T B B R R L L T T B B n R R L L T T B B mn i R R L L T T B B R R L L T T B B R R L L T T B B g V g V g V g V g V g V g V g V g V g V g V g V k g V g V g V g V g V g V g V g V g V g V g V g V (10)
2 has a minimum for the best fit gains (g R , g L , g T and g B ) and mn k . To make sure the value of the denominator is not zero, we fit the same data four times, each time we set one of the electrode gains to 1, and then average the results.
Fitting the calibration data
All the 19 HLS II injector stripline BPMs are calibrated at test bench, using a tungsten filament to simulate the beam [1]. The filament was moved in a 5 mm To verify the effectiveness above method, the table 1 shows the main geometric calibration parameters change of the LA-BD-BPM03 before and after gain fitting.
Compared to the geometric coefficient before gain fitting, the geometric coefficient are closer to the theoretical value 7.55mm after gain fitting. Thus, the above method is effective. Fig. 5. The distribution of fitted gains is shown in Fig. 6. We can see most electrodes gain errors are between 0.9 and 1.1. Note that the average value of parameter mn k is 0.530, which is a little bit larger than the theoretical value.
Conclusion
We have derived a relationship among the intensities of the four electrodes of orthogonal symmetrical type beam position monitor. The relationship is better than the previous study because it is irrelevant to the beam charge and the related coefficient can be theoretical calculated.
We analyze the effect of electrodes coupling on the relationship. We show how the relationship can be used to make a beam based measurement of the relative gains of the four electrodes. We have used the calibration data to fit the gain for all 19 injector beam position monitors. The standard deviation of the distribution of measured gains is about 5%, consistent with the specifications of the system electronics. We will use the real beam data of HLS II injector to fit the electrodes gain, this can be implemented as a part of the standard measurements of the HLS II injector BPM system.
Fig. 1 .
1et al. It also based on the fact that, in a four electrodes beam position monitor, the position of the beam is overdetermined. The relative gains of the electrodes can be calculated by measuring the electrode signal at many different beam positions. The method of Rubin is based on the image theory, which requires the geometry of the four BPM electrodes be diagonal symmetric. The geometry of a typical HLS II beam position monitor is as in Fig. 1. The four electrodes are orthogonal symmetric, which does not apply to Rubin's method, so we develop a new technique to measure the relative gains of this type of four electrodes beam position monitor. Through the analysis of the theoretical electrode signal induced by the beam, we find a new expression only related to the electrode signal. This expression can be used to fit the electrode gain errors, within each fitting procedure, four unknown parameters are fitted: three button gains and a geometry scaling factor. HLS II injector beam position monitor
Fig. 2 .Fig. 3 .
23Σ m Σ n vs Σ mn for points on a 5 mm 5 mm grid with simulated electrodes signal vs beam position. In practice, the four electrodes do not have the same gain, then the connection between electrodes defined by Eq. (6) will fail. We simulate the effect of gain errors by reducing the signal on electrodes 4 by 10%, that is, the gains (1:4) = 1.0, 1.0, 1.0, 0.9. Fig. 3 shows the Σ m Σ n vs Σ mn with the data under this condition, ╋ indicates the coordinate (0,0). The data is no longer linear and it is offset from zero. Σ m Σ n vs Σ mn for points on a 5 mm 5 mm grid with electrode intensity computed with the nonlinear map.4 Electrodes coupling effectEq. (6) is based on the assumption that the four electrodes are independent to each other. In fact, there is coupling effect between the electrodes. Each electrode can be induced to signals from other electrodes. We set K 1 as the coupling coefficient of opposite electrode, K 2 as the coupling coefficient of adjacent electrode. So the four
Fig. 4 .
4×5 mm square area with a step of 0.5 mm. We collect the electrodes signal data on each simulated beam position using Libera Brilliance Single Pass[5].An example of fitted data based on Eq. (10) at one BPM (LA-BD-BPM03) is shown in Fig. 4. In Fig. 4, the open circles are the raw electrode data, the crosses are the electrode data corrected with the fitted gains, the ╋ indicates the coordinate (0,0). The fitted gains(g R , g L , g T and g B ) respectively are 0.882, 1.122, 0.923 and 1.122. The resultshows the data has better linearity and passes through zero after gain fitting. m n vs mn for a calibration data at LA-BD-BPM03.
Fig. 5 .
5Fitted gains and parameter mn k from calibration data for all 19 injector beam position monitors.
Fig. 6 .
6Distribution of fitted gains for the data plotted inFig. 5.
shown, the four electrodes of a HLS II typical BPM are 90 degrees away from each other. By ignoring the influence of bunch size, the electrode signal of this type of BPM can be represented by[3]
1
2
3
4
1
2
3
4
1
2
3
4
1
2
3
4
1
...
2
1
. . .
2
1
. . .
2
1
. . .
2
beam
R
x
x
beam
L
x
x
beam
T
y
y
beam
B
y
y
Table 1 .
1The change of calibration parameters before and after gain fitting Gains for 19 BPMs are shown inbefore gain
fitting
after gain
fitting
Position
x
y
x
y
Offset/mm
-0.19
-0.15 -0.13
-0.01
Geometric
coefficient /mm
7.60
7.41
7.60
7.45
Calibration of Beam Position Monitors in the Injector of HLS II. Yang Zou Jun-Ying, Sun Yong-Liang, Bao-Gen, Proceedings of IPAC2013. IPAC2013Shanghai, ChinaZOU Jun-Ying, YANG Yong-Liang, SUN Bao-Gen, et al. Calibration of Beam Position Monitors in the Injector of HLS II. Proceedings of IPAC2013, Shanghai, China, 2013, 568-570
Beam Based Measurement of Beam Position Monitor Electrode Gains. D L Rubin, M Billing, R Meller, Phys. Rev. ST Accel. Beams. 139Rubin D L, Billing M, Meller R, et al. Beam Based Measurement of Beam Position Monitor Electrode Gains. Phys. Rev. ST Accel. Beams, 2010, 13(9): 092802-1~092802-6
New Methods of Beam Position Monitors for Measurement of Quadrupole Component. High Power Laser and Particle Beams. Li Peng, Bao-Gen, Luo Qing, 20LI Peng, SUN Bao-gen, LUO Qing, et al. New Methods of Beam Position Monitors for Measurement of Quadrupole Component. High Power Laser and Particle Beams, 2008, 20(4): 573-578
Matlab Code for BPM Button Geometry Computation. A Olmos, F Pérez, G Rehm, Proceedings of DIPAC 2007. DIPAC 2007Venice, ItallyOlmos A, Pérez F, Rehm G. Matlab Code for BPM Button Geometry Computation. Proceedings of DIPAC 2007, Venice, Itally, 2007, 186-188
Application of Libera Brilliance Single Pass at NSRL Linac BPM System. Zou Jun-Ying, Jia, Sun Bao-Gen, Proceedings of IPAC2011. IPAC2011San Sebastián, SpainZOU Jun-Ying, FANG Jia, SUN Bao-Gen, et al. Application of Libera Brilliance Single Pass at NSRL Linac BPM System. Proceedings of IPAC2011, San Sebastián, Spain, 2013, 1284-1286
| {'fraction_non_alphanumeric': 0.06442789849566935, 'fraction_numerical': 0.03168211518006382, 'mean_word_length': 3.0917003419334783, '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': 119, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The new beam position monitor (BPM) system of the injector at the upgrade project of Hefei Light Source (HLS II) has 19 stripline beam position monitors. Most consist of four orthogonal symmetric stripline electrodes. The differences in electronic gain and mismachining tolerance can cause the change of the beam response of the BPM electrodes. This variation will couple the two measured horizontal positions in order to bring the measuring error. To alleviate this effect, a new technique to measure the relative response of the four electrodes has been developed. It is irrelevant to the beam charge and the related coefficient can be theoretical calculated. The effect of electrodes coupling on this technique is analyzed. The calibration data is used to fit the gain for all 19 injector beam position monitors. The results show the standard deviation of the distribution of measured gains is about 5%.', 'arxivid': '1401.0136', 'author': ['Jun-Ying Zou \nUniversity of Science and Technology of China\n230029HefeiP. R. China\n', 'W U Fang-Fang \nUniversity of Science and Technology of China\n230029HefeiP. R. China\n', 'Yong-Liang Yang \nUniversity of Science and Technology of China\n230029HefeiP. R. China\n', 'Bao-Gen Sun \nUniversity of Science and Technology of China\n230029HefeiP. R. China\n', 'Zhou Ze \nUniversity of Science and Technology of China\n230029HefeiP. R. China\n'], 'authoraffiliation': ['University of Science and Technology of China\n230029HefeiP. R. China', 'University of Science and Technology of China\n230029HefeiP. R. China', 'University of Science and Technology of China\n230029HefeiP. R. China', 'University of Science and Technology of China\n230029HefeiP. R. China', 'University of Science and Technology of China\n230029HefeiP. R. China'], 'corpusid': 118370180, 'doi': '10.1088/1674-1137/38/12/127002', 'github_urls': [], 'n_tokens_mistral': 5597, 'n_tokens_neox': 4684, 'n_words': 2805, 'pdfsha': '9997750699d49e910d7dbda5a248eacbf8eb5f26', 'pdfurls': ['https://export.arxiv.org/pdf/1401.0136v1.pdf'], 'title': ['A new measurement method of electrode gains for orthogonal symmetric type beam position monitor*', 'A new measurement method of electrode gains for orthogonal symmetric type beam position monitor*'], 'venue': []} |
arxiv |
COMPARATIVE ANALYISIS ON SOME POSSIBLE PARTNERSHIP SCHEMES OF GLOBAL IP EXCHANGE PROVIDERS
March 2014
David Gunawan
Development of Infrastructure
Telkom Indonesia Research & Development Centre
Research, BandungIndonesia
Karno Budiono
Development of Infrastructure
Telkom Indonesia Research & Development Centre
Research, BandungIndonesia
COMPARATIVE ANALYISIS ON SOME POSSIBLE PARTNERSHIP SCHEMES OF GLOBAL IP EXCHANGE PROVIDERS
International Journal of Computer Networks & Communications (IJCNC)
62March 201410.5121/ijcnc.2014.620999IPX (IP eXchange)GRX (GPRS Roaming eXchange)LTEroaminginterconnectionpeeringhostedwhite label
IPX (IP eXchange) is GSMA's proposal for IP interconnection model which supports multi services to offerend-to-end QoS, security, interoperability, SLAs through a dedicated connection. It provides a commercial and technical solution to manage IP traffic and follows the GSMA's 4 key IP interworking principle such as openness, quality, cascading payments, and efficient connectivity. In order to get global IPX reachability, it is possible for an IPX provider to build partnership with other global IPX providers in business and network configuration. There are some possible partnership schemes between IPX providers such as peering mode, semi-hosted mode, full-hosted mode, or combination between these modes. The implementation of the schemes will be case-by-case basis with some considerations based on (but not limited to) IPX Provider's network asset & coverage, services & features offer, commercial offer, and customers. For an IPX provider to become competitive in IPX business and become a global IPX hub, somevalue added should be considered such as cost efficiency and great network performance. To achieve it, an IPX provider could implement some strategies such as build network sinergy between them and partners to develop IPX Service as single offering, offer their customers with bundled access network and services. An IPX provider should also consider their existing customer-based that can be a benefit to their bargaining position to other potential IPX provider partners to determine price and business scheme for partnership.
INTRODUCTION
Nowadays, as international telecommunication business increases in means of service types, traffic, and operator revenues, then IPX become one of telecommunication operator's option as an interconnection model that support multi services for their customers. A number of services such as roaming data 2G/ 3G/ LTE, roaming signaling 2G/ 3G/ LTE, SMS/ MMS interworking, RIM connection, WiFi roaming, bilateral IPX services, Voice over IPX, HD Voices, and RCS roaming can be delivered through IPX connection. Based on GSMA definition, IPX is a telecommunications interconnection model for the exchange of IP-based traffic between customers of separate mobile and fixed operators as well as other types of service provider (such as ISP), via IP based network-to-network interface. In the interconnection context, IPX is used to mean an interconnection at the service level (not at the network level). It also refers to the collection of all the interconnected IPX provider's networks, a subset of the inter-service provider IP backbone. The IPX network includes inter-service provider IP backbone which comprises the interconnected networks of various IPX providers. An IPX provider is a provider that offers IPX services, meanwhile a service provider is a mobile, fixed operator, or other types of operator connecting to inter-service provider IP backbone for roaming and/or interworking purposes. As the next generation interconnect solution, IPX have a number characteristics, such as:
• Openess, means that any potential players in the delivery of IP Services (MNOs, FNOs, Carriers and ISPs) has the freedom of choice to be involved • Quality, means reliable & secure delivery of services in conformance to agreed QoS levels ensures benefits for all player and end users • Cascade payments, means parties who meet their mutual obligations in the value chain will receive a fair commercial return • Efficient connectivity, means IPX is a gateway to managed IP network-managing data flow and commercial information and providing the benefits of multilateral connectivity to all players Generally, a service provider have two possibilities to interconnect with other service providers either by establishing an IPX connection via IPX providers (or GRX providers if only for the GRX service) or using direct connection with other service providers with leased lines, internet using IPSec protocol, or VPN connectivity. Interconnection using IPX is shown in Figure 1, which service provider A uses IPX provider X to interconnect with service provider B and C. IPX provider X have direct connection with service provider A and B as on-net subscriber means that it will be no problem to have interconnection between service provider A and B since they belongs to same IPX provider. However, IPX provider X should cooperate with IPX provider Y in order to service provider A possible to interconnect with service provider C since IPX provider X doesn't have direct cooperation with service provider C. This is the basic need for IPX providers cooperation. The main background of cooperation between IPX providers because the difficulties for one IPX provider to have a global and direct connectivity to all service providers in the world since the will takes time and strong effort in business and network infrastructure aspect. One of ideas for an IPX providers to solve that problem is through cooperation and partnership with other IPX providers. The goal of this paper is to analyze some possible partnership schemes of global IPX providers.
The remainder of the paper is structured as follows: In section 2, we analyze current condition of IPX providers from technical and business aspects, which include IPX capability, development drivers, barriers, potential business models and revenue stream from IPX. In section 3, we continue with some IPX interconnection model which consists of IPX bilateral transport only, IPX service transit, and IPX multilateral hub services. In section 4, we explain some possible IPX partnership models between IPX providers such as normal IPX peering, semi hosted, and full hosted. In section 5, we conclude the paper with some recommendations to choose the most suitable partnership models for IPX providers.
ANALYSIS OF CURRENT CONDITION OF IPX PROVIDERS
IPX basically is a technology evolution of GRX therefore the providers and market itself are already quite mature. An IPX provider is possible to offer multiple type of telecommunication services with single IP network connection and end-to-end network performance guarantees. In the other hand, network elements of IPX still similar with GRX but with addition of Diameter router to accommodate LTE roaming service. The emerging market for IPX is LTE-based roaming services (signalling, voice, and data). However, the OTT (Over The Top) providers markets still wait for the strong drivers to use IPX. Even, the bigger bandwidth in customer side make OTT can still use public internet network as happened today. IPX is also able to support a number of GRX services such as MMS interworking and WLAN (authentication) data roaming, as well as diagnostic protocols, for example ICMP (Ping), connectivity between any types of service providers, end-to-end QoS for roaming and interworking, and any IP services on a bilateral basis with end-to-end QoS and interconnect charging.
Some drivers for IPX development come from both IPX providers and service providers such as from technology background to migrate circuit-switched services to IP, LTE interoperability, LTE roaming, and some new retail services (HD voice, high quality video services). From business background, IPX bring opportunities in some aspects such as introducing new revenuegenerating services, increasing quality, the cost and operational advantages of the hub model for service interconnect, and could drive out cost by combining multiple services over a single connection.
Despite some drivers listed above, a number of IPX providers and service providers also consider some barriers to develop IPX. From IPX provider's point of view, they will face organizational barriers including operational splits between voice and data, fixed and mobile, commercial and network department, lack of critical mass means that many were not prepared to migrate of there only a few partners using an IPX, lack of LTE network and no visible time line for LTE launch, and services pricing issues that their potential customers didn't get detail pricing information clearly from them. From service provider's point of view, some barriers to develop IPX are lack of IPX understanding that many of them still not convince with IPX capability because of minimum IPX knowledge, uncertainty about the ability IPX to fix interoperability problems that a few IPX providers fail to make adequate information available. The barriers could also come from regulation and infrastructure perspective such as license of international service providers, restriction/ outright ban on VoIP, and infrastructure/ geographical barrier that lack of international IP connectivity/ capacity in many emerging countries.
There are a number of possible business model and role strategies for IPX such as IPX network operator that build PoPs in key geographic markets connected by an MPLS networks, IPX platform provider that lease network services and focus on providing interoperability platforms, white label reseller that focusing on selling access to third party networks and platformspossibly on a white label basis, VAS Provider that focusing on value added services for IPX providers to resell or build communities of application providers for them, Voice IPX specialist that ignore the data service market in the short term and focus on VoIPX only, Regional gateway that seek to build a strong regional IPX network and service offering, and non-IPX player that stay away form IPX altogether and focus on providing high quality voice, GRX, and signaling services, building on what carrier already does now.
IPX networks are being considered for, or used as, a platform for the delivery of a variety of new international or roaming services. The services which scored highest for both 'currently using' or 'already plan to use IPX' were GRX and enhanced GRX, roaming signalling for LTE and legacy services, SMS and MMS interworking, LTE voice, LTE data roaming, and content services.
Other services such as HD voice and TDM/ VoIP interoperability also possible to be implemented using IPX.
One example of IPX potential revenue streams come from managed access services that not only offer services, but also for access connection, another business model typically purchase of connection, port, and capacity. Other revenue streams are from roaming data transit services (CRX, GRX, and LTE roaming), roaming IP-based LTE voice (VoLTE) transit, roaming signalling (transport for 2G, 3G, roaming signalling, and LTE signalling), roaming messaging (roaming SMS and MMS), settlement and clearing (data, financial clearing, and settlement naturally as VAS for IPX providers), traffic steering for a variety of guises including traffic redirection using mobile number portability database & ENUM database, and analytics that helping IPX customers to improve their service and profitability. One of the example applications are route management and balancing based on QoS, pricing, and knowledge of the number of hops to end points, silent roamer identification and marketing services, fraud management services that enhanced with the use of analytics.
An IPX Provider is also possible to provide NRTRDE (Near Real Time Roaming Data Exchange) services, international voice break-in/ break-out that Provide termination for inbound services on to PSTN or mobile network (break-in), or transit & termination for outbound services, IP Transit with added QoS/ security which includes transit of IP traffic related to cloud services, content, and application. Another example that are already implemented are interconnection of operators' IPX network with RIM data centres to ensure more secure transit of BBM traffic, and hosted application for hosting of managed cloud-based RCS solutions, conferencing solutions, or hosting of enterprise cloud platform (PaaS) that operators can used to serve their end customer with guaranteed QoS assurance. It is also possible to deliver IPX advances telephony such as HD voice and conferencing video calling/ video conferencing (in SD and HD), IPX RCS and rich media by providing interconnection and interoperability for the services, content transcoding and transrating that can help operators to deliver content internationally using codecs. Roaming WiFi is one example of popular IPX services which enable mobile operators to take advantage of public WiFi infrastructure in other markets while retaining the ability to monitor customers' usage an to bill customers by using WiFi roaming exchange.
ANALYSIS ON SOME IPX INTERCONNECT MODELS
In this section, we describe three IPX interconnect models which are possible to be implemented by service providers that are free to choose on a per service basis:
1) Bilateral Transport Only
In this model, IPX provider provides transport at a guaranteed QoS and each service provider will pay their respective IPX provider costs for transport. The bilateral agreement is between end service providers and any payment of termination charges is a mater for the service providers. A bilateral connection between two service providers (SP-1 & SP-2) using the IPX transport layer with guaranteed QoS end-to-end. In this case, settlement is independent of the IPX domain but connectivity still operates within IPI key business principles. Cascading of responsibilities (such as QoS) applies but not cascading of payments (cascade billing). Each service provider will also pay their respective IPX provider for the transport capacity, potentially depending on the level of QoS provided. The IPX provides QoS-based transport and cascading interconnect payment facilities. This enables an originating service provider to make a single payment to their IPX Provider who passes on a payment on to the next IPX provider in the value chain who pays the final termination charge to the terminating service provider. Within service transit, traffic is transited though IPX providers but prices (termination charges) are agreed bilaterally between service providers and settlement of termination charges can be performed bilaterally between the service providers or via the IPX providers (upon the service provider's choice). Cascade billing (for transport and/or service layer) and other associated facilities provided by the IPX provider (on the service layer) may be applied depending on the service. A bilateral connection between two service providers (SP-1 & SP-2) using the IPX service layer and the IPX transport layer with guaranteed QoS end-to-end. Within service transit, traffic is transited though IPX providers but prices (termination charges) are agreed bilaterally between service providers and settlement of termination charges can be performed bilaterally between the service providers or via the IPX providers (upon the Service Provider's choice).
Cascade billing (for transport and/or service layer) and other associated facilities provided by the IPX Provider (on the Service layer) may be applied depending on the service. Therefore, through service transit, the following connections can be implemented:
• Bilateral connectivity with routing performed within the IPX domain and within IPI key business principles but settlement of termination charges performed bilaterally between the ending parties. • Bilateral connectivity with both routing and settlement of termination charges performed under the IPX Domain and within IPI key business principles.
The transit fee owed to the IPX Providers is always cascaded. Cascading of responsibilities and payments fully apply (on both IPX transport layer and IPX service layer).
3) Multilateral Hub Service
IPX provides QoS transport and cascading interconnect payments to a number of interconnect partners via a single agreement between the service provider and IPX. This "one-to-many" mode is operationally highly efficient for the service provider. Charging transparency is a requirement on both IPXs and service providers in this multilateral connection using hub functionality.
Hubbing or multilateral connectivity is where traffic is routed from one service provider to tens/ hundreds of destinations/ interworking partners through a single agreement but the cascading of responsibilities applies. Cascading of payments may be applied depending on the service (on both IPX transport layer and IPX service layer). The deployment scenarios are possible to be implemented using two alternatives. The first option is through direct investment. The benefit of this option are total operational control, access to new markets, flexibility in choosing geographic location. But, this option also have some drawbacks such as longer time to market, requires capital commitment, and high risk. IPX providers should consider some strategy before implement this option such as investing on data roaming services as the first stages of IPX deployment and initiate peering partnership with other potential IPX Provider.
The second option is through collaboration. The benefit of this option are sharing of risk, fast roll out, access to existing infrastructure, and geographic network. However, this option have some drawbacks in smaller profit margin, dependant on partner's strategy. The implementation strategies could be collaboration with leading IPX provider to resell (white label) under its own brand that enable immediate access to IPX services range and coverage, or collaborate without white label scheme.
Some existing IPX providers' background are experienced GRX (GPRS Roaming eXchange) providers and IPX implementation is executed with strategy to add IPX capability over their existing network. It means that currently all GRX operators are IPX-ready and they are in progress in partnership stages to extend their coverage area and potential customers. The partnership itself is already built from GRX that then developed to IPX. A number of customers and coverage areas become main considerations to choose partnership model, whether based on peering and/ or transit.
ANALYSIS OF POSSIBLE IPX PARTNERSHIP SCHEMES BETWEEN GLOBAL IPX PROVIDERS
The main idea of IPX partnership between IPX providers come from the limitation of one IPX provider to have global coverage to all their potential customers all over the world. The generic configuration for IPX partnership is shown in figure 5. • Partnership model could be peering and/ or hosted based on service • Network responsibilities L1/ L2 network peering at POP location with bandwidth capacity and QoS will be based on further agreement and requirements between partners • IPX services implementation could be implemented gradually based on agreements between partners • Business scheme and charging could be based on type of IPX customers (on-net, off-net, location) and traffic volumes. In most cases, all on-net customers will be opened and charged based on traffic activities From above generic configuration, there are at least 3 (three) possible IPX patnerships could be implemented comprises of normal IPX peering, semi hosted, and full hosted partnership schemes.
Normal IPX Peering
The normal partnership between IPX providers is based on IPX peering shown in Figure 6. In standard peering model between IPX providers defines NNI (Network to Network Interconnection) and the access is limited to on-net (direct) partner's IPX customers. In this model, VLAN should be separated based on service, and the traffic and charging will be consolidated for all MNOs per-service based. The reporting also should be based on service and there is no dedicated reporting per MNO. In normally commercial model, it is possible to add instalation fee and monthly fee parameters based on type of services and number of destinations between IPX providers.
The main advantage for this partnership model is both partners already have independent service node elements and system, and they will have same position level and can reach or access IPX partner's on-net customers. The challenges of this model are each IPX provider need to peer with more than one IPX providers to get global reach since majority of IPX providers will not open their off-net destinations. Some cases will be occured when a larger IPX provider peer with the smaller one means that the smaller IPX provider need to pay to the bigger one. Other notes for this partnership model are IPX network will be separated per-VLAN-based per-service and consolidated traffic & reporting per-service will be in IPX provider's level.
Semi Hosted IPX Partnership
The semi hosted IPX partnership is shown in Figure 7. In this partnership model, IPX provider 1 (the left one) doesn't need to invest their own service node equipments and IPX system since they will use elements and systems from IPX provider 2 (the right one). This model can be used as starting point and short-term scenario for a new established IPX provider that already network infrastructure and prospective customers and they want to deliver IPX services instantly to their customers without build their own IPX system. IPX provider 1 is possible to have access to IPX provider 2's complete IPX coverage (on-net and off-net) and it could minimize the possibility to have partnership with other IPX providers. IPX Provider 2 will manage IPX service node elements for all or specific services. In the other hand, IPX provider 1 will provide CPEs in customer's side and access network from customer to IPX provider 2's service node. It also should separate VLAN per service and consolidated traffic for all MNOs by service. The consolidated reporting will base on service and no dedicated reporting per MNO. In normally commercial model, this partnership scheme is often included installation fee and monthly fee per MNO (based on bandwidth size or per message) However, this partnership scheme will lead to exclusive partnership between both IPX providers and IPX provider 1 will depend on capability and coverage of IPX provider 2's. There are also possibility for some business issues such as price competitiveness and margin share between partners. Same with the first model, the IPX network will be separated per-VLAN-based perservice and consolidated traffic & reporting per-service in IPX provider's level.
Full Hosted IPX Partnership
The full hosted IPX partnership is shown in Figure 8. This partnership scheme almost similar with semi hosted partnership explained above that IPX provider 1 possible to access IPX provider 2's on-net and off-net customers and they don't need to invest on IPX service node equipment. The main difference between them is in full hosted partnership scheme, IPX providers 1's position and main task is to market IPX provider 2's IPX service since all IPX network infrastructure will be provided by IPX provider 2. However, there is a possibility for IPX Provider 1 to re-brand the IPX services using their own brand. IPX provider 1 can apply a direct price mark up for reselling.
By considering existing services, infrastructure, potential partnership, market, and implementation time, an IPX operator can have different types of partnership model, i.e. some IPX providers is come from voice services and signalling providers, they can migrate voice traffic to IPX-based. For example, an IPX provider that already have existing strong customers and partners can attract other IPX providers to peer with them. The implementation model could be started by migrating traffic from non-IPX to IPX environment without changing the existing business model. Some new services such as LTE (signaling, data, and voice) and diameter could become main drivers to do partnership between IPX providers since LTE is a green-field service that currently in initiation stages, with some IPX providers willing to have peering and do trial with other IPX providers.
For other services such as GRX, SMS/MMS, and RIM can be implemented using aggregation business model since GRX and SMS/MMS are mature services then the performance improvement resulted in IPX environment is still can not be a major driver for service providers to move to IPX. In some cases, a number of GRX providers offer aggregator partnership model, with consideration in implementation simplicity. A non-GRX provider is possible to facilitate access network from their existing and domestic customers to their IPX service node provider partner. By having partnership with them, they will become GRX/SMS/MMS/RIM hubber.
Consideration to choose IPX partnership scheme
The IPX partnership between IPX providers come from business and technical perspective. From business perspective, it could be seen 3 (three) issues, such as:
• Peering scheme (IPX transport with services) with equivalent IPX providers with a number of on-net customers and coverage areas as main consideration to build partnership • Transit scheme with non-equivalent IPX providers • For peering scheme, usually only include on-net customers of peering partners.
Therefore, peering will need more than 1 (one) IPX providers to reach global coverage. Based on experience from some IPX providers, they could build peering partnership with more than 5 (five) other IPX providers.
From technical perspective, there are several issues regarding IPX partnership:
• To maintain network performance, majority of IPX providers limit their end-to-end IPX customers to maximum 2 IPX providers (2 hops). • Several IPX providers stated that peering partnership could be a challenging task for service interoperability. • Although IPX offer single IP private connection for multi services, however the reporting mechanism is often done partially.
CONCLUSIONS
There are a number of possible partnership schemes can be implemented between IPX Providers such as peering, semi hosted, full hosted, or combination between with service-based implementation. However, deciding the best partnership scheme, IPX providers should consider some factors, related to (but not limited to) IPX providers' network asset, coverage, and ownership, IPX services and features offering whether they offer a part or full IPX-based services, and their support of tools and data analytic, financial data clearing, ENUM, CDN, or other IPX-related services. In the other hand, a number of a IPX Provider's on-net and off-net customers should become one main considerations, beside business scheme and pricing offer.
For an IPX provider to become competitive in IPX business and become a global IPX hubber, they should able to give value added to customers, such as cost efficiency and great network performance. To achieve it, an IPX provider could implement some strategies such as build network sinergy between them and partners to develop IPX Service as single offering, offer their customers with bundled access network and IPX Service with cheaper price than competitors. An IPX provider should also consider their existing customer-based that can be a benefit to their bargaining position to other potential IPX provider partners to determine price and business scheme for partnership.
Figure 1 .
1IPX basic network configuration
Figure 2 .
2IPX bilateral transport only interconnect model 2) Bilateral Service Transit
Figure 3 .
3IPX bilateral service transit interconnect model
Figure 4 .
4IPX multilateral hub service interconnect model
Figure 5 .
5IPX providers partnership generic configurationSome key points regarding IPX partnership model shown inFigure are:
Figure 6 .
6Normal IPX peering between 2 (two) IPX providers
Figure 7 .
7Semi hosted IPX partnership between 2 (two) IPX providers
Figure 8 .
8Full hosted IPX partnership between 2 (two) IPX providers
AuthorsDavid GunawanPosition:Engineer
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A Distributed IP-Based Telecommunication System Using SIP. C A Thompson, H A Latchman, N Angelacos, B K Pareek, IJCNC International Journal of Computer Networks and Communications. 56C.A.Thompson, H.A.Latchman, N.Angelacos, B.K.Pareek, (2013) "A Distributed IP-Based Telecommunication System Using SIP," IJCNC International Journal of Computer Networks and Communications, Vol.5, No.6, pp.121-136.
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arxiv |
GPA-Tree: Statistical Approach for Functional-Annotation-Tree-Guided Prioritization of GWAS Results
Aastha Khatiwada
Department of Public Health Sciences
Medical University of South Carolina
CharlestonSouth CarolinaUSA
Bethany J Wolf
Department of Public Health Sciences
Medical University of South Carolina
CharlestonSouth CarolinaUSA
Ayse Selen Yilmaz
Department of Biomedical Informatics
The Ohio State University
ColumbusOhioUSA
Paula S Ramos
Department of Public Health Sciences
Medical University of South Carolina
CharlestonSouth CarolinaUSA
Department of Medicine
Medical University of South Carolina
CharlestonSouth CarolinaUSA
Maciej Pietrzak
Department of Biomedical Informatics
The Ohio State University
ColumbusOhioUSA
Andrew Lawson
Department of Public Health Sciences
Medical University of South Carolina
CharlestonSouth CarolinaUSA
Kelly J Hunt
Department of Public Health Sciences
Medical University of South Carolina
CharlestonSouth CarolinaUSA
Hang J Kim
Division of Statistics and Data Science
University of Cincinnati
CincinnatiOhioUSA
Dongjun Chung
Department of Biomedical Informatics
The Ohio State University
ColumbusOhioUSA
GPA-Tree: Statistical Approach for Functional-Annotation-Tree-Guided Prioritization of GWAS Results
* To whom correspondence should be addressed ([email protected]).
Motivation: In spite of great success of genome-wide association studies (GWAS), multiple challenges still remain. First, complex traits are often associated with many single nucleotide polymorphisms (SNPs), each with small or moderate effect sizes. Second, our understanding of the functional mechanisms through which genetic variants are associated with complex traits is still limited. functional annotations related to risk-associated SNPs. To address these challenges, we propose GPA-Tree and it simultaneously implements association mapping and identifies key combinations of functional annotations related to risk-associated SNPs by combining a decision tree algorithm with a hierarchical modeling framework.Results: First, we implemented simulation studies to evaluate the proposed GPA-Tree method and compared its performance with existing statistical approaches. The results indicate that GPA-Tree outperforms existing statistical approaches in detecting risk-associated SNPs and identifying the true combinations of functional annotations with high accuracy. Second, we applied GPA-Tree to a systemic lupus erythematosus (SLE) GWAS and functional annotation data including GenoSkyline and GenoSky-linePlus. The results from GPA-Tree highlight the dysregulation of blood immune cells, including but not limited to primary B, memory helper T, regulatory T, neutrophils and CD8 + memory T cells in SLE.These results demonstrate that GPA-Tree can be a powerful tool that improves association mapping while facilitating understanding of the underlying genetic architecture of complex traits and potential mechanisms linking risk-associated SNPs with complex traits.Availability: The GPATree software is available at https://dongjunchung.github.io/GPATree/.
by multiple single nucleotide polymorphisms (SNPs), often with small or moderate effect sizes [33,31].
Such SNPs often do not meet the genome-wide p-value cutoff of 5 × 10 −8 and as a result, still many trait-associated SNPs remain unidentified. In theory, utilizing a large sample size will improve statistical power to detect these SNPs. However, traits of limited prevalence in the population often result in limited sample sizes, and recruiting a large sample size requires significant resources and is often not feasible.
Therefore, there is a critical need to find alternate ways to increase statistical power to detect SNPs with small and moderate effect sizes.
Second, over 85% of the genetic variants identified by GWAS are located in non-coding regions [15] and it is often difficult to understand their functional roles in the trait etiology. For example, in autoimmune diseases, about 90% of the causal genetic variants lie in non-coding regions, a bulk of which are located in regulatory DNA regions [27,11]. Utilizing tissue-and cell type-specific functional information can potentially improve our understanding of biological mechanisms through which SNPs may be associated with traits [36]. The general hypothesis is that functional roles relevant to trait-associated SNPs may influence the distribution of these SNPs in the GWAS summary statistics. Therefore, integrating GWAS summary statistics and functional annotation information might not only improve statistical power to detect SNPs , but also identify the mechanisms by which trait-associated SNPs may influence trait [28,39,36]. For example, in the case of autoimmune diseases like systemic lupus erythematosus (SLE) and multiple sclerosis (MS), risk-associated SNPs might be more enriched for those with roles in immunity, while for psychiatric disorders like bipolar disorder (BPD) and schizophrenia (SCZ), risk-associated SNPs might be more relevant to the central nervous system or brain function.
Recognizing the potential of such integrative approaches, several methods have been proposed to prioritize SNPs and identify relevant functional annotations by integrating GWAS data with functional annotation data [36,39,22,25,28,29]. Still currently available methods utilize functional annotations in a relatively simple form without considering interactions between them, i.e., only main effects terms are included in the model. However, this can be a critical limitation because valuable in-depth biological insight can often be obtained through investigating combinations of the functional annotations, e.g., different types of histone modifications, epigenetic marks in different immune cell subsets, and expression quantitative trait loci (eQTL) for different traits. In theory, some existing methods can be extended by including interaction terms to identify combinations of functional annotations. However, this requires strong prior scientific knowledge, which is often lacking, especially when a large number of functional annotations is considered in the analysis. Moreover, including all possible interactions can quickly become computationally taxing. Therefore, there is a critical need for a method that can efficiently identify relevant combinations of functional annotations without requiring strong prior knowledge.
To fill in this important research gap, we propose GPA-Tree, a novel statistical approach that simultaneously prioritizes trait-associated SNPs and identifies key combinations of functional annotations related to the mechanisms through which trait-associated SNPs influence the trait, within a unified framework.
Specifically, GPA-Tree is based on a hierarchical modeling approach integrated with a decision tree algorithm and facilitates easy interpretation of findings. GPA-Tree takes GWAS summary statistics as input, which allows wide applications and adaptations. Our comprehensive simulation studies and real data applications show that GPA-Tree consistently improves statistical power to detect trait-associated SNPs and also effectively identifies biologically important combinations of functional annotations.
GPA-Tree
Model
Let YM×1 = (Y1, Y2, . . . , YM ) be a vector of genotype-trait association p-values for i = 1, 2, · · · , M such that yi denotes the p-value for the association of the i th SNP with the trait. We also assume that we have K binary annotations (A).
A = (A.1, . . . , A.K ) = a11 . . . a1K . . . . . . . . . aM1 . . . aMK M ×K , where a ik = 0, if i th SNP is not annotated in the k th annotation 1, if i th SNP is annotated in the k th annotation
Here our ultimate goal is association mapping, i.e., identifying SNPs associated with the trait given both GWAS and functional annotation data. To accomplish this, we introduce the latent variable Z,
where zi indicates association of i th SNP with the trait. Then, the GWAS association p-values (yi) are assumed to come from a mixture of non-risk-associated (zi = 0) and risk-associated groups (zi = 1). As previously proposed by Chung and colleagues [7], if the i th SNP belongs to the non-risk-associated group (zi = 0), then its p-value is assumed to come from the Uniform distribution on [0, 1]. This is based on the rationale that U [0, 1] provides a p-value density corresponding to the non-risk-associated group [32].
If the i th SNP belongs to the risk-associated group (zi = 1), then its p-value is assumed to come from the Beta distribution with parameters (α, 1), where 0 < α < 1. We restrict α in the Beta distribution to be between 0 and 1 because the smaller α value corresponds to the higher density at lower p-values, while the α value closer to one resembles a U nif [0, 1] distribution.
We further integrate functional annotation data with the GWAS data by modeling the latent Z as a function of the functional annotation data A. Specifically, we define a function f that is a combination of functional annotations A and relate it to the expectation of latent Z as given in Equation (1).
P (Zi = 1; ai1, ..., aiK ) = f(ai1, ..., aiK )(1)
Let θ = (α, π), where π = {π1, π2, ..., πM } is a function of A and represents the prior probabilities that the SNPs belong to the risk-associated group, i.e., πi = P (Zi = 1). See Section 1 in the Supplementary Materials for the joint distribution of the observed data, and the incomplete and complete data loglikelihoods.
Algorithm
Given the approach described in Section 2.1, we implemented parameter estimation using an EM algorithm. The function f in Equation (1) is estimated by a decision tree algorithm and it allows to identify combinations of functional annotations related to risk-associated SNPs. To improve stability, we employed a two-stage approach for parameter estimation. Specifically, in Stage 1, we first estimate the parameter α without identifying a combination of functional annotations. Then, in Stage 2, we identify key combinations of functional annotations (f (A)) while the parameter α is kept fixed as the value obtained in the first step. We illustrate more detailed calculation steps below.
Stage 1:
For the i th SNP, the t th iteration of the E-step can be written as:
E − step : z (t) i = E[Z i ; Y, A, θ (t−1) ] = P r(Zi = 1; Y, A, θ (t−1) ) = P (Y i ; Z i =1, θ (t−1) )P (Z i =1; A i. , θ (t−1) ) d∈{1,0} P (Y i ; Z i =d, θ (t−1) )P (Z i =d; A i. , θ (t−1) ) = α (t−1) y α (t−1) −1 i π (t−1) i 1−π i (t−1) +α (t−1) y α (t−1) −1 i π (t−1) i (2)
In the t th iteration of the M-step, πi and α are updated as:
M − step :
Fit a linear regression model as
z (t) i = β (t) 0 + β (t) 1 ai1 + · · · + β (t) K aiK + (t) i Update π (t)
i as the predicted value from the linear regression model. Update
α (t) = − M i=1 zi (t) / M i=1 zi (t) log(yi),(3)
where β (t) k , k = 0, · · · , K are the regression coefficients and (t) i is the error term. The E and M steps are repeated until both the incomplete log-likelihood and the α estimate converge. The α and π estimated in this stage are used to fix α and initialize π, respectively, in Stage 2.
Stage 2:
In this stage, we implement another EM algorithm employing a decision tree algorithm (CART [2]), which allows to identify union, intersection, and complement relationships between functional annotations in estimating πi.
For the i th SNP, the t th iteration of the E-step can be written as:
E − step : z (t) i =α yα −1 i π (t−1) i 1−π i (t−1) +αyα −1 i π (t−1) i(4)
Note that here α is fixed asα, which is the final estimate of α obtained from Stage 1. In the t th iteration of the M-step, πi is updated as:
M − step : Fit a CART model as z (t) i = f (t) (ai1, · · · , aiK ) + (t) i Update π (t)
i as the predicted value from the CART model,
where i is the error term. In the M-step, the complexity parameter (cp) is the key tuning parameter and defined as the minimum improvement that is required at each node of the tree. Specifically, in the CART model, the largest possible tree (i.e., a full-sized tree) is first constructed and then pruned using cp. The pruned regression tree structure identified by the CART model upon convergence of the EM algorithm (Equation (5)) is used as f in Equation (1). This approach allows for the construction of the accurate yet interpretable regression tree that can explain relationships between functional annotations and genotype-trait associations. The E and M steps are repeated until the incomplete log-likelihood converges.
We note that unlike the standard EM algorithm, the incomplete log-likelihood in Stage 2 is not guaranteed to be monotonically increasing. Therefore, we implement Stage 2 as a generalized EM algorithm by retaining only the iterations in which the incomplete log-likelihood increases compared to the previous iteration.
Prioritization of risk-associated SNPs and identification of relevant combinations of functional annotations
Once the parameters are estimated as described in Section 2.2, we can now prioritize risk-associated SNPs and identify combinations of functional annotations relevant to these SNPs. First, SNPs are prioritized using the local false discovery rate, f dr, which is defined as the posterior probability that the i th SNP belongs to the non-risk-associated group given its GWAS p-value and functional annotation information,
i.e., f dr(Yi, Ai.) = P (Zi = 0; Yi, Ai.) = 1−P (Zi = 1; Yi, Ai.). We utilize the 'direct posterior probability' approach [30] to control the global false discovery rate, F DR, which is the expected ratio of the number of SNPs that are incorrectly predicted to be risk-associated SNPs (false positives) compared to the number of SNPs that are predicted to be risk-associated SNPs (positives). In this approach, SNPs are first sorted by their f dr in an ascending order, denoted as hi.
The threshold for f dr, κ, is then increased from 0 to 1 until
F DR = M i=1 hi 1{hi ≤ κ} M i=1 1{hi ≤ κ} ≤ τ,
ShinyGPATree: Shiny app for interactive analysis of risk-associated SNPs and the functional annotation tree
We implemented the forementioned GPA-Tree algorithm as an R package 'GPATree'. To further facilitate user's convenience, we developed 'ShinyGPATree', a Shiny app for interactive analysis of risk-associated SNPs and the functional annotation tree (Fig 1). This Shiny app can be open by sequentially running
Info tab: Association mapping and annotation selection
The 'Info' tab opens the user interface for association mapping and functional annotation characterization for SNPs as seen in Fig 1B.
Results
Simulation study
We conducted a simulation study to evaluate the performance of the proposed GPA-Tree approach. For each combination of the simulation parameters defined above, we simulated 100 datasets and compared the performance of GPA-Tree with LPM [29] and LSMM [28]. The metrics for comparing the methods include (1)
Real data analysis
We applied the GPA-Tree approach to the SLE GWAS data [20] sourced from the GWAS Catalog GenoSkyline (GS) [24] and GenoSkylinePlus (GSP) [23]. GS were generated by integrating epigenetic annotations from the Roadmap Epigenomics Consortium [19]. They predict tissue-specific functional relevance for SNPs, which are available for seven tissue clusters (brain, gastrointestinal/GI, lung, heart, blood, muscle and epithelium tissues). GSP added another layer of information to GS in the form of
Tissue-level investigation
We initially investigated the functional potential of all SNPs using seven tissue-specific GS annotations.
With a GS score cutoff of 0.5, 35.90% of SNPs were annotated in at least one of the seven tissue types Crohn's disease, ulcerative colitis and rheumatoid arthritis [24].
The original GPA-Tree model fit contained blood tissue at the root node and included 28 leaves. For easier interpretation, we used ShinyGPATree app to prune the tree so that it includes 7 leaf nodes ( Fig 3A). We note that although it is occasionally possible to obtain a large functional annotation tree that can be cumbersome to visualize and interpret, the ShinyGPATree app can be utilized to manage such cases as it allows users to investigate different layers of functional annotation trees in an interactive and dynamic manner. For example, the annotation combination for SNPs in leaf 7 can be written as Blood immune cell subtypes [34], and can be utilized to investigate a variant's functional role in previously defined associations between SLE, CLEC16A and IKZF 3 [37,9,14,21,26,4], among others.
Cell-type-level investigation
Conclusion
In this paper, we presented GPA-Tree, a novel statistical methodology that integrates GWAS summary statistics and functional annotation data within a unified framework. GPA-Tree simultaneously identifies risk-associated SNPs and combinations of functional annotations that potentially explain the mechanisms through which risk-associated SNPs are related with traits. GPA-Tree showed the higher AUC and statistical power to detect risk-associated SNPs compared to existing approaches. GPA-Tree also successfully identified the true combinations of functional annotations in most cases, facilitating understanding of potential biological mechanisms linking risk-associated SNPs with complex traits. The proposed GPA-Tree approach was implemented as the R package 'GPATree' and we also developed 'ShinyGPATree', a Shiny app for interactive and dynamic investigation of association mapping results and functional annotation trees. In the future, we plan to further improve the proposed GPA-Tree method to jointly analyze GWAS data for multiple traits [18,6]. Overall, the ability of GPA-Tree to improve SNP prioritization and attribute functional characteristics to risk-associated SNPs or gene locus can be powerful in facilitating our understanding of genetic susceptibility factors related to complex traits.
Funding
This work was supported in part by NIH/NIGMS grant R01-GM122078, NIH/NCI grant R21-CA209848, NIH/NIDA grant U01-DA045300, and NIH/NIAMS grants P30-AR072582 and R01-AR071947. RE is defined as the ratio of the proportion of SLE-associated SNPs that are annotated for a specific bloodrelated GSP cell type, relative to the the proportion of non-SLE-associated SNPs that are annotated for the same blood-related GSP cell type.
Conflict of
where τ is the predetermined level of F DR (e.g., τ ≤ 0.05). Finally, SNPs with hi ≤ κ are considered to be risk-associated SNPs. Second, relevant combinations of annotations are inferred based on the combination of functional annotations selected by the CART model upon convergence of the EM algorithm in Stage 2.
GPATree() and ShinyGPATree() functions. First, the GPATree() function takes 4 arguments: gwasPval, annMat, initAlpha and cpTry. gwasPval is a M × 1 matrix of GWAS association p-values for M SNPs, annMat is a M × K matrix of K binary functional annotations for M SNPs, initAlpha is the initial alpha value to be used to fit the GPA-Tree model (default value = 0.1), and cpTry is the cp parameter to be used to fit the GPA-Tree model (default value = 0.001). The GPATree() function generates a GPA-Tree model fit required for the ShinyGPATree app. The ShinyGPATree() function takes the output of GPATree() as an input and opens the ShinyGPATree app using the R code below.R> fit <-GPATree(gwasPval, annMat, initAlpha, cpTry) R> ShinyGPATree(fit) The ShinyGPATree app provides visualization of the GPA-Tree model fit, identifies risk-associated SNPs, and characterizes the combinations of functional annotations that can describe the risk-associated SNPs. The app also allows to improve the visualization of the GPA-Tree model fit by collating or separating layers of the model using the cp parameter. The number of non-risk-associated and risk-associated SNPs that can be characterized by combinations of functional annotations are also automatically updated based on user-selected cp, FDR type (global vs. local) and FDR level values. The interactive nature of the app allows users to effortlessly interact with the GPA-Tree model results to generate plots, prioritize risk SNPs, and make inferences about relevant combinations of functional annotations for the risk-associated SNPs. ShinyGPATree consists of two main tabs, namely 'Plot' and 'Info', which are explained in detail below.
Figure 1 :
1Screenshot of the ShinyGPATree app with (A) the 'Plot' tab and (B) the 'Info' tab open.2.4.1 Plot tab: Visualization of the GPA-Tree model Fig 1A shows the layout of the ShinyGPATree app, where the 'Plot' tab opens by default. In the displayed plot, each leaf (terminal node) is characterized by combinations of the functional annotations that are encountered as users move from the root node to the leaf. The summary information is provided for each leaf, including the number of SNPs that satisfy the combination of functional annotations specific to the leaf and the mean local FDR for these SNPs. The summary information displayed in each leaf is automatically updated as the user modifies the cp value on the left panel. Users can also improve visualization of the functional annotation tree plot using the Plot width and Plot height options on the left panel. The 'Download Plot' button on the top allows users to download the functional annotation tree plot as a PNG format file. Finally, a table titled 'Leaf Description' underneath the plot characterizes the functional annotations that are 0 or 1 for SNPs in specific leaves.
Figure 2 :
2Comparison of (A) AUC, (B) statistical power to detect true risk-associated SNPs when global FDR is controlled at the nominal level of 0.05, (C) estimated α parameter, and (D) proportion of times only true functional annotations A 1 − A 4 are simultaneously identified by GPA-Tree (red line) and the average proportion of noise annotations (A 5 − A 75 ) among the functional annotations identified by GPA-Tree (blue line). The results are presented for different proportions of SNPs annotated in A 1 − A 4 (u; x-axis) and proportions of the overlap between SNPs annotated in A 1 − A 2 and A 3 − A 4 (v; panel). M = 100, 000, K = 75, and α = 0.7 in Beta(α, 1) and results are summarized from 100 replications.
For all the simulation data, the number of SNPs was set to M = 100, 000, the number of annotations was set to K = 75, and risk-associated SNPs were assumed to be characterized with the combinations of functional annotations defined by L = (A1 ∩ A2) ∪ (A3 ∩ A4); all the remaining functional annotations (A k , k = 5, .., 75) were considered to be noise annotations. The percentage of annotated SNPs (u) for annotations A1 − A4 was set to 2%, 6%, 10%, 14% and 20%, while the percentage of overlap between the true combinations of functional annotations (v) was set to 12.5%, 25%, 50%, 75% and 87.5%. For noise annotations A5 − A75, approximately 20% of SNPs were annotated by first generating the proportion of annotated SNPs from U nif [0.1, 0.3] and then randomly setting this proportion of SNPs to one. TheSNPs that satisfy the functional annotation combination L were assumed to be risk-associated SNPs and their p-values were simulated from Beta(α, 1) with α = 0.7. The SNPs that do not satisfy L were assumed to be non-risk SNPs and their p-values were simulated from U [0, 1].
area under the curve (AUC), where the curve was created by plotting the true positive rate (sensitivity) against the false positive rate (1-specificity) to detect risk-associated SNPs when global FDR was controlled at various levels; (2) statistical power to identify risk-associated SNPs when global FDR was controlled at the nominal level of 0.05; and (3) estimation accuracy for α parameter in the Beta(α, 1) distribution used to generate the p-values of risk-associated group. For GPA-Tree, we also examined the accuracy of detecting the correct functional annotation tree, based on the proportion of simulation data for which all relevant functional annotations in L (A1−A4) were identified simultaneously, and the average proportion of noise functional annotations (A5 − A75) among the functional annotations identified by GPA-Tree. Here we especially investigate how the percentage of SNPs annotated in A1 − A4 (u) and the overlap between SNPs annotated in A1 − A2 and A3 − A4 (v) impact GPA-Tree's ability to separate functional annotations relevant to the risk-associated SNPs from noise annotations. • AUC: Fig 2A shows the AUC comparison between GPA-Tree, LPM, and LSMM. For all the combinations of u and v, GPA-Tree showed the consistently higher AUC relative to LSMM while performing comparably or better than LPM. The performance of LPM and LSMM improved as signal-to-noise ratio increases (i.e., as u and v increase), demonstrating performance closer to GPA-Tree. • Statistical power: Fig 2B compares the power to detect true risk-associated SNPs when global FDR is controlled at 0.05 for the three methods. GPA-Tree showed higher statistical power to detect true risk-associated SNPs relative to LPM and LSMM for almost all combinations of u and v. The estimated power for GPA-Tree was relatively more variable for u = 2% and v = 12.5% but it still outperformed LPM and LSMM. The statistical power of LPM increased as a function of u for all v, and the statistical power of LSMM increased as u increases for higher v. However, both LPM and LSMM showed greater variability in statistical power compared to GPA-Tree and on average they showed lower statistical power compared to GPA-Tree. • Estimation of parameter α: Fig 2C shows the α parameter estimates obtained from the three methods. GPA-Tree showed less variability in the α estimates compared to LPM and LSMM. LPM was on average more accurate than GPA-Tree in estimating α, however it still often underestimated α. LSMM showed decreased variability in estimation of α as u increases, and estimated α well for higher u and v levels. GPA-Tree generally overestimated α and this was most notable when u and v are small. As u and v increase, α estimates from GPA-Tree became closer to the true value.When u and v are large (u ≥ 10% and v ≥ 75%), GPA-Tree estimated α accurately. We note that overestimation of α by GPA-Tree did not impact the method's ability to identify the true combinations of functional annotations or the risk-associated SNPs, which are the main objectives of GPA-Tree.• Selection of relevant and noise annotations: The red line in Fig 2D shows the proportion of times only functional annotations in the true combination L (A1−A4) were simultaneously identified by GPA-Tree while the blue line shows the proportion of noise annotations (A5 − A75) that were also selected. Excluding instances when signal in the data is really weak (u ≤ 6% and v ≤ 25%), GPA-Tree successfully identified all functional annotations included in the true combination L more than 75% of the time. Moreover, GPA-Tree could identify all functional annotations included in the true combination approximately 100% of the time as u or v get larger(Fig 2D, red line). These results demonstrate the potential of GPA-Tree to correctly identify true annotations as long as signal in the data is not too weak. In instances where GPA-Tree did not identify all functional annotations included in L, it either identified one or more noise annotations in addition to the true annotations (false positives), or failed to identify one or more annotations in L (false negative) (Fig 2D, blue line).
[ 3 ]
3(https://www.ebi.ac.uk/gwas/). Summary statistics were originally obtained using the genotyped and imputed Immunochip, profiled for 18, 264 individuals (6, 748 cases and 11, 516 controls) of European ancestry. 336, 745 SNPs passed quality control criteria. After excluding SNPs located in the MHC region, 293, 976 SNPs were included in the final analysis and integrated with functional annotation data from
epigenomic and transcriptomic annotations, and are available for 127 annotation tracks. The Manhattan plot and p-value histogram for SLE GWAS data are presented in Fig S1 in the Supplementary Materials.
(
Fig S2A in the Supplementary Materials) and the percentage of annotated SNPs ranged from 8.66% for lung tissue to 19.14% for blood tissue (Fig S2B in the Supplementary Materials). We also measured the overlap in SNPs annotated in different tissue types using log odds ratio (Fig S2C in the Supplementary Materials). While the highest proportion of SNPs is annotated for blood tissue, SNPs annotated for blood tissue overlap less with other tissue types. On the contrary, SNPs annotated for heart, lung and muscle tissues overlap more with other tissue types. This is consistent with the literature indicating that blood shows the lowest levels of eQTL sharing with other tissue types while muscle and lung tissues show higher levels of eQTL sharing [38, 24]. Next, we applied GPA-Tree to the SLE GWAS and GS annotation data for association mapping and characterization of relevant functional annotations. GPA-Tree identified 8, 962 SLE-associated SNPs at the nominal global FDR level of 0.05. Among SLE-associated SNPs, 46.40% were annotated for at least one of the seven GS tissue type (Fig S3A in the Supplementary Materials), and the percentage of annotated SNPs ranged from 9.89% for lung tissue to 30.22% for blood tissue (Fig S3B in the Supplementary Materials). We also measured relative enrichment (RE), the ratio of the proportion of SLE-associated SNPs annotated for a specific tissue type, relative to the proportion of non-SLE-associated SNPs annotated for the same tissue type. RE was again highest for the blood tissue with the value of 1.61 (Fig S3C in the Supplementary Materials). These results are consistent with the dysregulation of blood immune cells that characterizes SLE and other autoimmune diseases like
Figure 3 :
3Functional annotation tree identified by GPA-Tree approach when (A) seven tissue-level GenoSkyline (GS) annotations and (B) 10 blood-related cell-type-level GenoSkylinePlus (GSP) annotations are considered. Both trees were generated by pruning the GPA-Tree model fit using cp = 2.5 × 10 −4 . Each leaf (terminal node) in the tree shows the total number of SNPs in the leaf and the mean local FDR for the SNPs in the leaf.∩ !Heart ∩ Epithelium, i.e., leaf 7 includes SNPs that are annotated for blood and epithelium tissues but not for heart tissue. The number of SNPs that are located in each leaf node, and the combination of functional annotations that describe SNPs in each leaf node are displayed inFig 3A.Further investigation of the GPA-Tree model fitting results showed that, among the 8, 962 SLE-associated SNPs, 578 are concurrently annotated for blood and epithelium tissues while not being annotated for heart tissue as represented in leaf 7; 609 are concurrently annotated for both blood and heart tissues as represented in leaf 4; and 230 are concurrently annotated for epithelium and GI tissues while not being annotated for blood tissue as represented in leaf 2. Blood, epithelium, GI and heart also have the largest RE (Fig S3Cin the Supplementary Materials). In general, our results are consistent with the literature indicating relevance of blood tissue in SLE, and further add genomic-level support to the relevance of other tissues concurrently with blood[5,13,16,10].
on the observed relationship between GS annotation for blood tissue and SLE, in the second phase of the real data analysis, we considered 10 blood-related GSP functional annotations. With a GSP score cutoff of 0.5, 25.29% were annotated in at least one of the 10 GSP blood annotations (Fig S4A inthe Supplementary Materials) and the highest enrichment was observed for primary regulatory T cells(12.13%) (Fig S4B in the Supplementary Materials). The highest overlaps were observed between SNPs annotated with primary memory helper T, effector memory T and CD8 + memory T cells (Fig S4C in the Supplementary Materials). Next, we applied GPA-Tree to the SLE GWAS and GSP blood annotations. At the nominal global FDR level of 0.05, GPA-Tree identified 8, 993 SLE-associated SNPs, where 8, 723 among those overlapped with the SNPs prioritized in the first phase using GS annotations. Among the SLE-associated SNPs prioritized in the second phase, 37.54% were annotated for at least one of the 10 GSP blood annotations (Fig S5A in the Supplementary Materials). The largest proportion of SLE-associated SNPs was annotated for primary B cells (19.47%), followed by primary regulatory T cells (18.45%) (Fig S5B in the Supplementary Materials). Primary B cells also showed the highest RE with the value of 2.12 (Fig S5C in the Supplementary Materials). Since SLE is characterized by the production of autoantibodies, the involvement of B cells, which produce antibodies, is consistent with disease pathology. The original GPA-Tree model with GSP blood annotations identified primary B cells at the root node and included 172 leaves. Again, to improve interpretability and visualization, we used ShinyGPATree to prune the tree so that it includes 10 leaf nodes (Fig 3B). In addition to primary B cells, other blood-related GSP functional annotations identified as important included primary memory helper T, regulatory T, neutrophils, natural killer, effector memory T, and CD8 + memory T cells. Among the 8, 993 SLE-associated SNPs, 613 are concurrently annotated for primary B and helper memory T cells as represented in leaf 8; 68 are concurrently annotated for primary B and CD8 + memory T cells while not being annotated for memory helper T cells as represented in leaf 10; and 108 are concurrently annotated for primary regulatory T, neutrophils and effector memory T cells while not being annotated for primary B cells as represented in leaf 4. Overall, these results are consistent with previous literature indicating connections between SLE and B cells, regulatory T cells, neutrophils and CD8 + memory T cells [8, 35, 17, 1, 12]. These results also provide several new insights for future investigations. For instance, among the SLE-associated SNPs, 43 SNPs located in the CLEC16A gene and 41 SNPs located in the IKZF 3 gene are in leaf 8 and concurrently regulate primary B and memory helper T cells; however, an additional 16 SNPs in the CLEC16A gene are in leaf 4 and concurrently regulate primary regulatory T, neutrophils and effector memory T cells while not regulating B cells. These results provide further evidence that multiple independent SNPs in the same gene locus can have different effects on the levels of different
; 0 ,[
0SNPs are independent, we can write the joint distribution of the observed data P r(Y,A) Zi = 1, Ai.) + (Zi = 0)P (Yi; Zi = AiThe incomplete data log-likelihood ( IC ) and the complete data log-likelihood ( C ) for GPA-Tree are also shown below. Zi (log πi + log α + (α − 1) log yi) + (1 − Zi) log(1 − πi)]
Figure 4 :
4Characteristics of the SLE GWAS data. (A) Manhattan plot. Genome-wide significance level (5 × 10 −8 ) is indicated by the dashed red line. (B) GWAS association p-value histogram.
Figure 5 :
5Characteristics of 293, 976 SNPs when integrated with seven GenoSkyline (GS) annotations. (A) Number of GS tissues in which SNPs are annotated. (B) Proportion of SNPs that are annotated for each GS tissue type. (C) Overlap of SNPs annotated by seven GS tissue types, calculated using log odds ratio.
Figure 6 :
6Characteristics of the 8, 962 GPA-Tree identified SLE-associated SNPs when integrated with seven GenoSkyline (GS) annotations. (A) Number of GS tissues in which SLE-associated SNPs are annotated. (B) Proportion of SLE-associated SNPs annotated in each GS tissue type. (C) Relative enrichment (RE)of GS tissue types for SLE-associated SNPs. RE is defined as the ratio of the proportion of SLE-associated SNPs that are annotated for a specific GS tissue type, relative to the the proportion of non-SLE-associated SNPs that are annotated for the same GS tissue type.
Figure 7 :
7Characteristics of the 293, 976 SNPs when integrated with 10 GenoSkylinePlus (GSP) bloodrelated annotations. (A) Number of blood-related GSP annotation type in which SNPs are annotated. (B) Proportion of SNPs annotated for each blood-related GSP annotation type. (C) Overlap of SNPs annotated by 10 blood-related GSP cell types, calculated using log odds ratio.
Figure 8 :
8Characteristics of the 8, 993 GPA-Tree identified SLE-associated SNPs when integrated with 10 blood-related GSP annotations. (A) Number of blood-related GSP annotations in which SLE-associated SNPs are annotated. (B) Proportion of SLE-associated SNPs annotated in each of the blood-related GSP annotation type. (C) Relative enrichment (RE) of blood-related GSP cell type for SLE-associated SNPs.
Under this tab, users can find more information on specific SNPs driving the visualization. The top of the panel provides multiple options to control association mapping, including FDR level and FDR type (global vs. local). It also provides options to select which SNPs to display, e.g., choosing SNPs that fall on specific leaves of the GPA-Tree model and/or selecting SNPs with specific association status (non-risk-associated vs. risk-associated SNPs). The 'SNPTable'at the bottom of the 'Info' tab panel shows information about the SNPs that satisfy these options. Each row of the table represents a SNP, where columns include SNP ID, local FDR value, GWAS association p-value, the leaf ID in which the SNP is located, and the corresponding complete functional annotation information. The 'Download SNPTable'button allows users to download the 'SNPTable'as a CSV format file.
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| {'fraction_non_alphanumeric': 0.04715099459129216, 'fraction_numerical': 0.024042694650590286, 'mean_word_length': 4.912249017316477, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 2, 'https://': 2, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 1, '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': 'Motivation: In spite of great success of genome-wide association studies (GWAS), multiple challenges still remain. First, complex traits are often associated with many single nucleotide polymorphisms (SNPs), each with small or moderate effect sizes. Second, our understanding of the functional mechanisms through which genetic variants are associated with complex traits is still limited. functional annotations related to risk-associated SNPs. To address these challenges, we propose GPA-Tree and it simultaneously implements association mapping and identifies key combinations of functional annotations related to risk-associated SNPs by combining a decision tree algorithm with a hierarchical modeling framework.Results: First, we implemented simulation studies to evaluate the proposed GPA-Tree method and compared its performance with existing statistical approaches. The results indicate that GPA-Tree outperforms existing statistical approaches in detecting risk-associated SNPs and identifying the true combinations of functional annotations with high accuracy. Second, we applied GPA-Tree to a systemic lupus erythematosus (SLE) GWAS and functional annotation data including GenoSkyline and GenoSky-linePlus. The results from GPA-Tree highlight the dysregulation of blood immune cells, including but not limited to primary B, memory helper T, regulatory T, neutrophils and CD8 + memory T cells in SLE.These results demonstrate that GPA-Tree can be a powerful tool that improves association mapping while facilitating understanding of the underlying genetic architecture of complex traits and potential mechanisms linking risk-associated SNPs with complex traits.Availability: The GPATree software is available at https://dongjunchung.github.io/GPATree/.', 'arxivid': '2106.06877', 'author': ['Aastha Khatiwada \nDepartment of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA\n', 'Bethany J Wolf \nDepartment of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA\n', 'Ayse Selen Yilmaz \nDepartment of Biomedical Informatics\nThe Ohio State University\nColumbusOhioUSA\n', 'Paula S Ramos \nDepartment of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA\n\nDepartment of Medicine\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA\n', 'Maciej Pietrzak \nDepartment of Biomedical Informatics\nThe Ohio State University\nColumbusOhioUSA\n', 'Andrew Lawson \nDepartment of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA\n', 'Kelly J Hunt \nDepartment of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA\n', 'Hang J Kim \nDivision of Statistics and Data Science\nUniversity of Cincinnati\nCincinnatiOhioUSA\n', 'Dongjun Chung \nDepartment of Biomedical Informatics\nThe Ohio State University\nColumbusOhioUSA\n'], 'authoraffiliation': ['Department of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA', 'Department of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA', 'Department of Biomedical Informatics\nThe Ohio State University\nColumbusOhioUSA', 'Department of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA', 'Department of Medicine\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA', 'Department of Biomedical Informatics\nThe Ohio State University\nColumbusOhioUSA', 'Department of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA', 'Department of Public Health Sciences\nMedical University of South Carolina\nCharlestonSouth CarolinaUSA', 'Division of Statistics and Data Science\nUniversity of Cincinnati\nCincinnatiOhioUSA', 'Department of Biomedical Informatics\nThe Ohio State University\nColumbusOhioUSA'], 'corpusid': 235421781, 'doi': '10.1093/bioinformatics/btab802', 'github_urls': [], 'n_tokens_mistral': 16305, 'n_tokens_neox': 13449, 'n_words': 8246, 'pdfsha': '4df76ca15df6b7d78747ebc01ac2a06e69797c4c', 'pdfurls': ['https://arxiv.org/pdf/2106.06877v1.pdf'], 'title': ['GPA-Tree: Statistical Approach for Functional-Annotation-Tree-Guided Prioritization of GWAS Results', 'GPA-Tree: Statistical Approach for Functional-Annotation-Tree-Guided Prioritization of GWAS Results'], 'venue': []} |
arxiv |
4 Feb 2016
Alexander Braverman
Department of Mathematics
Department of Mathematics
Department of Mathematics
Department of Mathematics
WITH AN APPENDIX BY R. BEZRUKAVNIKOV
Perimeter Institute for Theo-retical Physics
BERNSTEIN COMPONENTS VIA BERNSTEIN CENTER
Massachusetts Institute of Technology
University of Toronto
Brown University
Hebrew University of Jerusalem
David Kazhdan
Department of Mathematics
Department of Mathematics
Department of Mathematics
Department of Mathematics
WITH AN APPENDIX BY R. BEZRUKAVNIKOV
Perimeter Institute for Theo-retical Physics
BERNSTEIN COMPONENTS VIA BERNSTEIN CENTER
Massachusetts Institute of Technology
University of Toronto
Brown University
Hebrew University of Jerusalem
R B
Department of Mathematics
Department of Mathematics
Department of Mathematics
Department of Mathematics
WITH AN APPENDIX BY R. BEZRUKAVNIKOV
Perimeter Institute for Theo-retical Physics
BERNSTEIN COMPONENTS VIA BERNSTEIN CENTER
Massachusetts Institute of Technology
University of Toronto
Brown University
Hebrew University of Jerusalem
A B
Department of Mathematics
Department of Mathematics
Department of Mathematics
Department of Mathematics
WITH AN APPENDIX BY R. BEZRUKAVNIKOV
Perimeter Institute for Theo-retical Physics
BERNSTEIN COMPONENTS VIA BERNSTEIN CENTER
Massachusetts Institute of Technology
University of Toronto
Brown University
Hebrew University of Jerusalem
D K
Department of Mathematics
Department of Mathematics
Department of Mathematics
Department of Mathematics
WITH AN APPENDIX BY R. BEZRUKAVNIKOV
Perimeter Institute for Theo-retical Physics
BERNSTEIN COMPONENTS VIA BERNSTEIN CENTER
Massachusetts Institute of Technology
University of Toronto
Brown University
Hebrew University of Jerusalem
4 Feb 2016Dedicated to J. Bernstein on the occasion of his 70th birthday
Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that Φ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modification of the argument from Section 2 of [6], where our result is proven in one direction.
1. Introduction 1.1. Components of Irr(G). In this paper G denotes the set of point of a connected reductive algebraic group over a local non-archimedian field K. We shall denote by M(G) the category of smooth complex representations of G. This category is equivalent to the category of unital modules over the Hecke algebra H(G). We let Irr(G) denote the set of isomorphism classes irreducible objects of M(G). Bernstein and Zelevinsky defined a decomposition of the set Irr(G) of irreducible representations of G into a union of certain components Ω; this decomposition in fact defines a decomposition of M(G) into a product of the corresponding categories. The set C(G) of components of Irr(G) is in one-to-one correspondence with pairs (M, σ) where M is a Levi subgroup of G and σ is a cuspidal representation of M (the data of (M, σ) is uniquely determined by Ω up to natural equivalence relation generated by conjugation and multiplying σ by an unramified character of M ). An element π ∈ Irr(G) lies in Ω(M, σ) if and only if there exists a parabolic subrgoup P containing M as a Levi subgroup and an unramified character χ of M such that π is a subquotient of the induced representation Ind G P (σ⊗χ) (here we use the natural map P → M in order to view σ ⊗ χ as a representation of P ). Every Ω(M, σ) is equipped with a map to an irreducible affine algebraic variety Ω(M, σ). The variety Ω(M, σ) is in fact a quotient of the torus of unramified characters of M by a finite group. The above map has finite fibers and is generically one-to-one. We shall say that a function f : Ω(M, σ) → C is regular if it comes from a regular function on Ω(M, σ). We shall say that a function f : Irr(G) → C is regular iff it is regular when restricted to every component.
Bernstein center.
Let Z(G) be the center of the category M(G). It is easy to see that it consists of all invariant distributions Φ on G such that for any h ∈ H(G) we have Φ ⋆ h ∈ H(G). It is enough to test the above condition for all h = e K where K is an open compact subgroup of K and e K is the Haar measure on it.
By Schur-Quillen lemma any Φ ∈ Z(G) defines a function on the set Irr(G). Bernstein proved (cf. [1]) that in this way we get an isomorphism between Z(G) and the algebra of regular functions on Irr(G). Thus Z(G) has both "geometric" (in terms of distributions on G) and "spectral" (in terms of functions on Irr(G)) description. The relationship between these two descriptions tends to be quite non-trivial. This note is devoted to one particular aspect of this relationship. Namely, we are going to prove the following Theorem 1.3. Let Z comp (G) denote the subspace of Z(G) consisting of distributions supported on compact elements. Similarly, let Z lc (G) denote the subalgebra of Z(G) consisting of those elements Φ for which f (Φ) is a locally constant function (i.e. a constant function when restricted to every Bernstein component of Irr(G)). Then Z comp (G) = Z lc (G). Theorem 1.3 has the following surprising corollary (in fact, technically we are first going to prove the corollary and then deduce Theorem 1.3 from it, but historically our starting conjectural point was the assertion of Theorem 1.3):
Corollary 1.4. Z comp (G) is a subalgebra of Z(G).
Corollary 1.4 is surprising since the set of compact elements of G is not closed under multiplication. We believe that Corollary 1.4 is actually a part of a more general statement. While we are not sure what this statement really is, at least we believe in the following: Namely, it is shown in loc. cit. that every idempotent in Z(G) is supported on compact elements. Hence if for every Ω ∈ C(G) we denote by E Ω the element of Z(G) for which the function f (E Ω ) is equal to 1 on Ω and is equal to 0 on any other component, then E Ω ∈ Z comp (G). On the other hand, any Φ ∈ Z(G) such that f (Φ) is constant on every Ω is locally on G a linear combination of the distributions E Ω , hence Φ is supported on compact elements. The main observation of this note is that a mild adaptation of Dat's argument also proves the converse statement.
A variant.
In fact the inclusion Z lc (G) ⊂ Z comp (G) has the following stronger version. Given an element g ∈ G we can define in a standard way a parabolic subgroup P g of G and a strictly dominant element λ g ∈ Z(M g )/Z(M g ) 0 (the latter group is always a lattice and we shall denote the multiplication there by +; also, "strictly dominant" means that the adjoint action of λ g contract the unipotent radical of P g to the unit element). Here we denote by M g the Levi group of P g ; also Z(M ) stands for the center of M g and Z(M g ) 0 is its maximal compact subgroup. Namely, P g consists of all x ∈ G such that lim n→∞ g n xg −n exists. Also the image of g under the natural map P g → M g must be compact modulo center and hence g defines an element in Z(M )/Z(M ) ∩ M 0 = Z(M )/Z(M ) 0 which we call λ g . Note that P g = G if and only if g is compact modulo center. Moreover, we have P g = G, λ g = 0 if and only if g is compact.
Let now P(G) denote the set of conjugacy classes of pairs (P, λ) as above. Then the above construction produces a decomposition Let now D(G) denote the space of distributions on G; let also D inv (G) ⊂ D(G) be the space of invariant distributions. Then (1.1) produces a decomposition
D inv (G) = (P,λ)∈P(G) D inv P,λ . (1.2)
Here D inv P,λ consists of all invariant distributions supported on G P,λ . We can now formulate Theorem 1.8. Let Φ ∈ Z lc (G). Then convolution with Φ preserves the decomposition (1.2) (i.e. preserves each D inv P,λ ). This result is due to R. Bezrukavnikov and we reproduce its proof in the Appendix. Theorem 1.8 implies the inclusion Z lc (G) ⊂ Z comp (G). Namely let δ denote the deltadistribution at the unit element of G. Obviously δ ∈ D inv G,0 , hence Φ = Φ ⋆ δ ∈ D inv G,0 , i.e. Φ is supported on compact elements. Our proof of Theorem 1.8 is somewhat simpler than the proof of the inclusion Z lc (G) ⊂ Z comp (G) from [6]. However, we still need the arguments of [6] in order to prove the opposite inclusion.
1.9. An example. For a rational number r ∈ Q ≥0 Moy and Prasad (cf. [7]) define a subset Irr ≤r (G) of Irr(G) called "representations of depth ≤ r". The set Irr ≤r (G) is a union of components of Irr(G). Let Φ r ∈ Z G be the projector to Irr ≤r (G); in other words Φ r is the element of Z(G) such that f (Φ r )(π) = 1 if π ∈ Irr ≤r (G) and f (Φ r )(π) = 0 otherwise. According to Theorem 1.3 Φ r should be concentrated on compact elements. In [3] the authors give an explicit formula for Φ r which indeed shows this explicitly. In fact, the main result of [3] implies a much stronger restriction to on the support of Φ r . It would be interesting to include this restriction into a general theorem in the style of Theorem 1.3.
1.10. Geometric and spectral support. We conclude the introduction with yet another conjecture which contains Theorem 1.3 as a special case. To simplify the discussion we shall assume that G is a split.
Let us assume that G = G(K) where G is the corresponding split algebraic group defined over Z. Let Λ denote the coweight lattice of G; we shall denote by Λ + the set of dominant coweights. Also, for λ, µ ∈ Λ we shall write λ ≥ µ if λ − µ is a sum of positive coroots of G. Let K = G(O). Then by Cartan decomposition the double quotient K\G/K is in natural bijection with the set Λ + of dominant coweights of G. For each λ ∈ Λ + we shall denote by G λ the corresponding double coset. We set For an open subset X of G we denote by 1 X the characteristic function of X. Then multiplication (not convolution!) by 1 X is an endomorphism of H which (abusing the notation) we shall also denote by the same symbol. Moreover, if X is invariant under conjugation, then multiplication by 1 X descends to endomorphism of H, which we shall denote by1 X . Then1 G 0 ,1 Gc and1 G 0 c are well-defined and we have1 G 0 c =1 Gc •1 G 0 . Also, all of these endomorphisms commute with each other.
G ≤λ = µ∈Λ + ,µ≤λ G µ . We now define Z ≤λ geom (G) = {Φ ∈ Z(G)| supp(Φ) ⊂ AdG · (G ≤λ )}. (1.3) Note that Z ≤0 geom (G) = Z comp (G).
Below is the main result of this Section. Let us now assume that we are given Φ ∈ Z(G) such that Φ commutes with1 G 0 c . Then
for any h ∈ H we have Φ ⋆ (h| G 0 c ) = (Φ ⋆ h)| G 0 c . Let h(g) = h(g −1 ). Then we have Φ( h) = (Φ ⋆ h)(e) = (Φ ⋆ h)| G 0 c (e) = (Φ ⋆ (h| G 0 c ))(e). Hence Φ( h) = 0 if supp( h) = supp(h) ⊂ G\G 0 c , which means that supp Φ ⊂ G 0 c .
2.4. Let us now start proving the opposite direction. Namely, let Φ ∈ Z comp (G). We want to show that Φ commutes with1 G 0 c . For this it is enough to prove that Φ commutes with 1 G 0 and1 Gc . Let us first prove that Φ commutes with1 G 0 . For this it is enough to prove that Φ commutes with 1 G 0 . In other words, we need to prove that for any h ∈ H we have Φ ⋆ (h| G 0 ) = (Φ ⋆ h)| G 0 . But this is obvious since G 0 is a subgroup of G and supp(Φ) ⊂ G 0 .
Φ ⋆ī H GM (f ) =ī H GM (r Z GM (Φ) ⋆ f ) 2) For any Φ ∈ Z(G), h ∈ H(G) we have r H M G (Φ ⋆ h) = r Z M G (Φ) ⋆ r H M G (h) 3)
Let U P denote the unipotent radical of a standard parabolic subgroup P with the standard Levi subgroup M . Let π P denote the natural projection from G to G/U P ; clearly M is a closed subset of G/U P . Let now Φ ∈ Z(G). The direct image (π P ) * Φ makes sense -by the definition for any locally constant compactly supported function φ on G/U P we set (π P ) * Φ(φ) = (e K ⋆ Φ)(π * P φ), where K is any open compact subgroup of G such that φ is K-invariant and e K is the Haar measure on K. Then the distribution (π P ) * (Φ) is concentrated on M . Moreover, the resulting distribution on M is equal to r Z GM (Φ) up to multiplication by an unramified character of M . Property 3) above implies that if Φ ∈ Z comp (G) then r Z GM (Φ) ∈ Z comp (M ). Indeed, this follows from the fact that an element g ∈ P ⊂ G is compact if and only if its projection to M = P/U P is compact.
2.6.
Clozel's formula. The main ingredient of the argument of Section 2 of [6] (and also of our proof of Theorem 2.2) is the following formula due to Clozel. Moreover, χ G = 1.
On the other hand, by Plancherel formula for every Ω ∈ C(G) there exists a measure dπ Ω on Ω such that
δ G = Ω∈C(G) π∈Ω ch π dπ Ω . (3.1)
Here δ G denotes the δ-distribution at the unit element of G.
Convolving this with a central element Φ ∈ Z(G) we get
Φ = Ω∈C(G) π∈Ω f (Φ)(π) ch π dπ Ω . (3.2) If Φ ∈ Z Ω,comp we get Φ = π∈Ω f (Φ)(π) ch π dπ Ω . (3.3)
Since the LHS of (3.3) is concentrated on G 0 c , the same is true for RHS. Thus we get
Φ = π∈Ω f (Φ)(π) ch π | G 0 c dπ Ω . (3.4)
Hence Φ ∈ D Ω , i.e. Z Ω,comp ⊂ D Ω , which implies that dim Z Ω,comp is finite-dimensional. which is compatible with (1.2) by means of the above pairing. Namely, we let H(G) P,λ to be the image of H(G) P,λ where the latter consists of functions supported on G P,λ . The fact that (4.1) holds is clear.
Spectral description of H(G).
The space H(G) admits the following well-known description. Let π ∈ M(G) be a finitely generated representation and let E be an endomorphism of π. It is well-known (cf. e.g. [8]) that we can associate to the pair (π, E) and element [π, E] of H(G). Moreover, H(G) is isomorphic to the C-span of symbols [π, E] subject to the relations: a) Let π 1 , π 2 ∈ M(G) and let u ∈ Hom(π 1 , π 2 ), v ∈ Hom(π 2 , π 1 ).
Then [π 1 , vu] = [π 2 , uv]. b) [π 1 , E 1 ] + [π 3 , E 3 ] = [π 2 , E 2 ] for a short exact sequence 0 → π 1 → π 2 → π 3 → 0 which is compatible with the endomorphisms E i ∈ End (π i ). c) [π, c 1 E 1 + c 2 E 2 ] = c 1 [π, E 1 ] + c 2 [π, E 2 ]
, where c i ∈ C and E i ∈ End (π). The action of Z(G) on H(G) can also be described in these terms. Namely, let Φ ∈ Z(G).
Then Φ · [π, E] = [π, E • π(Φ)].
In addition, let ρ be an admissible representation of G. Then we have [π, E], ch ρ = i (−1) i Tr(E, Ext i (π, ρ)).
(4.2)
In view of the Trace Paley-Wiener theorem (cf. [2]), (4.2) defines [π, E] uniquely.
4.3.
Spectral description of H P,λ . Let now P , P be a pair of opposite parabolic subgroups with M = P ∩ P . Let λ ∈ Z(M )/Z(M ) 0 such that (P , λ) ∈ P(G). Let also σ be a finitely generated representation of M . Set
π = i GP (σ). (4.3)
Let us now choose a uniformizer t of our local field. Then any λ ∈ Z(M )/Z(M 0 ) lifts naturally to an element t λ ∈ Z(M ). Hence it defines an endomorphism of σ and thus also of π. We shall denote this endomorphism by E λ .
Theorem 4.4. The subspace H P ,λ is spanned by elements [π, E λ ] as above (here again P denotes a parabolic subgroup which is opposite to P ).
Remark. The element [π, E λ ] actually depends on the choice of t; however, it is easy to see that the span of all the [π, E λ ] does not.
Proof. For (P, λ) = (G, 0) this is the "abstract Selberg principle" (cf. [4]). The case P = G and arbitrary λ is completely analogous. Let us now take arbitrary P and λ. Let σ be a finitely generated representation of the Levi group M as above and λ -a strictly dominant cocharacter of π. Now to prove (4.4) it is enough ( by Trace Paley-Wiener theorem) to check that both the LHS and the RHS of (4.4) have the same inner product with ch ρ where ρ stands for a generic irreducible representation of G. But we have Ext i G (i GP (σ), ρ) = Ext i M (σ, r GP (ρ)). Hence [π, E λ ], ch ρ = [σ, λ], r GP (ρ) and (4.4) follows from (4.5) and from the Casselman formula for the character of r GP (ρ) (cf. [5]) which says for any g ∈ G such that P g = P we have ch ρ (g) = ch r GP (ρ) (g). Proof. It is enough to show that the action of any Φ ∈ Z lc (G) preserves each H P ,λ . Let us consider an element [π, E] as above; without loss of generality we may assume that all irreducible subquotients of σ lie in one component of C(M ). But then all irreducible subquotients of π as in lie in one component Ω ∈ C(G) and it follows that Φ ⋆ [π, E λ ] = [π, f (Φ)| Ω · E λ ] = f (Φ)| Ω · [π, E λ ] (note that f (Φ)| Ω ∈ C as Φ ∈ Z lc (G)). Hence the span of all the [π, E λ ] is preserved by the convolution with Φ.
Conjecture 1 . 5 .
15Let Z tunip (G) denote the subspace of Z(G) consisting of distributions supported on topologically unipotent elements. Then Z tunip (G) is a subalgebra of Z(G).
1. 6 .
6Relation to the work of J.-F. Dat. Theorem 1.3 is in fact not completely newthe inclusion Z lc (G) ⊂ Z comp (G) was essentially proved by J.-F. Dat (cf. Section 2 of [6]).
G P,λ is an open subset of G invariant under conjugation.
On the hand, let P, M be a parabolic subgroup of G and its Levi subgroup (both defined over K). Let Λ M be the cocharacter lattice of M/[M, M]. Then Λ M is a sublattice of Λ and C[Λ M ] is the algebra of regular functions on the set of unramified characters of M (for an element λ ∈ Λ M we shall denote by e λ the corresponding element of C[Λ M ]). Fix a cuspidal representation σ of M with unitary central character. Then every Φ ∈ Z(G) defines anelement f σ (Φ) ∈ C[Λ M ].Namely, for an unramified ψ : M → C * we define f σ (Φ)(ψ) to be the scalar by which Φ acts in i GM (σ ⊗ ψ) where i GM stands for unitary induction from P to G. We now defineZ ≤λ spectral (G) = {Φ ∈ Z(G)|f σ (Φ) = µ≤λ a µ e µ . (1.4) It is easy to see that Φ ∈ Z ≤0 spectral (G) ifand only if f (Φ) is constant on every component of Irr(G). We can now formulate Proposition 1.11. For any λ ∈ Λ + we have Z ≤λ geom (G) = Z ≤λ spectral (G). Proposition 1.11 reduces to Theorem 1.3 when λ = 0. The proof of Proposition 1.11 easily follows from Theorem 4.4. 1.12. Acknowledgements. A.B. and R.B. were partially supported by the National Science Foundation. D.K. was partially supported by the European Research Council. 2. Proof of Corollary 1.4 As was mentioned above, for the most part our proof is a repetition of the arguments of Section 2 of [6]. 2.1. Z comp and H. Let H(G) denote the Hecke algebra of G. In what follows we shall choose a Haar measure on G and we are going to identify H(G) with the space of locally constant compactly supported functions on G. Let H(G) = H(G)/[H(G), H(G)]. Obviously Z(G) acts on H. For any Φ ∈ Z(G) we shall denote by Φ the corresponding endomorphism of H.Following[6] let us denote by G 0 the subgroup of G generated by all the open compact subgroups of G. We also denote by G c the set of compact elements modulo center. Then G 0 c = G 0 ∩ G c is the set of compact elements of G.
2. 5 .
5Induction and restriction. Let us choose a split Cartan subgroup T of G and a Borel subgroup B of G with unipotent radical U . We denote by B the opposite Borel subgroup of G. Then we have a notion of standard Levi subgroup M of G. We shall use the notation M < G to indicate that M is a standard Levi subgroup of G. To any such M there corresponds a pair of parabolic subgroups P, P , where P ∩ P = M and B ⊂ P, B ⊂ P . Also for any such M we have maps (cf. [6] and references therein): i Z GM ,ī Z GM : Z(M ) → Z(G), r Z GM , r Z GM : Z(G) → Z(M ), i H GM ,ī H GM : H(M ) → H(G), r H GM , r H GM : H(G) → H(M ). These maps satisfy the following properties: 0) All these maps are equal to identity when M = G. 1) For any Φ ∈ Z(G), f ∈ H(M ) we have
Proposition 2 . 7 .
27For any standard Levi M there exists a function χ M : M/M 0 → C such that for any h ∈ H(G) we have h = M <Gī H GM (χ M · (1 Mc (r H GM (h))). (2.1)
4 .
4Appendix: Proof of Theorem 1.8 (by R. Bezrukavnikov) 4.1. Decomposition of H(G). We have an obvious perfect pairing between D inv (G) and H(G). We claim that there is a decomposition
Then we have a natural identification t λ M 0 c /Ad(M ) = G P,λ /Ad(G). Hence we get a natural isomorphism between H(M ) M,0 and H(G) P,λ . Indeed, if an element of H(M ) is represented by some h ∈ H(M ) supported on M 0 c = M M,0 , then let us denote by h λ the corresponding element of H(M ) supported on M M,λ = t λ · M 0 c . For an open compact subgroup K of G let us denote by h λ,K the result of averaging of h λ with respect to the adjoint action of K. Its image in H(G) is independent of K and the assignment h mod[H(M ), H(M )] → h λ,K mod[H(G), H(G)] is the desired isomorphism. Let us denote it by η P,λ . Now in order to finish the proof it is enough to show that for π as in (4.3) we have [π, E λ ] = η P ,λ ([σ, Id]). (4.4) Let h = [σ, Id]. Then it is easy to see that [σ, t λ ] = h λ . (4.5)
Corollary 4 . 5 .
45Theorem 1.8 holds.
Theorem 2.2. We have Φ ∈ Z comp (G) if and only if Φ commutes with1 G 0 c .It is clear that Theorem 2.2 implies Corollary 1.4.
2.3. Proof of Theorem 2.2: the "if" direction. This is the easy part of Theorem 2.2.
Note that two elements h 1 , h 2 ∈ H have the same image in H iff for any invariant distribution
E on G we have E(h 1 ) = E(h 2 ). In particular, if h 1 = h 2 , then h 1 (e) = h 2 (e) where e is the
unit element of G.
2.8.End of the proof. We can now finish the proof of Theorem 2.2. By induction we can assume that for any Levi subgroup M of G which is different from G and any Φ ∈ Z comp (M ) we have1McThen we have1Here 1 = follows from the fact that r Z GM (Φ) ∈ Z comp (M ) and from Section 2.4. The equality 2 = follows from the induction hypothesis.3. Proof of Theorem 1.33.1.To finish the proof of Theorem 1.3 we need to show that for any Φ ∈ Z comp (G) the function f (Φ) is constant on every Ω ∈ C(G). Let us recall that we denote by E Ω the element of Z(G) for which the function f (E Ω ) is equal to 1 on Ω and is equal to 0 on any other component. Then by[6]we have E Ω ∈ Z comp (G) and by Corollary 1.4 we also have Φ ⋆ E Ω ∈ Z comp (G) and it is enough to prove that f (Φ ⋆ E Ω ) is constant on Ω. Let Z Ω (G) denote the subalgebra of Z(G) consisting of elements Φ such that f (Φ) is equal to 0 on any Ω ′ = Ω and let Z Ω,comp (G) = Z comp (G) ∩ Z Ω (G). We already know that E Ω ∈ Z Ω,comp (G). Clearly, our assertion follows from the followingWe claim that Proposition 3.2 follows from the following:Indeed, Z Ω,comp (G) is a subalgebra of Z Ω (G) which is isomorphic to the algebra of functions on an irreducible algebraic varietyΩ. Hence the only finite-dimensional subalgebra of it consists of constants. So, it remains to prove Lemma 3.3.Proof. Let D(G 0 c ) denote the space of distributions on G 0 c . Consider the subspace D Ω of D(G 0 c ) generated by distributions of the form ch π | G 0 c where π is an irreducible representation in Ω and ch π is its character. Then it follows immediately from the Corollary in Section 3.1 in[2]that D Ω is finite-dimensional.
J Bernstein, P Deligne, Le Centre De Bernstein, Representations des groups reductifs sur un corps local. Hermann, ParisTraveaux en cours (P.Deligne ed.J. Bernstein and P. Deligne, Le centre de Bernstein, In Representations des groups reductifs sur un corps local, Traveaux en cours (P.Deligne ed.), Hermann, Paris, 1-32 (1984).
Trace Paley-Wiener theorem for reductive p-adic groups. J Bernstein, P Deligne, D Kazhdan, J. Analyse Math. 47J. Bernstein, P. Deligne and D. Kazhdan, Trace Paley-Wiener theorem for reductive p-adic groups, J. Analyse Math. 47 (1986), 180-192.
R Bezrukavnikov, D Kazhdan, Y Varshavsky, arXiv:1504.01353On the depth r Bernstein projector. R. Bezrukavnikov, D. Kazhdan and Y. Varshavsky, On the depth r Bernstein projector, arXiv:1504.01353.
Cyclic homology and the Selberg principle. P Blanc, J.-L Brylinsky, J. Func. Anal. 109P. Blanc and J.-L. Brylinsky, Cyclic homology and the Selberg principle, J. Func. Anal. 109, (1992) 289-330.
Characters and Jacquet modules. W Casselman, Math. Ann. 2302W. Casselman, Characters and Jacquet modules, Math. Ann. 230 (1977), no. 2, 101-105.
Quelques propriétés des idempotents centraux des groupes p-adiques. J.-F Dat, J. reine angew. Math. 554J.-F. Dat, Quelques propriétés des idempotents centraux des groupes p-adiques, J. reine angew. Math. 554 (2003), 69-103
Unrefined minimal K-types for p-adic groups. A Moy, G Prasad, Invent. Math. 116393408A. Moy and G. Prasad, Unrefined minimal K-types for p-adic groups, Invent. Math. 116 (1994) 393408.
On formal dimensions for reductive p-adic groups, Israel Math. M.-F Vignéras, Conf. Proc. 2M.-F. Vignéras, On formal dimensions for reductive p-adic groups, Israel Math. Conf. Proc. 2, 225-265, 1990.
| {'fraction_non_alphanumeric': 0.07322953700298312, 'fraction_numerical': 0.018389113644722323, 'mean_word_length': 3.3943257317292153, 'pattern_counts': {'":': 0, '<': 2, '<?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': 5, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Let G be a reductive p-adic group. Let Φ be an invariant distribution on G lying in the Bernstein center Z(G). We prove that Φ is supported on compact elements in G if and only if it defines a constant function on every component of the set Irr(G); in particular, we show that the space of all elements of Z(G) supported on compact elements is a subalgebra of Z(G). Our proof is a slight modification of the argument from Section 2 of [6], where our result is proven in one direction.', 'arxivid': '1512.08637', 'author': ['Alexander Braverman \nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n\n', 'David Kazhdan \nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n\n', 'R B \nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n\n', 'A B \nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n\n', 'D K \nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n\n'], 'authoraffiliation': ['Department of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n', 'Department of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n', 'Department of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n', 'Department of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n', 'Department of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nDepartment of Mathematics\nWITH AN APPENDIX BY R. BEZRUKAVNIKOV\nPerimeter Institute for Theo-retical Physics\nBERNSTEIN COMPONENTS VIA BERNSTEIN CENTER\nMassachusetts Institute of Technology\nUniversity of Toronto\nBrown University\nHebrew University of Jerusalem\n'], 'corpusid': 119273782, 'doi': '10.1007/s00029-016-0277-3', 'github_urls': [], 'n_tokens_mistral': 8458, 'n_tokens_neox': 7614, 'n_words': 4750, 'pdfsha': '1d569841bd762c6b029481de7484551344a3885a', 'pdfurls': ['https://arxiv.org/pdf/1512.08637v3.pdf'], 'title': [], 'venue': []} |
arxiv |
Tightly Integrated Motion Classification and State Estimation in Foot-Mounted Navigation Systems
Isaac Skog [email protected]
Dept. of Electrical Engineering
Dept. of Electrical Engineering
Uppsala University Uppsala
Sweden
Gustaf Hendeby [email protected]
Delft Center for Systems and Control Delft
Linköping University Linköping
Sweden
Manon Kok
University of Technology Delft
The Netherlands
Tightly Integrated Motion Classification and State Estimation in Foot-Mounted Navigation Systems
Index Terms-Inertial navigationZero-velocity detectionConstant height detectionFilter bankMotion-constraints
A framework for tightly integrated motion mode classification and state estimation in motion-constrained inertial navigation systems is presented. The framework uses a jump Markov model to describe the navigation system's motion mode and navigation state dynamics with a single model. A bank of Kalman filters is then used for joint inference of the navigation state and the motion mode. A method for learning unknown parameters in the jump Markov model, such as the motion mode transition probabilities, is also presented. The application of the proposed framework is illustrated via two examples. The first example is a foot-mounted navigation system that adapts its behavior to different gait speeds. The second example is a foot-mounted navigation system that detects when the user walks on flat ground and locks the vertical position estimate accordingly. Both examples show that the proposed framework provides significantly better position accuracy than a standard zero-velocity aided inertial navigation system. More importantly, the examples show that the proposed framework provides a theoretically well-grounded approach for developing new motionconstrained inertial navigation systems that can learn different motion patterns.
I. INTRODUCTION
Current state-of-the-art technology for zero-velocity aided inertial navigation systems is based upon a strategy of loose integration between the zero-velocity detector and the inertial navigation filter [1]. That is, the zero-velocity detector and the inertial navigation filter are treated as separate functions, with a one-directional flow of information from the former to the latter. From an information theoretical perspective this is suboptimal because: (a) the estimated navigation states carry information about the system's motion mode that is not used in the zero-velocity detector, and (b) the test statistic used in the zero-velocity detector is quantized before it is used in the navigation filter. Thus, information that could be used both to improve the detection of zero-velocity events and to control how the zero-velocity updates are performed, is lost. The same argumentation also applies to other motion constraints commonly applied to foot-mounted inertial navigation systems and where an external motion classifier is used to determine when to apply these constraints. A few examples of commonly used motion constraints are that the system keeps a constant height [2], constant heading [3], or constant speed [4].
The fact that the estimated navigation state carries information that can be used to improve the zero-velocity detection process is utilized in [5], where the velocity estimates from the inertial navigation filter are used as a prior for a Bayesian zero-velocity detector. And in [6], the test statistics from the zero-velocity detector are used to control the magnitude of the measurement covariance in the inertial navigation filter. Thereby, quantization of the zero-velocity test statistics is avoided and the zero-velocity update process can adapt to different gait conditions. Still, both [5] and [6] employ a loose integration strategy where the zero-velocity detector and the inertial navigation filter are treated as separate functions.
This paper instead proposes a framework for tightly integrated motion mode classification and navigation state estimation, of which tightly integrated zero-velocity aided inertial navigation is a special case. The core of the framework is a jump Markov model that includes both the navigation state and the motion mode. That is, a single model is used to describe both the kinematics of the system under various motion modes and the probability of transitioning between the motion modes. A bank of Kalman filters is then used to jointly estimate the navigation state and the motion mode. A method to automatically learn unknown parameters in the jump Markov model is also presented. The proposed framework is evaluated on two data sets consisting of various gait conditions and motion modes.
Reproducible research: The data and code used to produce the presented results can be downloaded at www.openshoe.org.
II. SIGNAL MODEL
Let the system state x k and input vector u k at time instant k be defined as
x k = r k v k q k ξ k and u k = s k ω k ,(1)
respectively. Here r k , v k , and q k denote the position, velocity, and attitude quaternion, respectively. Further, ξ k denotes potential auxiliary states, such as sensor biases, needed to describe the behavior of the system. Moreover, s k and ω k denote the specific force and angular velocity, respectively.
Next, define the discrete state
δ k ∈ {1, . . . , L}(2)
that indicates the current motion mode of the navigation system, e.g., the system is stationary, keeping a constant height, or moving at constant velocity. A jump Markov nonlinear model that can be used to the describe the dynamics of the inertial navigation system is then given by [7] x
k+1 = f (x k , δ k , u k , η k ) (3a) y k = h(x k , δ k , u k ) + e k (3b) η k i.i.d. ∼ N n k ; 0, Q k (3c) e k i.i.d. ∼ N e k ; 0, R(δ k ) (3d) p(δ k+1 |δ k ) = Π δ k+1 ,δ k .(3e)
Here y k 0 is used as a pseudo-observation to impose a set of motion mode dependent "stochastic constraints", such as zerovelocity constraints, on the navigation state x k . The exact form of these constraints, defined by h(·), depends on the assumed motion modes and will be discussed later. The "hardness" of the imposed constraints is controlled via the motion mode dependent covariance R(δ k ). Further, N · ; µ, Σ denotes a multivariate normal distribution parameterized by the mean vector µ and covariance matrix Σ. The function f (·) describes the system dynamics and is given by the inertial navigation equations and the dynamics of the auxiliary states ξ k . Moreover, Q k denotes the covariance of the process noise, and Π i,j denotes the i:th row and j:th column entry of the mode transition probability matrix. Finally, p(a|b) denotes the probability density (mass) function of a given b. Note that it is straightforward to extend the model (3) to also include real observations from various sensors, but it is out of the scope of this paper, as the focus is on motion-constrained inertial navigation.
III. STATE ESTIMATION USING A FILTER BANK A variety of inference techniques for jump Markov models exist. Here a commonly used filter banks solution to the inference problem will be recapitulated; for details the reader is referred to [8] and [9]. Thereafter, necessary approximations needed to adapt the filter bank solution to the considered navigation problem will be presented.
A. Linear model with normal distributed noise
The goal of the inference process is to estimate the a posteriori distribution p(x k |y 1:k ) of the state x k given all the observations up until time k, denoted as y 1:k . If the motion mode sequence δ 1:k is known, the state-space model in the Markov jump system is linear, and the process and measurement noise are normally distributed, then the a posteriori distribution can be estimated with the Kalman filter. That is, p(x k |δ 1:k , y 1:k ) = N x k ;x δ 1:k k|k , P δ 1:k k|k ,
wherex δ 1:k k|k and P δ 1:k k|k denote the Kalman filter state estimate and state covariance given the mode sequence δ 1:k , respectively. In reality, the motion mode sequence is unknown and must be estimated from the measurements. Let p(δ 1:k |y 1:k ) denote the a posteriori distribution of the motion mode sequences δ 1:k given the measurements y 1:k . The sought-after a posteriori distribution of the state x k can then be found as
p(x k |y 1:k ) = L k i=1 p(x k |δ i 1:k , y 1:k )p(δ i 1:k |y 1:k ) = L k i=1 w i k N x k ;x δ i 1:k k|k , P δ i 1:k k|k ,(5)
where w i k p(δ i 1:k |y 1:k ) is probability of the i:th motion mode sequence. Hence, the posteriori distribution of the state x k is given by a mixture of L k weighted normal distributions where the expected value and covariance of each mixture component are calculated via a Kalman filter. That is, the posteriori distribution can be calculated via a bank of Kalman filters where the size of the filter bank grows according to a tree structure. The weights w i k in the mixture, i.e., the probability for each branch in the tree, can be recursively calculated as
w i k ∝ p(y k |y 1:k−1 , δ i 1:k )p(δ i k |δ i k−1 )w i k−1 , (6a) where L k i=1 w i k = 1,(6b)
and
p(y k |y 1:k−1 , δ i 1:k ) = N y k ;ŷ δ i 1:k k|k−1 , S δ i 1:k k . (6c)
Hereŷ δ 1:k k|k−1 and S δ 1:k k denote the Kalman filter measurement prediction and innovation covariance, respectively.
From the posteriori distribution, the minimum variance estimate of the state x k can be calculated aŝ
x mv k = L k i=1 w i kx δ i 1:k k|k (7a)
and the associated conditional covariance matrix as
P mv k = L k i=1 w i k P δ i 1:k k|k + (x δ i 1:k k|k −x mv k )(x δ i 1:k k|k ) −x mv k ) . (7b)
B. Approximative solution
The outlined filter bank solution to the inference problem can in general not be used without modifications due to the growing number of Kalman filters needed. Therefore, many strategies have been developed for pruning and merging branches in the growing filter bank tree so that a fixed complexity is achieved [10].
Beyond the challenges with the exponentially increasing complexity, the outlined general solution cannot still be applied to the Markov jump system model in (3). The system is nonlinear and the attitude states belong to a manifold of Euclidean space. This implies that p(x k |δ 1:k , y 1:k ) in (4) is generally not a normal distribution, neither can the minimum variance estimate of the attitude q k be calculated as in (7a). A common way to handle that the system is nonlinear and the attitude not belonging to Euclidian space in the filtering process is to use an error state Kalman filter [11]. The a posteriori distribution p(x k |δ 1:k , y 1:k ) is then, at every time instant, approximated as normal distributed with the mean and covariance given by the error state Kalman filter [12].
One way to calculate a point estimate of the attitude in terms of the Euler angles φ is viâ
φ mv k = arg min φ∈Ω φ L k i w i k C(q δ i 1:k k|k ) − C(φ) 2 F ,(8)
where C(q) denotes the rotation matrix that transforms a vector from the body frame to the navigation frame [13]; to simplify the notation, a slight misusage of notation is admitted, and the rotation matrix is here interchangeably parameterized by the attitude quaternion q and the corresponding Euler angles φ. Further, the set
Ω φ = [0, 2π) × [−π/2, π/2) × [0, 2π).
The covariance ofφ mv k can be approximately calculated as
Cov(φ mv k ) ≈ L k i w i k Σ δ i 1:k k|k + ∆φ δ i 1:k k (∆φ δ i 1:k k ) .(9)
Here Σ δ i 1:k k|k denotes the covariance of the attitude (in terms of Euler angles) calculated by the error state Kalman filter for the branch corresponding to the motion sequence δ i 1:k . Further, the difference between the attitude estimate calculated by the error state Kalman filter and the minimum variance attitude estimate in (8) is given by
∆φ δ i 1:k k|k = {φ ∈ Ω φ s.t. C(φ mv k ) = C(φ)C(q δ i 1:k k|k )}.
(10) Note that the optimization problem in (8) can be efficiently solved using a polar decomposition [13].
IV. LEARNING OF UNKNOWN MODEL PARAMETERS
The signal model in (3) may have unknown parameters, such as the transition probability matrix Π, whose values can be hard to specify accurately. Instead of resorting to cumbersome hand-tuning of these parameters, they may be learned from data. A variety of methods to learn model parameters in jump Markov models has been suggested [14], [15]. Here a maximum likelihood method, similar to that presented in [12], for learning the parameters from the data will be outlined.
Let θ denote the unknown parameters in the model. The maximum likelihood estimate of these parameters is then given byθ = arg max θ p(y 1:k ; θ),
where p(y 1:k ; θ) = k n=2 p(y n |y 1:n−1 )p(y 1 ), (11c)
Unconstrained motion δ = 1 Almost stationary δ = 2 Stationary δ = 3 Π 1,1 Π 2,2 Π 3,3 Π 2,1 Π 3,2 Π 2,3 Π 1,2
(a) Varying gait speed example.
Unconstrained motion δ = 1 Stationary & new height δ = 2 Stationary & same height δ = 3 Π 1,1 Π 3,1 Π 3,3 Π 2,1 1 Π 1,3
(b) Return to same height example. Here p(y 1 ) denotes the a priori probability of the observation y 1 . Note that the number of modes in the full mixture distribution, which subsequently must be summed to marginalize away the mode dependence, increases exponentially in n. However, in practice, the actual number of modes to consider is reduced using pruning or merging in the filter bank. Hence, the number of modes to consider when evaluating (11c) can be kept tractable.
V. APPLICATION EXAMPLES
Next, the proposed inference framework is used to realize two foot-mounted inertial navigation systems that incorporate different motion modes. The first system adaptively selects the covariance matrix used in the zero-velocity update, and the second system automatically detects if the system returns to the same height after a step. For both systems, the motion mode state transition matrix is learned from training data using the method outlined in Sec. IV. The performances of both systems are compared to the OpenShoe system presented in [16].
A. Example: Varying gait speed
The challenge of designing and tuning a foot-mounted zerovelocity aided inertial navigation system so that it works well for multiple gait speeds is well-known [1]. To use the proposed framework to design a system that adaptively selects the detection threshold, as well as the covariance matrix used in the zero-velocity updates, we define the following three motion modes: the unconstrained motion mode (δ k = 1), the almost stationary motion mode (δ k = 2), and the stationary motion mode (δ k = 3). A motion mode transition diagram illustrating how the system is assumed to transition between these motion modes is shown in Fig. 1a. The associated motion mode transition probabilities Π i,j are also shown.
Next, introduce the following system dynamics
f (x k , δ k , u k , η k ) = r k + ∆tv k v k + ∆t C(q k )(s k + η s k ) + g Ω ∆t (ω k + η ω k ) q k .
(12a) Here ∆t and g denote the sampling period and gravity vector, respectively. Further, Ω(·) denotes the quaternion update matrix (see [17] for details). Moreover, η s k and η ω k denote the process noise associated with the accelerometers and gyroscopes, respectively.
Finally, define the stochastic motion constraints imposed during the different motion modes as follows
h(x k , u k , δ k ) = 0 , δ k = 1 h 0 (x k , u k ), δ k = {2, 3} ,(12b)
where
h 0 (x k , u k ) = v k ω k C(q k )s k + g .
(12c)
The "hardness" of the constraints is controlled by the covariance matrix
R(δ k ) = σ 2 nc I, δ k = 1 R 0 (δ k ), δ k = {2, 3} ,(12d)
where
R 0 (δ k ) = σ 2 v (δ k )I ⊕ σ 2 ω (δ k )I ⊕ σ 2 s (δ k )I.(12e)
Here ⊕ denotes the direct sum matrix operator and I denotes an identity matrix of appropriate size. Further, σ 2 v (δ k ), σ 2 ω (δ k ), and σ 2 s (δ k ) denote the mode-dependent variance associated with the stochastic constraints on the velocity, angular velocity, and acceleration, respectively. Moreover, σ 2 nc is a design parameter that controls the measurement likelihood associated with no constraints.
A filter bank with the jump Markov model defined by (12) was designed and two data sets were recorded with a sensor array consisting of 32 InvenSense MPU9150 IMUs while a person walked and ran back and forth along a straight line. From the recorded data, two data sets with 50 measurement sequences each were created by repeatedly drawing four random IMUs and averaging their measurements. These measurement sequences were then processed using the OpenShoe system algorithm with the zero-velocity detector threshold tuned to generate a detection at least once per gait cycle. The measurements were also processed using the designed filter bank. Pruning was used to keep the tree size to a maximum of nine leaves, and a pruning strategy where the most probable leaves were retained was used.
The measurement sequences in the first data set were used to learn the mode transition matrix Π. The initial value and end result of the learning process were respectively, and the optimization converged in less than 10 iterations. The matrix Π end corresponds to a mode transition system with low-pass characteristics. That is, the probability of staying in a mode is much higher than transitioning to another mode. On average, the percentage of time spent in the motion modes one, two, and three is 57%, 5%, and 38%, respectively. The learned mode transition matrix was then used in the filter bank when processing the measurements in the second data set. The result is shown in Fig. 2. In Fig. 2a the speed and most likely mode sequenceδ 1:k estimated by the filter bank from one of the measurement sequences are shown. And in Fig. 2b the horizontal plane errors at the end of the trajectory are shown.
From the figures, the following things can be observed. The filter bank adaptively selects different motion modes depending on the gait speed. That is, when walking, the filter bank frequently selects the stationary motion mode as the most likely mode, whereas when running, the filter bank never selects the stationary mode. Comparing the horizontal positioning error of the OpenShoe system and the filter bank the systems have about the same cross-track error, whereas the filter bank has a significantly smaller along-track error.
The OpenShoe system has an along-track bias error of about minus one meter. This is likely due to the high detection threshold used in the zero-velocity detector, causing zerovelocity updates to be applied even when the system is moving. This, in turn, causes part of each step to be cut away, especially when the user walks and the foot's transition from stationary to moving is less distinct. Thanks to the ability of the filter bank to adaptively select the detection threshold and the covariance used in the zero-velocity update, this effect is reduced. However, looking at the vertical root mean square (RMS) error, both systems have an error of several meters. How to reduce this error will be illustrated next.
B. Example: Return to same height
The vertical position error often grows faster than the horizontal position error in foot-mounted zero-velocity aided inertial navigation systems. This is because the beginning and end of each step, when the foot is mainly moving in the vertical direction, are cut away by the zero-velocity updates. However, the vertical error growth can be significantly reduced by noting that most of the time a person is walking on flat ground and the foot should return to the same height [3]. To design a system using the proposed framework that incorporates this information, we introduce the following three motion modes: the unconstrained motion mode (δ k = 1), the stationary at new height motion mode (δ k = 2), and the stationary at the same height motion mode (δ k = 3). A motion mode transition diagram illustrating how the system is assumed to transition between these motion modes is shown in Fig. 1b. Here it has been assumed that if the system becomes stationary at a new height, the system will be stationary for at least two samples. Hence the probability of transitioning from mode δ k = 2 to mode δ k = 3 is one. (b) Horizontal position error at end of the trajectory. Also shown are the mean error (blue dot) and 95% confidence interval (blue circle) calculated from the filter covariance, assuming the error to be normally distributed. Fig. 2: Results from varying gait speed application example.
Next, we introduce the following extended system dynamics
f ext (x k , δ k , u k , η k ) = f (x k , δ k , u k , η k ) 1(δ k = 1)ξ k + 1(δ k = 1)[p k ] 3 ,(13a)
where 1(·) denotes the indicator function and [a] j denotes j:th element of the vector a. Furthermore, the auxiliary state ξ k stores the height from the time when the system was last stationary.
The stochastic motion constraints are as follows (b) Height versus horizontal position error at end of the trajectory. Also shown are the mean error (blue dot) and 95% confidence interval (blue circle) calculated from the filter covariance, assuming the error to be normally distributed. Fig. 3: Results from return to same height application example.
h ext (x k , u k , δ k ) = 0 , δ k = 1 h 0 (x k , u k ), δ k = 2 h 0 (x k , u k ) [p k ] 3 − ξ k , δ k = 3 .(13b
The "hardness" of these constraints is controlled by the covariance matrices
R ext (δ k ) = σ 2 nc I, δ k = 1 R 0 (δ k ), δ k = 2 R 0 (δ k ) ⊕ σ 2 h (δ k )I, δ k = 3 ,(13c)
where σ 2 h (δ k ) denotes the variance associated with the stochastic constraints on height.
The jump Markov model defined by (13a), (13b), and (13c) was used to design a filter bank, and two data recordings were collected while a person walked on flat ground and then climbed and descended a stair. Following the same procedure as in the previous example, the data recordings were used to create two data sets with 50 measurement sequences each. The measurement sequences in the first data set were used to learn the mode transition matrix Π. The initial value and end result of the learning process were respectively, and the optimization converged in less than 10 iterations. Once again the matrix corresponds to a mode transition system with low-pass characteristics, where the average percentages of time spent in motion modes one, two, and three is 56.2%, 0.2%, and 43.6%, respectively.
The learned mode transition matrix was then used in the filter bank when processing the measurements in the second data set. Once again, pruning was used to keep the tree size to a maximum of nine leaves and a pruning strategy where the most leaves were retained was used. The result is shown in Fig. 3. In Fig. 3a the height and the most likely mode sequenceδ 1:k estimated by the filter bank is shown. And in Fig. 3b the height versus horizontal plane errors at the end of the trajectory are shown.
From the figures, the following things can be observed. The filter bank can detect whether the user walks on flat ground or not, i.e., whether the foot returns to the same height as in the previous step or not. Therefore it effectively reduces the height error from several meters to a few decimeters. A minor increase in the horizontal position error is observed with the filter bank.
VI. DISCUSSION AND CONCLUSIONS
A framework for tightly integrated motion classification and state estimation in motion-constrained inertial navigation systems has been presented. The framework provides a structured and theoretically sound way to design motion-constrained inertial navigation systems that can learn different motion patterns. The application of the framework has been illustrated via two examples of foot-mounted zero-velocity aided inertial navigation systems. The examples show that a significant performance gain can be achieved compared to a standard footmounted inertial navigation system. However, the price paid is an increased computational complexity (proportional to the number of filters in the filter bank) and an increased number of system parameters to tune. The challenges with tuning the system parameters can to some extent be alleviated via the proposed parameter learning method. Still, tuning the system parameters can be challenging, especially if many motion modes are included in the system model.
VII. OUTLOOK AND FUTURE RESEARCH
A basic version of the filter bank framework has been presented and tested with models that have a few motion modes. Many possible improvements and open research questions exist. First and foremost, the framework should be compared against other adaptive zero-velocity detector frameworks. More complex models that include a variety of motion modes should also be explored. Related to that, hieratical model structures constructed of several linked small hidden Markov models for the motion states are of special interest to obtain a computationally attractive algorithm. Further, to obtain a smoother transition between motion modes and a more robust algorithm framework, the feasibility of substituting the normal distribution in (6c) with a more heavytailed distribution, such as the Student-t distribution, should be explored. Moreover, since the probability of transitioning between motion modes varies both with time and the motion dynamics, the use of a constant mode transition probability matrix Π is clearly suboptimal. Therefore, the feasibility of learning a time-varying and motion-dynamic dependent transition matrix should be explored.
and p(y n |y 1:n−1 ) =
Fig. 1 :
1Mode transition diagrams for the application examples.
speed and most likely motion mode versus time.
ACKNOWLEDGMENT This work has been partially funded by the Swedish Research Council project 2020-04253 Tensor-field based localization and the Dutch Research Council (NWO) research program Veni project 18213 Sensor Fusion For Indoor Localisation Using The Magnetic Field.
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| {'fraction_non_alphanumeric': 0.050261780104712044, 'fraction_numerical': 0.01799738219895288, 'mean_word_length': 4.2036437936318745, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, 'xml': 0}, 'pii_count': 2, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 28, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "A framework for tightly integrated motion mode classification and state estimation in motion-constrained inertial navigation systems is presented. The framework uses a jump Markov model to describe the navigation system's motion mode and navigation state dynamics with a single model. A bank of Kalman filters is then used for joint inference of the navigation state and the motion mode. A method for learning unknown parameters in the jump Markov model, such as the motion mode transition probabilities, is also presented. The application of the proposed framework is illustrated via two examples. The first example is a foot-mounted navigation system that adapts its behavior to different gait speeds. The second example is a foot-mounted navigation system that detects when the user walks on flat ground and locks the vertical position estimate accordingly. Both examples show that the proposed framework provides significantly better position accuracy than a standard zero-velocity aided inertial navigation system. More importantly, the examples show that the proposed framework provides a theoretically well-grounded approach for developing new motionconstrained inertial navigation systems that can learn different motion patterns.", 'arxivid': '2305.09363', 'author': ['Isaac Skog [email protected] \nDept. of Electrical Engineering\nDept. of Electrical Engineering\nUppsala University Uppsala\nSweden\n', 'Gustaf Hendeby [email protected] \nDelft Center for Systems and Control Delft\nLinköping University Linköping\nSweden\n', 'Manon Kok \nUniversity of Technology Delft\nThe Netherlands\n'], 'authoraffiliation': ['Dept. of Electrical Engineering\nDept. of Electrical Engineering\nUppsala University Uppsala\nSweden', 'Delft Center for Systems and Control Delft\nLinköping University Linköping\nSweden', 'University of Technology Delft\nThe Netherlands'], 'corpusid': 258714674, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 8575, 'n_tokens_neox': 7517, 'n_words': 5092, 'pdfsha': '0fd398898db6b7eba9b76bb88fb56dc75c2c0f4e', 'pdfurls': ['https://export.arxiv.org/pdf/2305.09363v1.pdf'], 'title': ['Tightly Integrated Motion Classification and State Estimation in Foot-Mounted Navigation Systems', 'Tightly Integrated Motion Classification and State Estimation in Foot-Mounted Navigation Systems'], 'venue': []} |
arxiv |
Comptes Rendus Mathématique On the disentanglement of Gaussian quantum states by symplectic rotations Spectral Theory / Théorie spectrale On the disentanglement of Gaussian quantum states by symplectic rotations Sur la désintrication des états quantiques Gaussiens par des rotations symplectiques
2020
Maurice A De Gosson [email protected].
Universität Wien Fakultät für Mathematik (NuHAG)
Oskar-Morgenstern-Platz 11090WienAustria
Maurice A De Gosson
Universität Wien Fakultät für Mathematik (NuHAG)
Oskar-Morgenstern-Platz 11090WienAustria
Comptes Rendus Mathématique On the disentanglement of Gaussian quantum states by symplectic rotations Spectral Theory / Théorie spectrale On the disentanglement of Gaussian quantum states by symplectic rotations Sur la désintrication des états quantiques Gaussiens par des rotations symplectiques
35842020Manuscript received 19th March 2020, revised 23rd April 2020, accepted 24th April 2020.<https://doi.org/10.5802/crmath.57> This article is licensed under the Creative Commons Attribution 4.0 International License. Comptes Rendus Mathématique 2020, 358, n 4, p. 459-462
We show that every Gaussian mixed quantum state can be disentangled by conjugation with a unitary operator corresponding to a symplectic rotation via the metaplectic representation of the symplectic group. The main tools we use are the Werner-Wolf condition for separability on covariance matrices and the symplectic covariance of Weyl pseudo-differential operators.Résumé. Nous montrons que chaque état quantique Gaussien peut-être rendu séparable (= « désintriqué ») par conjugaison avec un opérateur unitaire associé via le groupe métaplectique à une rotation symplectique. Pour cela nous utilsons la condition de séparabilité de Werner et Wolf sur la matrice de covariance ainsi que la covariance symplectique des opérateurs pseudo-différentiels de Weyl.
Introduction
Gaussian states play an ubiquitous role in quantum information theory and in quantum optics because they are easy to manufacture in the laboratory, and have in addition important extremality properties [12]. Of particular interest are the separability and entanglement properties of Gaussian states; the literature on the topic is immense; two excellent texts whose mathematical setup is rigorous are [1,2]. It turns out that even if major advances have been made in the study of the separability of Gaussian quantum states in recent years (one of the milestones being Werner and Wolf's paper [10] about the covariance matrices of bipartite states), the topic is still largely open. The aim of this Note is to show that every Gaussian state can be made separable by using a symplectic rotation and of the corresponding metaplectic operator. (We note that physicists use the terminology "passive symplectic transformations" in place of "symplectic rotation"). This result can be viewed as closing a problem originally posed in Wolf et al. [11], who asked which Gaussian states can be entangled by symplectic rotations. A full answer has recently been given in [8] et al. where the Gaussian states that are separable for all symplectic rotations are characterized. Our result (Theorem 1) shows that, conversely, every entangled Gaussian state can be separated ("disentangled") by metaplectic transformations corresponding to symplectic rotations.
We will use the following notation. Let R 2n = R 2n A ⊕ R 2n B be the phase space of a bipartite system (n A ≥ 1, n B ≥ 1). We will use the following phase space variable ordering:
z = (z A , z B ) = z A ⊕z B with z A = (x 1 , p 1 , . . . , x n A , p n A ) and z B = (x n A +1 , p n A +1 , . .
. , x n , p n ). We equip the symplectic spaces R 2n A and R 2n B with their canonical bases. The symplectic structure on R 2n is then
σ(z, z ) = J z · z with J = J A ⊕ J B where J A = n A k=1 J k , J k = 0 1 −1 0
and likewise for J B . Thus J A (resp. J B ) determines the symplectic structure on the partial phase space R 2n A (resp. R 2n B ).
Result: Statement and Proof
Let Σ be a real positive definite symmetric 2n × 2n matrix (to be called "covariance matrix" from now on) and consider the associated normal probability distribution
ρ(z) = 1 (2π) n det Σ e − 1 2 Σ −1 z 2 .(1)
If the covariance matrix satisfies in addition the condition
Σ + i ħ 2 J ≥ 0(2)
(J the standard symplectic matrix) then ρ is the Wigner distribution of a mixed quantum state, identified with its density operator ρ. We notice that property (2) crucially depends on the numerical value of ħ (see [4,7]). We will say that ρ is "AB -separable" if there exist sequences of density operators ( ρ A j ) and ( ρ B j ) on L 2 (R n A ) and L 2 (R n B ), respectively and coefficients λ j ≥ 0 summing up to one, such that
ρ = j λ j ρ A j ⊗ ρ B j(3)
where the convergence is for the trace-class norm. The problem of determining necessary and sufficient conditions for a density operator to be separable is still very largely open; while there exist necessary conditions, no simple sufficient condition for separability is known in the general case; for a recent up to date discussion see Lami et al. [8]. Werner and Wolf [10] have proven that in the Gaussian case ρ is separable if and only if there exists a 2n A × 2n A covariance matrix Σ A and a 2n B × 2n B covariance matrix Σ B such that the following conditions hold:
Σ A + i ħ 2 J A ≥ 0 (4) Σ B + i ħ 2 J B ≥ 0 (5) Σ ≥ Σ A ⊕ Σ B .(6)
The aim of this Letter is to prove that for every Gaussian density operator there exists a unitary transform U such that U ρ U −1 is a separable Gaussian state: (1). There exists a symplectic rotation U ∈ U (n) (= Sp(n) ∩ O(2n, R)) such that U ρ U −1 is separable where U ∈ Mp(n) is any of the two metaplectic operators covering U .
Theorem 1. Let ρ be a density operator with Gaussian Wigner distribution
Proof.
We begin by recalling [5,6] that the quantum condition (2) is equivalent to the statement:
There exists S ∈ Sp(n) such that SB 2n ( ħ) ⊂ Ω Σ (7) where Sp(n) is the symplectic group of the phase space R 2n ≡ R n x ×R n p equipped with the standard symplectic form σ = dp 1 ∧ dx 1 + · · · + dp n ∧ dx n , B 2n ( ħ) is the phase space ball defined by |z| ≤ ħ and Ω Σ the covariance ellipsoid of ρ:
Ω Σ = {z ∈ R 2n : 1 2 Σ −1 z 2 ≤ 1} . Let S = P R (P = (S T S) 1/2 , R = (S T S) −1/2 S)
be the symplectic polar decomposition [5] of S ∈ Sp(n), that is P ∈ Sp(n), P > 0, and
R ∈ U (n) = Sp(n) ∩ O(2n, R) .
We have SB 2n ( ħ) = P B 2n ( ħ) by rotational symmetry of the ball B 2n ( ħ). There exists a symplectic rotation U ∈ U (n) diagonalizing P [5]:
P = U T ∆U(8)
where ∆ ∈ Sp(n) is a diagonal matrix whose form will be discussed in a moment. The inclusion SB 2n ( ħ) ⊂ Ω Σ in (7) is thus equivalent to ∆B 2n ( ħ) ⊂ U (Ω Σ ), that is
∆B 2n ( ħ) ⊂ Ω Σ U(9)
where Σ U = U ΣU T . This inclusion is equivalent to the matrix inequality
ħ 2 ∆ 2 ≤ Σ U(10)
(A ≤ B meaning that B − A is positive semidefinite). We next note that Σ U is the covariance matrix of the density operator ρ U with Wigner distribution ρ U (z) = ρ(U T z) that is
ρ U (z) = 1 (2π) n detU ΣU T e − 1 2 Σ −1 U T z·U T z .
Recall now the following symplectic covariance property: if A = Op W (a) is a Weyl operator with symbol a and S ∈ Mp(n) a metaplectic operator covering S ∈ Sp(n) then
S Op W (a) S −1 = Op W (a • S −1 )(11)
(see for instance [9] or [5,Ch. 7]). Applying this covariance formula to ρ = (2πħ) n Op W (ρ) yields
since U T = U −1 , ρ U = U ρ U −1(12)
where U is anyone of the two metaplectic operators ± U covering U . We claim that ρ U is separable. To see this, let us come back to the diagonal matrix ∆ appearing in the factorization P = U T ∆U (8). Its diagonal elements are the eigenvalues λ 1 , . . . , λ 2n of the positive definite symplectic matrix P and therefore appear in pairs (λ, 1/λ) with λ > 0 [3,5]. In fact, in the ABordering we are using, the matrix ∆ has the form ∆ = ∆ A ⊕ ∆ B with
∆ A = n A k=1 ∆ k , ∆ B = n k=n A +1 ∆ k
and ∆ k = λ k 0 0 λ −1 k for k = 1, . . . , n. Clearly ∆ A ∈ Sp(n A ) and ∆ B ∈ Sp(n B ). The symmetric matrices
Σ A = ħ 2 ∆ 2 A , Σ B = ħ 2 ∆ 2 B
trivially satisfy Σ A + i ħ 2 J A ≥ 0 and Σ B + i ħ 2 J B ≥ 0. In view of (10) we have Σ A ⊕ Σ B ≤ Σ U and the theorem now follows using the Werner-Wolf conditions (4)- (6).
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Extremality of Gaussian quantum states. M M Wolf, G Giedke, J I Cirac, ID 080502Phys. Rev. Lett. 968M. M. Wolf, G. Giedke, J. I. Cirac, "Extremality of Gaussian quantum states", Phys. Rev. Lett. 96 (2006), no. 8, article ID 080502.
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arxiv |
DPMLBench: Holistic Evaluation of Differentially Private Machine Learning
November 26-30, 2023
Chengkun Wei
Zhejiang University
Minghu Zhao
Zhejiang University
Zhikun Zhang
Stanford University
Min Chen
CISPA Helmholtz Center for Information Security 4 DBAPPSecurity
Wenlong Meng
Zhejiang University
Bo Liu
Yuan Fan
Zhejiang University
Wenzhi Chen
Zhejiang University
DPMLBench: Holistic Evaluation of Differentially Private Machine Learning
the 30th ACM SIGSAC Conference on Computer and Communications Security
November 26-30, 2023* Corresponding authors. 1 The implementation can be found at https://github.com/ DmsKinson/DPMLBench
Differential privacy (DP), as a rigorous mathematical definition quantifying privacy leakage, has become a wellaccepted standard for privacy protection. Combined with powerful machine learning techniques, differentially private machine learning (DPML) is increasingly important. As the most classic DPML algorithm, DP-SGD incurs a significant loss of utility, which hinders DPML's deployment in practice. Many studies have recently proposed improved algorithms based on DP-SGD to mitigate utility loss. However, these studies are isolated and cannot comprehensively measure the performance of improvements proposed in algorithms. More importantly, there is a lack of comprehensive research to compare improvements in these DPML algorithms across utility, defensive capabilities, and generalizability.We fill this gap by performing a holistic measurement of improved DPML algorithms on utility and defense capability against membership inference attacks (MIAs) on image classification tasks. We first present a taxonomy of where improvements are located in the machine learning life cycle. Based on our taxonomy, we jointly perform an extensive measurement study of the improved DPML algorithms, over twelve algorithms, four model architectures, four datasets, two attacks, and various privacy budget configurations. We also cover state-of-the-art label differential privacy (Label DP) algorithms in the evaluation. According to our empirical results, DP can effectively defend against MIAs, and sensitivity-bounding techniques such as per-sample gradient clipping play an important role in defense. We also explore some improvements that can maintain model utility and defend against MIAs more effectively. Experiments show that Label DP algorithms achieve less utility loss but are fragile to MIAs. Machine learning practitioners may benefit from these evaluations to select appropriate algorithms. To support our evaluation, we implement a modular re-usable software, DPMLBench, 1 which enables sensitive data owners to deploy DPML algorithms and serves as a benchmark tool for researchers and practitioners.
Introduction
As machine learning (ML) continues to evolve, numerous fields are leveraging its power to advance their development [1,2]; however, this often involves the use of private data, such as medical records. Previous studies have revealed that the models trained on private data can leak information through a bunch of attacks, such as membership inference [3], model inversion [4], and attribute inference [5], which raises critical privacy and security concerns.
Differential privacy (DP) is a widely used notion to rigorously formalize and measure the privacy guarantee based on a parameter called privacy budget. Abadi et al.
[6] proposed a general DPML algorithm called differentially private stochastic gradient descent (DP-SGD) by integrating per-sample clipping and noise perturbation to the aggregated gradient in the training process. However, models trained by DP-SGD normally perform badly with respect to model utility. Recently, researchers proposed many improved algorithms with better privacy-utility trade-off [7,8,9,10,11,12,13,14,15]. In the rest of this paper, we refer to DP-SGD as vanilla DP-SGD to distinguish between DP-SGD and the improved algorithms.
The improved algorithms modify the vanilla DP-SGD from different aspects but are evaluated in isolation with various settings, which cannot reveal the differences between each other. Furthermore, existing studies [16,17,18,19] fail to report a complete and practical analysis of general DPML algorithms in practical scenarios. This motivates us to perform a holistic evaluation and analysis of these improved DPML algorithms.
Our Contributions
Algorithm Taxonomy. We first propose a new taxonomy for the state-of-the-art DPML algorithms based on their improved component in the ML pipeline. Concretely, we divide the ML pipeline into four phases: Data preparation, model design, model training, and model ensemble (see Section 2.1 for details), and categorize the DPML algorithms into each phase. We then perform a theoretical and empirical analysis to obtain an extensive view of the impact of differential privacy on machine learning. Experimental Evaluation.
In this paper, we concen- RQ3. What is the impact of dataset and model architecture on algorithms focusing on different stages?
In addition, our measurement covers two state-of-the-art label differential privacy (Label DP) algorithms, which is a variant DPML notion by relaxing the protection of the whole data sample to only protect the label. To the best of our knowledge, we are the first to analyze the Label DP algorithms on utility and defense empirically.
DPMLBench.
We implement a toolkit called DPML-Bench to support the comprehensive evaluation of DPML algorithms with respect to model utility and MIA defense. With a modular design, DPMLBench can easily integrate additional DPML algorithms, attacks, datasets, and model architectures by implementing new functional codes to the relevant modules. Our code will be publicly available, facilitating researchers to leverage existing DPML algorithms to provide DP guarantee or benchmark new algorithms. Main Findings. Our work reveals several interesting findings:
• Different improvement techniques can affect the privacyutility trade-offs of the algorithm from different perspectives. For example, we find that reducing the dimension of the parameter improves the performance of DPML on large models but may impair utility when the privacy budget is large. In addition, DP synthetic algorithms and algorithms in the model ensemble category are the most robust in defending against MIAs.
• DP can effectively defend against MIAs. Also, sensitivitybounding techniques such as per-sample gradient clipping play an important role in defense.
• Some model architecture design choices for non-private ML models are ineffective for private ML models. For instance, using Tanh as the activation function and Group-Norm can reduce the utility loss on vanilla DP-SGD. However, we also find that using Tanh and GroupNorm together would have a negative effect.
• Compared to standard DP, Label DP has less utility loss but is more fragile to MIAs. Figure 1 illustrates a typical machine learning pipeline, which consists of four phases: Data preparation, model design, model training, and model ensemble.
Preliminaries
Machine Learning Pipeline
The data preparation phase aims to explore the underlying distribution of data for learning algorithms. Commonly used techniques in this phase include data cleaning [27], data labeling [28], and feature extraction [29]. Feature extraction transforms the input data into a low-dimensional subspace that reveals the most relevant information [30]. Low dimensional information can downgrade the difficulty of the following training procedures [31,7,32].
In the model design phase, we aim to select components such as the model architectures, loss functions, and optimization algorithms that are appropriate for the task. There are plenty of studies on this topic [20,21].
The model training phase is the process of computing the following optimization objective: arg min where (x, y) is the data sample in the training dataset D train ; L and M represent loss function and model architecture, respectively. The parameters θ in model M are optimized to minimize the objective function L on training data during the model training phase.
The model ensemble phase combines multiple models while deploying the model. Previous studies show that aggregating multiple models' predictions can obtain better generalization performance than a single model [33].
Differential Privacy
Differential privacy (DP) [34] is a rigorous mathematical definition quantifying how much privacy preservation a mechanism can provide. DP provides a privacy guarantee by bounding the impact of a single input on the mechanism's output.
Definition 2.1 ((ε, δ )-Differential Privacy). Given two neighboring datasets D and D differing by one record, a mechanism M satisfies (ε, δ )-differential privacy if
Pr[M(D) ∈ S] ≤ e ε · Pr[M(D ) ∈ S] + δ ,
where ε is the privacy budget, and δ is the failure probability.
The privacy budget quantifies the maximum information a mechanism M can expose. A smaller privacy budget indicates better privacy preservation. δ indicates the probability that M fails to satisfy ε-DP. When δ = 0, we achieve pure ε-DP, a stronger notion, and a more rigorous privacy guarantee. Bounded DP and Unbounded DP. How to interpret neighboring datasets distinguishes between bounded DP and unbounded DP [35]. In unbounded DP, D and D are neighbors if D can be obtained from D by adding or removing one element. In bounded DP, D and D are neighbors if D can be obtained from D by replacing one element. When using bounded DP, two datasets should have the same number of elements. Furthermore, any algorithms that satisfy εunbounded DP also satisfy 2ε-bounded DP because replacing one element can be achieved by removing then adding one element. All algorithms in Table 1 satisfy the unbounded DP. Gaussian Mechanism. Adding noise sampled from Gaussian distribution is a commonly used approach to achieve (ε, δ )-DP, known as Gaussian mechanism [36]. Formally, applying the Gaussian mechanism to a function f can be defined as:
M(d) = f (d) + N(0, S 2 f · σ 2 ),
where N(0, S 2 f · σ 2 ) is the Gaussian distribution with mean 0 and standard deviation S 2 f · σ 2 , where σ is called noise multiplier and S f is the sensitivity of function f .
Definition 2.2. (Sensitivity).
Given two neighboring datasets D and D , the global sensitivity of a mechanism M, denoted by S M , is given below Composition. The composition theorems calculate the total privacy budget when we apply DP on the private dataset multiple times. The most straightforward composition strategy is summing up the privacy budget of each individual DP algorithm. Formally, for k DP algorithms with privacy budget ε 1 , ε 2 , ε 3 , · · · , ε k , the total privacy budget is ε = ε 1 + ε 2 + ε 3 + · · · + ε k . Mironov [37] et al. propose Rényi differential privacy to achieve a tighter analysis of cumulative privacy budgets. Note that larger α leads to more weight being assigned to worst-case events, e.g., (∞, ε)-RDP is equivalent to ε-DP.
S M = max
If M satisfies (α, ε)-RDP, it also satisfies (ε + log 1 δ α−1 , δ )-DP. Applying k algorithms with (α, ε 1 )-RDP, (α, ε 2 )-RDP, · · · ,(α, ε k )-RDP on same dataset sequentially leads to an algorithm with (α, ε 1 + ε 2 + · · · + ε k )-RDP. By selecting α delicately, accumulating privacy loss in RDP and then converting to DP can derive a tighter upper bound than composite (ε, δ )-DP directly. Post-processing. The post-processing property guarantees that no matter what additional processing one performs on the output of an algorithm that satisfies (ε,δ )-DP, the composition of the algorithm and the post-processing operations still satisfy (ε,δ )-DP.
Differentially Private Machine Learning
Abadi et al.
[6] integrated differential privacy with stochastic gradient descent (SGD) and proposed a general learning algorithm named differential privacy stochastic gradient descent (DP-SGD). Compared to SGD, DP-SGD introduced a few modifications to make the algorithm satisfy differential privacy. Firstly, the sensitivity of each gradient is bounded by clipping each gradient in the l 2 norm.
clip(g,C) = g/max(1, ||g|| 2 C ).(1)
Per-sample clipping bounds the contribution of each sample to model parameters to C. Moreover, DP-SGD applies a Gaussian mechanism to the aggregated clipped gradient. Formally,g
= g + N(0,C 2 σ 2 ),(2)
whereg is the noisy gradient used to update parameters and σ controls privacy level. After the above two steps, the gradients used to update the parameters satisfy DP. Nevertheless, gradient clipping and noise perturbation introduce deviation in the training process, which impairs the model's utility. Recently, researchers proposed a number of improved DPML algorithms to reduce the utility loss incurred by vanilla DP-SGD [38,11,31,12]. However, these improved DPML algorithms were evaluated on different models and datasets with different assumptions. Therefore, it is a pressing need to design a holistic benchmark to comprehensively evaluate these DPML algorithms to gain a deeper insight.
Taxonomy
In this section, we provide an overview of our taxonomy and give survey-style descriptions of the DPML algorithms.
Overview
We first propose a new taxonomy for the DPML algorithms based on the component they improve in the ML pipeline discussed in Section 2.1. We introduce this taxonomy due to the following reasons: (1) The training phases of ML are independent, meaning the improvements in different phases might be combined to achieve better model utility. (2) It provides future researchers with a clear roadmap to improve the DPML algorithms, which we hope can benefit the community.
(3) It is domain-agnostic and can be easily extended to evaluate the DPML algorithms in other domains, such as graph and NLP data. Table 1 summarizes all the improved DPML algorithms and their corresponding categories. We also discuss the properties of all the DPML algorithms. For instance, vanilla DP-SGD falls in the model training category and modifies the gradient to provide the DP guarantee, whereas PATE belongs to the model ensemble category and leverages auxiliary data to provide a DP guarantee. Auxiliary data generally refers to data with the same distribution as sensitive data but is publicly available, which is a common assumption in DPML [31,9,32]. Data Preparation. The algorithms in this category preprocess the original training data. Feature extraction and DP synthetic data are two typical approaches in this category. Feature extraction aims to reduce the difficulty of private training. Using a pre-trained network before classifier [6, 13, 42] can be seen as a variant of feature extraction. DP synthetic data aims to provide a DP guarantee for training data. Applying DP mechanisms to data directly, such as the Gaussian mechanism, downgrades the utility of data, especially when data is in high dimension (e.g., image). DP synthetic data is an alternative that aims to generate data in a DP manner with a similar distribution as sensitive data. Training models on synthetic data with traditional machine learning algorithms can derive a model with DP guaranteed according to post-processing property [43,15,44]. In this category, we pick three algorithms, of which Hand-DP [12] leverages a feature extractor, and the other two (PrivSet [45] and DP-GEN [15]) belong to DP synthetic data algorithms. Model Design. Algorithms in this category focus on designing more adapted model designs to DPML. Deep learning in non-private settings has been widely studied, and many rules have been summarized to train a standard neural network. However, these design guidelines do not perform well in vanilla DP-SGD [42] due to gradient clipping and noise perturbation. For instance, larger models often mean better performance in non-private settings. However, smaller models tend to get better performance on vanilla DP-SGD. Some existing studies focus on exploring more adapted model design rules to DPML [11,46]. We select two algorithms in this category, and they propose improvements from the activation function (TanhAct [11]) and loss function (FocalLoss [8]), respectively.
Model Training. Algorithms in this category explore DP mechanisms with less impact on model utility in the DP-SGD training phase. The vanilla DP-SGD [6] bounds the l 2 -norm of gradient g by clipping the gradient to the threshold C; thus, a straightforward improvement strategy is to find an optimal clipping strategy [47,48]. On the other hand, the noise perturbation leads to bias during model updating, which impairs the model's utility. Therefore, designing a better noise perturbation mechanism to alleviate the noise effect is another optimization option [31,32,13]. In this category, we select four algorithms, excluding vanilla DP-SGD. AdpClip [48] proposes an improved clipping strategy, and the rest of them (RGP [7], GEP [31], and AdpAlloc [13]) explore better noise perturbation mechanisms.
Model Ensemble. This category contains algorithms providing DP guarantee through the model ensemble. The vanilla DP-SGD has poor scalability because it requires modifications to the training process. Papernot et al. [9] propose Private Aggregation of Teacher Ensemble (PATE) by leveraging model ensemble. PATE treats the training phase of the model as a black box so that it has better scalability than vanilla DP-SGD for less modification to the training process. DP mechanism is applied while aggregating the prediction of multiple models. Since then, many DPML algorithms based on the model ensemble have emerged [38, 10]. We select PATE and Priv-kNN in our measurement. PrivSet [45]. It leverages dataset condensation to generate data in a differentially private manner. It directly optimizes for a small set of samples promising to derive approximate results under downstream tasks instead of imitating the complete data distribution. More specifically, they use DP-SGD to optimize a gradient-matching objective for the downstream task that minimizes the difference between the gradient on the real data and the generated data.
Data Preparation
Model Design
TanhAct [11]. Considering the need for DP to bound sensitivity, Papernot et al. [11] replace ReLU with tempered sigmoid as the activation function. The authors found that the bounded property of tempered sigmoid functions, especially Tanh, can effectively limit the l 2 -norm of the gradient while training models with DP-SGD. Thus, less information can be lost in gradient clipping.
FocalLoss [8].
It introduces a loss function adapted to vanilla DP-SGD, which combines three terms: The summed squared error L Focal , the focal loss L SSE [54], and a regularization penalty on the intermediate pre-activations L Reg . Finally, they proposed loss function L:
L = αL Focal + (1 − α)L SSE + (1 − α) β L Reg ,(3)
where α = Sigmoid(e c − e t ) (current epoch e c , and threshold epoch e t ), β is the hyperparameter controlling the strength of the regularization. These terms consider convergence speed, emphasis on complex samples, and sensitivity during training. The new loss function can better control the gradient sensitivity in the training procedure.
Model Training
RGP [7]
. It adopts a reparametrization scheme to replace the model weight in each layer with two low-dimensional weight matrices and a residual weight matrix:
W → LR +W.stop gradient () .(4)
By making the gradient carriers {L, R} consist of orthonormal vectors, a projection of the gradient of W can be constructed from the noisy gradients ofL andR. {L, R} are trained by DP-SGD separately to achieve the DP guarantee and finally combined to obtain the gradient for updating the model. Note that the dimensionality of L and R is much smaller than that of W. Thus RGP can reduce the storage consumption and the noise added to the model. , to reduce the dimension of the gradient before adding noise. GEP first computes an anchor subspace that contains some gradients of public data via the power method. Then, it projects the gradient of private data into the anchor subspace to produce a low-dimensional gradient embedding and a small-norm residual gradient. The two parts are applied with the DP mechanism separately and combined to update the original weight. Compared to RGP, GEP leverages public data to decompose the original model parameters for dimensionality reduction.
AdpAlloc [13]. It proposes a dynamic noise-adding mechanism instead of keeping noise multiplier σ constant every training epoch in vanilla DP-SGD. It replaces the variance in the Gaussian mechanism with a function of the epoch:
M(d) = f (d) + N(0, S 2 f · σ 2 t ),(5)
the value of σ t depends on the final privacy budget, epoch, and schedule function. The schedule function defines how the noise scale is adjusted during training. Yu et al. proposed several pre-defined schedules. We select Exponential Decay in our evaluation, which has the best average performance in [13]. The mathematical form of Exponential Decay is σ t = σ 0 e −kt , where k(k > 0) is decay rate and σ 0 is the initial noise scale.
AdpClip [48]. It uses an adaptive clipping threshold mechanism, which sets the clip threshold to a specified quantile of the update norm distribution every epoch. Formally, clipping threshold C t in epoch t can be computed as
C t = C t−1 · exp(−η C (b − γ)) , where γ ∈ [0, 1] is a quantile to be matched, b 1 m ∑ i∈[m] I x i ≤C
is the empirical fraction of samples with value at most C , and η C is the learning rate with default value of 0.2 in [48]. To address the issue that b reveals private information, Gaussian mechanism is applied to
b:b t = 1 m ∑ i∈Q t b t i + N O, σ 2 b .
The method consumes a negligible privacy budget to track the quantile closely. Adp-Clip was originally designed for federated learning (FL) but can be extended to traditional centralized learning scenarios.
Model Ensemble
PATE [9]
. It first trains multiple teacher models with disjoint private data. The teacher ensemble is later used to label the public data, and the noise perturbation is applied to the voting aggregation before generating a prediction. The student model, which gives the final output, is trained from labeled public data and cannot directly access private data. The privacy budget is determined by the noise added to the votes and the number of queries to the teacher ensemble. Additionally, PATE leverages a semi-supervised learning method to reduce the queries to the teacher ensemble. Priv-kNN [10]. In PATE, a larger number of teacher models lead to a larger absolute lead gap while aggregating votes, potentially allowing for a larger noise level. At the same time, splitting data makes each teacher model hold only partial original training data, which causes a model utility drop. Thus, Zhu et al.
[10] propose a data-efficient scheme based on the private release of k-nearest neighbor (kNN) queries to replace teacher ensemble, which avoids splitting the training dataset. For every given data sample from the public domain, Priv-kNN subsamples a random subset from the entire private dataset. Then it picks the k nearest neighbors from the subset in feature space, equivalent to k teachers' prediction in vanilla PATE.
DPMLBench
This section introduces DPMLBench, a modular toolkit designed to evaluate DPML algorithms' performance on utility and privacy. Figure 2 illustrates the four modules of DPML-Bench.
1. Input. This module prepares the dataset and model for the following modules. For dataset, it involves dataset partition and preprocessing e.g., normalization. For the model, it constructs model architectures and does necessary modifications for private training (see Section 5.1).
2.
Training. This module performs the DPML algorithms to train DPML models. It currently supports twelve different DPML algorithms into four categories (see Section 3).
3. Attack. This module performs two MIAs on models trained from the training module.
4.
Analysis. This module evaluates the performance of DPML algorithms on utility and privacy.
DPMLBench follows a modular design that makes it flexible to integrate new algorithms, attacks, datasets, and models. We envisage that DPMLBench can be used for the following purposes:
• As we have implemented twelve representative DPML algorithms, DPMLBench enables data owners to train their privacy-preserving models with these DPML algorithms efficiently.
• DPMLBench comprehensively assesses different DPML algorithms in utility and privacy. Researchers can re-use DPMLBench as a benchmark tool to evaluate other DPML algorithms and attacks in the future.
• Since DPMLBench follows a modular design, modules are connected through abstract interfaces. To integrate a new DPML algorithm and attack or to extend DPMLBench into different domains, users can re-implement processing functions in the corresponding modules and reuse other modules directly.
Experiments
Based on the proposed taxonomy, we present a series of comprehensive experiments to answer the following questions:
RQ1. What improvements in DPML algorithms are most effective in maintaining model utility?
RQ2. What improvements in DPML algorithms are most robust in defending MIAs?
RQ3. What is the impact of dataset and model architecture on algorithms focusing on different stages? , which are widely used in evaluating privacypreserving machine learning [6, 31, 9, 10]. We resize all images to 32x32 in our evaluation.
Since our attacks are all under the assumption that the attacker has an auxiliary dataset that shares similar distribution with the training data, we split each dataset into four disjoint parts: shadow training set, shadow testing set, target training set, and target testing set. Additionally, we allocate 90% of the data originally used for testing as public data for the algorithms in the ensemble category.
Model Architectures.
We focus on four model architectures, including ResNet20 [20], VGG16 [21], Inception-Net [22], and a simple three convolution layer network as SimpleCNN. Batch normalization makes each sample's normalized value depend on its peers in a batch, making it hard to restrict a single data contribution to the output. To adapt differential privacy, we replace all batch normalization [57] with group normalization [58]. We regard the models trained with the same hyperparameters without DP as the baseline to evaluate utility loss. Table 2 shows the performance of the baseline model across datasets, including testing accuracy and tailored AUC against black-box/white-box MIAs.
We use MLPs for black-box and white-box model architecture for the attack implementation as in [59,40]. A detailed description of the model architecture can be found in Appendix E.
Hyperparameters.
We use Rényi DP to accumulate the overall privacy budget and precompute the required noise scale (σ in DP-SGD) numerically [6, 60]. We keep δ = 10 −5 and use different privacy budgets: ε = {0.2, 0.3, 0.4, 0.5, 1, 2, 4, 8, 100, 1000}. All algorithms' clipping threshold C are fixed to 4 unless the algorithm has special clipping strategies.
We use the hyperparameters obtained by grid search on DP-SGD if the original paper does not mention the setting. While searching hyperparameters, we refer to the guides of recent studies on hyperparameter settings for private training [61,42]. For simplicity, we ignore the privacy leakage caused by hyperparameter tuning in our experiment [62]. For the attack models, we follow the settings in [59], where the batch size is 64, the epoch is 50, the optimizer is Adam, and the learning rate is 10 −5 . Appendix A shows the detailed hyperparameter settings. Metrics. Following previous studies [16,59,19, 18], we use accuracy ACC to evaluate the models' utility and the area under ROC curve (AUC) to evaluate the defense ability of the model. In MIAs, AUC lower than 0.5 indicates that the inference attack performs worse than a random guess and tends to infer non-members as members. Thus we set the lower bound of AUC to 0.5 for analysis convenience, indicating that AUC=0.5 implies no privacy leakage. We process the AUC metric as follows:
AUC = max(AUC, 0.5),
We name AUC as tailored AUC, which is always between 0.5 and 1.
To compare the performance of DPML algorithms and non-private algorithms more directly, we define proportional metric utility loss and privacy leakage, respectively:
Utility Loss = 1 − ACC M pri ACC M base ,(6)Privacy Leakage = AUC M pri − 0.5 AUC M base − 0.5 ,(7)
where M pri presents a private model trained by a DPML algorithm and M base presents a non-private model trained by vanilla SGD with the same settings as M pri . The utility loss denotes the percentage loss in accuracy of the DP model on the same test set relative to the normal model. The private leakage denotes the proportion of privacy models' privacy leakage compared to the normal model.
Evaluation on Utility Loss
Overview. Table 3 reports an overview of algorithms' utility loss across model architectures, datasets, and privacy budgets. Due to space limitations, we only show part of the experimental results. The rest results can be found in Appendix D (Table 9, which shows the similar trend as Table 3.). The experimental results for GEP on InceptionNet and VGG are unavailable due to memory limit. For brevity, we use a Alg, Model, Dataset, ε tuple to denote the Model trained with Alg on Dataset in the case of privacy budget ε. For instance, RGP, ResNet, MNIST, 0.2 indicates the ResNet model trained by RGP with a privacy budget of 0.2 on MNIST. We observe that the utility loss decreases with increasing privacy budget for all algorithms, which intuitively shows that the noise scale hurts the model's utility. However, the utility loss varies widely across algorithms for the same privacy budget. We analyze improved DPML algorithms' utility loss across four categories in the following. NonPrivate in figures denotes the model trained by normal SGD without DP.
Data Preparation.
Initially, in [15], the classifier was trained on private data in order to label the synthetic data, and then the labeled dataset was used to train the target model. This is similar to labeling public data through teacher ensemble in [9], which will consume additional privacy budgets. However, [15] does not count this part. In our implementation, we use data that does not overlap with private data to train the labeling model. Figure 3a illustrates the accuracy comparison between algorithms in the data preparation category and vanilla DP-SGD. The plot shows that Hand-DP outperforms DPGEN and PrivSet in low privacy budget generally. Hand-DP's accuracy is equivalent to vanilla DP-SGD and has a slight advantage on VGG. The performance of DPGEN and PrivSet is highly relative to the quality of synthetic data. When manually inspecting the generated data, we observe that there exist images with wrong labels and many similar, even identical images (e.g., mode collapse). More effort on hyperparameter tuning and manual data filtering for DP synthetic algorithms can improve the performance.
Moreover, Tramer et al. propose using the non-learned handcrafted feature to train a linear model with DP-SGD [12]. Thus, we perform the same experiment for Hand-DP on simple MLP. The experiment results on CIFAR-10 are shown as Table 7 in Appendix D. Comparing other model architectures, we observe that the simple MLP only has an advantage when the privacy budget is relatively small (e.g., ε < 0.5 ). Thus, we exclude the MLP in subsequent experiments to maintain uniformity with other algorithms. Model Design. Figure 3b illustrates the performance of algorithms in the model design category and vanilla DP-SGD.
In general, TanhAct outperforms vanilla DP-SGD and Fo-calLoss on SimpleCNN and VGG. However, neither Tan-hAct nor FocalLoss performs better than vanilla DP-SGD on ResNet and InceptionNet, TanhAct's performance is even much worse than vanilla DP-SGD on ResNet. [11] shows that TanhAct has a better utility-privacy trade-off on their models, whose architecture is similar to SimpleCNN. The difference among the architectures is that ResNet and Incep-tionNet both have GroupNorm layers while the others do not.
To figure out the impact of the GroupNorm layer and ac-tivation function, we add the GroupNorm layer before the activation function of the SimpleCNN and evaluate the performance of the vanilla DP-SGD (DP-SGD with ReLU) and TanhAct (DP-SGD with Tanh) respectively (in Figure 4). We observe that the GroupNorm layer improves the accuracy of the model overall. However, the improvement gap shrinks as the privacy budget increases when using Tanh as an activation function, e.g. DP-SGD (Tanh) w/o GroupNorm outperforms DP-SGD (Tanh) with GroupNorm when the privacy budget is greater than 10. The connection between the activation function and the normalization layer needs further exploration.
Model Training. Figure 3c illustrates the accuracy comparison of algorithms in the model training category and vanilla DP-SGD. When the privacy budget is large, the accuracy of GEP exceeds the baseline in some settings (e.g. GEP, ResNet, CIFAR-10, 1000 ) because of leveraging public data.
When the privacy budget is small, RGP is the only algorithm in this category to achieve acceptable performance on VGG. Model parameter dimensionality reduction is an effective technique to solve large models' inability to adapt to DP. Nevertheless, there is a significant performance degradation when the privacy budget is large for RGP. We suspect the reason is that reparameterization not only reduces the noise scale in private training but also leads to information loss in the gradient. When the noise scale is small, the information loss caused by reparametrization is higher than the mitigation effect on noise perturbation. Table 8 in Appendix D reports the accuracy of RGP (w/o DP) and vanilla SGD, and the difference between them is whether using reparametrization. We train models by using RGP (w/o DP) and vanilla SGD, respectively, , and the difference between them is whether using reparametrization. Overall, the accuracy of RGP (w/o DP) is lower than that of SGD under the same settings across all datasets and model architectures. The results can be found in Table 8 in Appendix D. The results echo our previous speculation that reparametrization reduces noise scale in private training but impairs performance in non-private settings. Model Ensemble. Figure 3d illustrates the accuracy of the algorithms in the model ensemble category and vanilla DP-
SGD.
Note that Priv-kNN and PATE use noise screening technique [10, 38], which ignores the data with low confidence in teacher ensembles to improve the utility-privacy tradeoff. We do not use this technique in our implementation because the privacy budget is given in our settings and the noise scale is precomputed, which requires a fixed number of queries.
When implemented on VGG, Priv-kNN can preserve an equivalent performance as other models, whereas PATE's performance plunges to random guesses. A large number of teachers can impair the noise effect, while the amount of data allocated to each teacher model is too small for a large model such as VGG to converge. The results echo the introduction Hand-DP 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.01 PrivSet 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 DPGEN 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 TanhAct 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.55 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.54 ± 0.00 FocalLoss 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.52 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.00 DP-SGD 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.01 RGP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 GEP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 0.57 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.00 0.54 ± 0.00 -----AdpAlloc 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.00 0.53 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.53 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.54 ± 0.00 AdpClip 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.00 0.56 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.54 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.00 PATE 0.50 ± 0.00 0.52 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 Priv-kNN 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 in Section 3.5, PATE is hard to get a good trade-off on the number of teacher models.
When implemented on other model architectures, Priv-kNN outperforms PATE with a low privacy budget and vice versa with a high privacy budget. PATE and Priv-kNN both show higher accuracy at some specific settings [9, 10]. However, they both fail to obtain a better utility-privacy tradeoff than vanilla DP-SGD at most settings in our measurements. We suspect that semi-supervised training techniques introduce more randomness and require fine-grained hyperparameter tunning, which leads to a high standard deviation as our experimental results show.
Evaluation on Defensive Capabilities
We report the tailored AUC of the black-box MIAs on CIFAR-10 in Table 4, and put the results of other datasets and more settings into Appendix D for having the similar trends (Table 10 for more datasets under black-box MIAs, Table 11 for white-box MIAs). Note that the tailored AUC of attacking the non-private model on the MNIST dataset is already very close to 0.5, so we omit the results on MNIST in this section. Generally, all algorithms' tailored AUC is around 0.5, which means a strong defense against the MIA compared to the baseline results Table 2. Figure 5 illustrates the privacy leakage of models trained by algorithms in a per-category manner. Compared to vanilla DP-SGD, the modification of RGP and FocalLoss change the feature of confidence vectors, resulting in training and testing data having a different distribution for the attack model. Thus, RGP and FocalLoss have a remarkable advantage over black-box and white-box attacks in general. Refer to Figure 5d and Figure 5a. We observe that PATE, Priv-kNN, DP-GEN, and PrivSet remain nearly free of privacy leakage. It is because the target models do not access private data. PATE and Priv-kNN use the knowledge transferred from teacher ensemble, and DPGEN and PrivSet only access generated data. Role of Sensitivity-bounding Techniques. To explore the role of sensitivity-bounding techniques in defending MIAs, we conduct attacks on a model trained with normal SGD and per-sample clipping to explore the impact of per-sample clipping on the defense. The results are shown in Table 5.
We observe that the per-sample clipping has a strong defense ability against MIAs with acceptable accuracy degradation compared to the non-private model. Moreover, the defensive effects and accuracy degradation are model dependent. For example, Inf(clip) performs comparably to ε = 8 on SimpleCNN, but when applied to other models, the performance is worse than when ε = 1000. We suspect the reason why the per-sample clipping technique can defend against MIAs is that it reduces the overfitting of the model. During the training process, applying gradient descent without clipping guides the model to the direction that overfits the training samples; while clipping the gradient makes the model move more conservatively and less overfit to the training samples. Note that the models trained by SGD with per-sample clipping have a defense ability against MIAs but do not satisfy the DP guarantee.
The Role of the Architecture
Architecture Complexity. According to baseline accuracy in Table 2, the model's performance can be ordered as Incep-tionNet > VGG ≈ ResNet > SimpleCNN. Architecture versus Utility Loss. To figure out the impact of model architecture on algorithm performance, we illustrate the boxplot for the utility loss overall algorithms, network, and dataset jointly vary with the privacy budget as Figure 6a.
We observe that the utility loss is similar for ResNet and InceptionNet across different privacy budgets. When the privacy budget is small (ε ≤ 1), the performance of SimpleCNN and VGG is worse than that of ResNet and InceptionNet. As the noise amount becomes smaller (ε > 1), the performance gap between SimpleCNN, ResNet, and InceptionNet narrows. The performance of VGG, the largest model in our assessment, is still poor unless perturbed noise is negligible (ε ≥ 100), while the privacy protection provided by DP is also meaningless. Further, we explored the test accuracy of We observe no strong correlation between privacy leakage and model architecture. VGG has the lowest privacy leakage because many algorithms fail to converge on VGG, leading to the following attack failure.
The Role of the Datasets
Dataset Complexity. As mentioned before, we resize all the samples in each dataset to 32 × 32 pixels. MNIST and FMNIST are simpler than SVHN and CIFAR10 as they only contain gray-scale images. When the number of channels is the same, MNIST and SVHN are easier than FMNIST and CIFAR10, respectively, because the contents of MNIST and FMNIST are digital numbers. The accuracy of baseline models in Table 2 shows the same conclusion.
Dataset versus Utility Loss. To explore the impact of the dataset on the DPML algorithm, we plot the relationship between dataset complexity and model utility loss in Figure 7a. As shown in the plots, the algorithm's performance on these datasets is correlated with the dataset complexity, with worse performance on the harder dataset. Even with a very large privacy budget (ε = 100), nearly half of the private models had a utility loss of more than 30% on CIFAR10 compared to the non-private setting. Dataset versus Privacy Leakage. We plot the relationship between dataset complexity and model privacy leakage in Figure 7b. We observe that more complex datasets lead to less privacy leakage. One reason is that a complex dataset is harder to converge under private settings, and attackers cannot obtain enough information to infer. Additionally, more complex datasets lead to better MIA performance [59] under non-private settings, leading to a smaller privacy leakage value. The tailored AUCs of MIAs on MNIST is around 0.5, whether with or without DP, which leads to privacy leakage close to 100%.
Comparison with Label DP
Label Differential Privacy (Label DP) is a variant of DP where the data labels are considered sensitive and must be protected. The definition of label differential privacy is: If δ = 0, then M is said to be ε-label differentially private (ε-LabelDP). Label DP and DP synthetic algorithms share similar paradigms but differ in generating synthetic datasets by satisfying Label DP instead of standard DP. Our evaluation covers two state-of-the-art Label DP algorithms: LP-MST [14] and ALIBI [63], to explore the difference between Label DP and standard DP algorithms. It is worth noting that the Label-DP satisfies bounded DP. We convert the privacy budget for equivalence while comparing it with other algorithms, and the figure shows the privacy budget in unbounded DP (e.g. RGP, ResNet, CIFAR-10, 1000 and LP-MST, ResNet, CIFAR-10, 2000 share the same horizontal coordinate, 1000). The concrete algorithm description can be found in Appendix C Figure 8a illustrates the comparison of accuracy between Label DP algorithms and vanilla DP-SGD, TanhAct, RGP, Priv-kNN, and DPGEN. We notice that the accuracy of LP-MST and ALIBI can approach or even exceed baseline when the privacy budget is not very large, e.g. the accuracy of LP-MST, ResNet, CIFAR-10, 4 is 71.82 larger than the baseline of 66.56. There are two reasons behind this. One is that noise only affects labels. The training process gradually becomes the same as non-private training as the private budget increase. The other is that the techniques used to mitigate the effects of wrong labels usually also improve the model's generalization, such as mixup [?] used in LP-MST [14]. Figure 8b illustrates the comparison of black-box MIA on Label DP algorithms and vanilla DP-SGD, TanhAct, RGP, Priv-kNN, and DPGEN with the metric of privacy leakage. We observe that Label DP algorithms have higher privacy leakage than standard DP algorithms, which is natural for Label DP because of no protection provided to data.
Takeaways
In the following, we summarize important insights obtained from our measurements and provide some actionable advice to future DPML practitioners. • Different improvement techniques can affect the privacyutility trade-offs of the algorithm from different perspectives. Concretely, parameter dimension reduction in the model training category improves the performance of DPML on large models but impairs utility when the privacy budget is large. Thus, RGP is a good choice for those who want to provide a DP guarantee for large models. On the other hand, algorithms in the model ensemble category and DP synthetic algorithms can be used when stronger defense against MIAs is desired. However, more effort on manual data filtering for DP synthetic algorithms is needed for better utility. • In general, the DPML algorithms provide an effective defense against practical MIAs in both black-box and whitebox manner. The defense performance hardly decreases when the privacy budget increases. The reason is that sensitivity-bounding techniques such as gradient clipping play an important role in defense. More specifically, improved algorithms that do not directly access private data are better at defending against attacks, such as algorithms in the model ensemble category and DP synthetic algorithms. In addition, improved algorithms that affect the attack features of MIAs can achieve additional defensive capabilities. For instance, the confidence vector distribution of FocalLoss is different from that of shadow models, which causes FocalLoss to be more robust to attacks. All algorithms that provide the standard DP guarantee can defend MIAs effectively. • Some model architecture design choices for non-private ML models are ineffective for private ML models. More specifically, a large model scale degrades utility for most DPML algorithms. In addition, using Tanh and Group-Norm can reduce the utility loss on vanilla DP-SGD. However, we also find that using both Tanh and GroupNorm has a negative effect. What model architectures are suitable for DPML is still a research question to be explored. When applying DP to ML models, ResNet and InceptionNet are preferred architectures to attempt. • In general, learning data distribution from more complex datasets is more difficult than that from easier datasets for all DP algorithms. Compared with the non-private setting, applying DP makes it even more difficult to learn from complex datasets. Leveraging external datasets (e.g., pretrain on public dataset [6] and public data embedding [31]) can be helpful to improve the utility of the model on complex datasets. Therefore, designing DPML algorithms to better learn from complex datasets is an interesting future research direction. • Label DP algorithms achieve better model utility than standard DP algorithms, which is expected since label DP algorithms loosen the constraint on adjacent datasets. However, the defense effectiveness of label DP algorithms is worse than that of standard DP algorithms since they only protect the privacy of the label instead of the privacy of the training sample. Label DP should only be used when the label is sensitive, not the data itself, and there is no need to defend against MIAs.
Discussion
In this section, we discuss several potential research directions to inspire interested readers to explore relavent domains.
Emsembled DPML Algorithms.
As discussed in Section 3.1, the improved DPML algorithms in different phases of our taxonomy are independent of each other; thus, one interesting future work is to combine the improvements in different phases to achieve better performance. Shamsabadi et al.
[8] take the first step and show that combining a handcrafted feature extractor [12] in the data preparation phase and optimal loss function in the model design phase can effectively improve the model utility. It would be exciting to follow our taxonomy and combine algorithms at different phases to achieve even better performance. Extension to Other Domains. Our current measurement primarily focuses on image classification tasks, it would be interesting to leverage DPMLBench to measure the performance of DPML algorithms in other domains, such as natural language processing (NLP) and graph neural networks (GNN). DPML Algorithms for Large Models. With the development of deep learning, the model scale increases rapidly, especially in the NLP field. For instance, the famous GPT-3 model contains 175B parameters [64]. However, our measurements show that most of the current DPML algorithms suffer from low model utilities. Furthermore, DP-SGDbased algorithms require calculating per-sample clipping of the gradients, which significantly increases the training time and memory consumption. Therefore, designing high-utility and efficient DPML algorithms for large models is of significant importance in the future. analyze the performance of naive noise perturbation in different stages of the training pipeline.
ML-Doctor [59] also investigates the defenses and attacks against ML models. However, we have different objectives. ML-Doctor aims to evaluate the effectiveness of different types of defenses against attacks. For DPML, they only evaluate the vanilla DP-SGD, and their only conclusion is that DP-SGD can defend against MIAs while failing for other attacks without considering the impact on model utility. On the other hand, DPMLBench conducts more fine-grained taxonomy and evaluation on different DPML algorithms and aims to evaluate the trade-off between model utility, privacy guarantee, and defense effectiveness. This can better facilitate future research on DPML. As such, we obtained more insights on how to design proper DPML algorithms to trade off the above triangle, as stated in Section 5.7.
Conclusion
This paper establishes a taxonomy of improved DPML algorithms along the ML life cycle for four types: data preparation, model design, model training, and model ensemble. Based on taxonomy, we propose the first holistic measurement of improved DPML algorithms' performance on utility and defense capability against MIAs on image classification tasks. Our extensive measurement study covers twelve DPML algorithms, two attacks, four model architectures, four datasets, and various privacy budget configurations. We also cover state-of-the-art label DP in the evaluation.
Among other things, we found that different improvement techniques can affect the privacy-utility trade-off of the algorithm from different perspectives. We also show that DP can effectively defend against MIAs and sensitivity-bounding techniques such as per-sample gradient clipping play an important role in defense. Moreover, some model architecture design choices for non-private ML models are ineffective for private ML models. In addition, label DP has less utility loss but is fragile to MIAs.
We implement a modular re-usable software, DPML-Bench, which contains all algorithms and attacks. DPML-Bench enables sensitive data owners to deploy DPML algorithms and serves as a benchmark tool for researchers and practitioners. Currently, while DPMLBench focuses on image classification models, we plan to extend other types of DP models, such as language models [80,81], graph neural networks [82,83,84], and generative models [85,44] [50] Yilun Du and Igor Mordatch. Implicit generation and modeling with energy based models. Advances in Neural Information Processing Systems, 32, 2019.
[51] Yann LeCun, Sumit Chopra, Raia Hadsell, M Ranzato, and Fujie Huang. A tutorial on energy-based learning.
Predicting structured data, 1(0), 2006.
[52] Laurent Younes. On the convergence of markovian stochastic algorithms with rapidly decreasing ergodicity rates. A Hyperparameter Settings Table 6 reports the detailed hyperparameter settings. Settings of DPGEN and PrivSet are for classifier training. We follow the author's setting for generated algorithms. based algorithm, to perform label perturbation, to determine whether the label of each data sample is obtained by the RR mechanism or randomly generated. To mitigate the effects of mislabeling, LP-MST leverages a multi-stage training strategy.
• ALIBI Malek et al.
[63] provide label DP guarantee by applying additive Laplace noise to a one-hot encoded label. To mitigate the effects of the perturbed label, they apply Bayesian post-processing to the output of the Laplace mechanism to mitigate the effect of mislabeling.
D Additional Results
Results of Hand-DP on MLP. Table 7 shows the results comparison of Hand-DP among 5 model architectures on CIFAR-10. Impact of Model Parameter Amounts on Accuracy. Figure 9 shows the test accuracy of ResNet with the different numbers of parameters trained by vanilla DP-SGD under different privacy budgets. Accuracy of RGP without DP. To figure out the effect of reparametrization in a non-private setting, Table 8 reports the accuracy comparison between RGP without DP and vanilla DP-SGD. Accuracy on FMNIST and SVHN. Figure 10 shows the accuracy of DPML algorithms on FMNIST and SVHN in terms of categories. As a supplementary to Figure 3. Utility loss on FMNIST and SVHN. Table 9 shows the results of utility loss on FMNIST and SVHN as the supplement of Table 3. Extra Results of MIAs.
E Model Architectures
Target Model. Table 12, Table 13, Table 15 and Table 16 show target model architecture, respectively. For simplicity, the details of the block used in the network are shown in Table 14 and Table 17. Attack Model. We present implementation details of attack models as follows:
• Black-Box. We refer to the model architecture of Liu et al.
[59]. The attack model receives two inputs: the target sample's sorted posteriors and a binary indicator on whether the target sample is predicted correctly. The attack model consists of three MLPs (Multi-layer Perceptron). Two processes the input to extract features and concatenated output features are fed into the third MLP to obtain the final prediction. Hand-DP 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 PrivSet 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 DPGEN 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.52 ± 0.00 0.52 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.52 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.01 0.51 ± 0.00 TanhAct 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.52 ± 0.00 FocalLoss 0.50 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.52 ± 0.00 DP-SGD 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 RGP 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.52 ± 0.00 0.52 ± 0.00 GEP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 -AdpAlloc 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 AdpClip 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 PATE 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.52 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 Priv-kNN 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 SVHN Hand-DP 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.53 ± 0.01 0.53 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.01 0.53 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 0.50 ± 0.00 0.53 ± 0.01 PrivSet 0.51 ± 0.01 0.53 ± 0.03 0.52 ± 0.01 0.53 ± 0.01 0.52 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.54 ± 0.02 0.53 ± 0.02 0.52 ± 0.01 0.54 ± 0.01 0.54 ± 0.01 DPGEN 0.50 ± 0.00 0.50 ± 0.01 0.52 ± 0.01 0.50 ± 0.01 0.50 ± 0.00 0.54 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.52 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.52 ± 0.01 0.52 ± 0.02 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.53 ± 0.00 0.51 ± 0.01 TanhAct 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.00 0.54 ± 0.01 0.53 ± 0.01 0.50 ± 0.00 0.52 ± 0.01 0.53 ± 0.00 0.56 ± 0.00 0.57 ± 0.01 0.51 ± 0.02 0.50 ± 0.00 0.51 ± 0.01 0.52 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.00 FocalLoss 0.50 ± 0.00 0.51 ± 0.01 0.52 ± 0.01 0.53 ± 0.00 0.52 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.51 ± 0.02 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.52 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.53 ± 0.00 DP-SGD 0.50 ± 0.00 0.52 ± 0.00 0.51 ± 0.00 0.53 ± 0.01 0.53 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 0.53 ± 0.01 0.51 ± 0.02 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 RGP 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.02 0.51 ± 0.02 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 GEP 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.52 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.52 ± 0.01 0.51 ± 0.01 -AdpAlloc 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.52 ± 0.01 0.53 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.52 ± 0.00 0.53 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.00 AdpClip 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.01 0.53 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.01 0.53 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 PATE 0.52 ± 0.01 0.52 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.53 ± 0.01 0.53 ± 0.01 Priv-kNN 0.52 ± 0.01 0.51 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.02 0.51 ± 0.00 0.52 ± 0.01 0.52 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 Hand-DP 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 PrivSet 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 DPGEN 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 TanhAct 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 FocalLoss 0.50 ± 0.00 0.50 ± 0.01 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.01 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 DP-SGD 0.50 ± 0.00 0.50 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 RGP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 GEP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 -AdpAlloc 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 AdpClip 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 PATE 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 Priv-kNN 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 SVHN Hand-DP 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.52 ± 0.01 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.01 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 0.53 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 PrivSet 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.02 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 DPGEN 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.02 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 TanhAct 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.02 0.55 ± 0.03 0.55 ± 0.03 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.02 0.54 ± 0.01 0.56 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.02 0.52 ± 0.01 0.53 ± 0.02 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 FocalLoss 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.52 ± 0.02 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 DP-SGD 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.02 0.53 ± 0.02 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.53 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.01 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 RGP 0.50 ± 0.00 0.50 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 GEP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.54 ± 0.02 0.51 ± 0.02 0.50 ± 0.00 0.52 ± 0.01 0.53 ± 0.01 0.52 ± 0.01 -AdpAlloc 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.53 ± 0.01 0.53 ± 0.01 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 0.52 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.52 ± 0.01 0.52 ± 0.01 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.53 ± 0.01 AdpClip 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.53 ± 0.02 0.53 ± 0.02 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.53 ± 0.01 0.50 ± 0.00 0.50 ± 0.01 0.51 ± 0.00 0.52 ± 0.01 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 PATE 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 Priv-kNN 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00
CIFAR-10
Hand-DP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.51 ± 0.00 PrivSet 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 DPGEN 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 TanhAct 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.01 0.53 ± 0.01 0.51 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.55 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.01 FocalLoss 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 DP-SGD 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 RGP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 GEP 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.54 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.53 ± 0.00 0.54 ± 0.00 -AdpAlloc 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.51 ± 0.01 0.52 ± 0.00 0.53 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 AdpClip 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.01 0.51 ± 0.00 0.51 ± 0.00 0.51 ± 0.00 0.52 ± 0.00 0.54 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.52 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.52 ± 0.00 PATE 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.01 0.50 ± 0.00 0.51 ± 0.00 0.51 ± 0.01 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 Priv-kNN 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.50 ± 0.00 0.51 ± 0.00 0.50 ± 0.00
Figure 1 :
1Machine learning pipeline.
Definition 2.3 ((α, δ )-Rényi Differential Privacy (RDP) [37]). A randomized mechanism M is said to satisfy ε-Rényi differential privacy of order α (which can be abbreviated as (α, δ )-RDP), if for any adjacent datasets D, D , it holds that D α (M(D)||M(D )) ≤ ε, where D α (M(D)||M(D )) is the α-Rényi divergence between the distribution of M(D) and the distribution of M(D ). Parameter α controls the momentum of the privacy loss random variable.
GEP [31]. Yu et al. observe that the number of noise increases with the growth of model size in vanilla DP-SGD and figure out a solution, GEP [31]
Figure 2 :
2Overview of DPMLBench.
Figure 3 :
3Accuracy comparison of the DPML algorithms in four categories, where the x-axis represents privacy budgets.
Figure 4 :
4Accuracy of SimpleCNN models with/without Group-Norm layer trained by DP-SGD with ReLU and Tanh activation function across varies privacy budget.
Figure 5 :Figure 6 :
56Privacy leakage (under MIA) of DPML algorithms in four categories when given different privacy budgets. Boxplot of utility loss and privacy leakage on all DPML algorithms with various privacy budgets and four network architectures .ResNet with different numbers of parameters under different privacy budgets. Due to space limitaions, detailed results can be viewed atFigure 9in Appendix D. Generally, the smaller the privacy budget and the more model parameters, the worse the model accuracy when training with vanilla DP-SGD.Architecture versus Privacy Leakage. We also present a boxplot for the privacy leakage of all algorithms on different network architectures across privacy budgets asFigure 6b.
Leakage. The tailored AUCs of MIAs on MNIST is around 0.5, whether with or without DP, which leads to privacy leakage close to 100%.
Figure 7 :
7Boxplot of utility loss and privacy leakage on all DPML algorithms with various privacy budgets for different datasets.
Definition 5.1. (Label Differential Privacy). A randomized training algorithm M taking a dataset as input is said to be (ε, δ )-label differentially private, if for any two training datasets D and D that differ in the label of a single example, Pr[M(D) ∈ S] ≤ e ε Pr[M(D ) ∈ S] + δ .
Figure 8 :
8Comparison of Label DP algorithms (LP-MST, ALIBI) and vanilla DP-SGD, TanhAct, RGP, Priv-kNN, and DPGEN under different privacy budgets.
•
MNIST comprises 60000 training samples and 10000 test samples. Each sample is a 28x28 pixel gray handwritten numeral picture. • Fashion-MNIST (FMNIST) has the same size, format, and train /test set division as the MNIST. It covers front images of products from 10 different clothing categories. It has 60000 training samples and 10000 test samples. • CIFAR-10 consists of 10 categories of real-world objects of color images, and the size of each picture is 32×32. There are 50000 training images and 10000 test images in the dataset. • Street View House Number (SVHN) is the house number extracted from the Google Street view image. It can be seen as a colorful and more realistic version of MNIST. It
Figure 9 :
9Accuracy of ResNet with the different number of parameters trained by vanilla DP-SGD under different ε levels.
•Figure 10 :
10White-Box. We use a similar model architecture as the one used by Nasr et al.[40]. There are four inputs for this attack model, including the target model's posteriors, classification loss, gradients of the parameters of the target model's last layer, and true labels in one-hot encoding. Each input is fed into a different neural network to extract the features respectively, and then the features are passed to the classifier after concatenation. Accuracy comparison on FMNIST and SVHN. As a supplementary toFigure 3.
Table 1 :
1Overview and comparison of DPML algorithms. *: Evaluation is based on subsequent private model training on generated data. : same as non-private training. : with modification but no noise adding. : with modification and noise adding.where I is the auxiliary knowledge of adversary, M is the model to be attacked, and x is a data sample. A can be seen as a binary classifier, where 1 means the data sample x is used for training model M, namely a member, and 0 otherwise. It is natural to use MIAs to evaluate the defensive capabilities of DPML algorithms, as in many previous studies[16, 19, 41].Based on the information an attacker can obtain, MIAs can be classified into two categories: White-box and black-box. The white-box attacks have full access to the target model, while black-box attacks only have query access to the target model and obtain the prediction confidence vector. We adopt both types of MIAs to comprehensively evaluate the defensive capabilities of the DPML algorithms (in Section 5.3).Algorithms
Auxiliary Data
Private Data
Model Architecture
Gradient
Loss Function
Perturbation
Data Preparation
Hand-DP
Gradient
PrivSet *
Input
DPGEN *
Input
Model Design
TanhAct
Gradient
FocalLoss
Gradient
Model Training
Vanilla DP-SGD
Gradient
RGP
Gradient
GEP
Gradient
AdpAlloc
Gradient
AdpClip
Gradient
Model Ensemble
PATE
Input
Priv-kNN
Input
2.4 Membership Inference in Machine Learn-
ing Models
The MIAs have become one of the most widely studied [39,
40] attacks against ML models after Shokri et al. proposed
in [3]. The MIA aims to infer whether a data sample is used
to train the target ML model. Formally, MIA A can be de-
fined as:
A : I, M, x −
→ {0, 1},
Hand-DP [12]. Tramer et al. leverage Scattering Network (ScatterNet) [49], a feature extractor that encodes images using a cascade of wavelet transforms to extract the features. To achieve the DP guarantee, they fine-tuned a model on top of extracted features through DP-SGD. DPGEN [15]. It is an instantiation of the DP variant of the Energy-based Model (EBM) [50, 51], which aims to privatize Langevin Markov Chain Monte Carlo (MCMC) sampling method [52] to synthesize images, of which an energy-based network guides the movement directions. DPGEN achieves DP by using Randomized Response (RR) in movement direction selection. Compared to other DP-SGD based synthesis methods [53, 44], DPGEN can generate higher-resolution images.
Table 2 :
2The testing accuracy, tailored AUC of MIAs in black-
box/white-box of baseline models. The number of parameters
follows each model name. (Accuracy(%)/black-box/white-box)
Target Model
MNIST
FMNIST
SVHN
CIFAR-10
SimpleNet (0.17M)
98.42/0.50/0.50 88.04/0.54/0.54 87.69/0.64/0.53 69.50/0.78/0.72
ResNet (0.26M)
99.12/0.50/0.50 89.16/0.52/0.54 92.88/0.57/0.59 66.56/0.77/0.63
InceptionNet (1.97M) 99.18/0.51/0.50 90.92/0.56/0.53 95.08/0.55/0.57 83.52/0.71/0.68
VGG (128.8M)
98.70/0.50/0.52 90.74/0.59/0.56 91.91/0.62/0.56 72.96/0.78/0.73
5.1 Experimental Setup
DPML Algorithms. We implement twelve DPML algo-
rithms; their details can be found in Section 3. For GEP,
RGP, Priv-kNN, DPGEN, and PrivSet, we use implementa-
tions of authors and modify codes to adapt for our evaluation.
The rest of the algorithms are implemented by PyTorch [55]
and Opacus [56].
Datasets.
We conduct experiments on four datasets:
MNIST [23], FashionMNIST [24], CIFAR-10 [25], and
SVHN [26]
Table 3 :
3Overview of algorithms' utility loss on different model architectures, datasets, and privacy budgets. For each privacy budget, we bold the value with the best performance (with the smallest value of utility loss). The experimental results of GEP on VGG are unavailable due to memory limits.SimpleCNN
ResNet
VGG
0.2
1
4
100
1000
0.2
1
4
100
1000
0.2
1
4
100
1000
MNIST
Hand-DP 89.19 ± 1.04 18.98 ± 1.67 8.46 ± 0.35 4.19 ± 0.17 3.73 ± 0.14 24.38 ± 1.93 11.58 ± 0.76 5.73 ± 0.46 2.88 ± 0.50 2.01 ± 0.68 88.63 ± 1.25 88.89 ± 1.23 90.46 ± 0.16 7.98 ± 0.15
3.80 ± 0.78
PrivSet
77.26 ± 5.89 47.54 ± 9.23 30.86 ± 2.26 19.72 ± 2.28 17.58 ± 2.60 89.70 ± 0.20 82.24 ± 2.70 57.01 ± 6.11 20.83 ± 3.60 17.86 ± 2.67 81.36 ± 11.11 58.61 ± 5.03 51.32 ± 22.10 52.40 ± 27.64 66.52 ± 18.65
DPGEN 60.99 ± 8.05 88.26 ± 0.83 14.95 ± 0.58 2.48 ± 0.43 2.70 ± 0.26 70.75 ± 4.84 84.07 ± 3.96 73.73 ± 3.77 3.64 ± 0.36 3.79 ± 0.25 90.08 ± 0.04 90.08 ± 0.04 58.28 ± 30.47 1.76 ± 0.14
2.20 ± 0.28
TanhAct 85.19 ± 2.58 18.30 ± 1.29 3.13 ± 0.39 1.74 ± 0.14 1.74 ± 0.16 38.16 ± 3.27 18.61 ± 3.42 7.05 ± 0.54 3.89 ± 0.44 3.12 ± 0.19 90.21 ± 0.04 89.59 ± 0.12 90.15 ± 0.31 6.14 ± 0.08
1.91 ± 0.03
FocalLoss 87.99 ± 1.97 29.11 ± 2.59 7.44 ± 0.40 3.32 ± 0.07 2.40 ± 0.01 43.09 ± 4.17 10.99 ± 1.70 6.32 ± 0.85 2.72 ± 0.12 1.78 ± 0.12 82.10 ± 1.77 88.16 ± 0.01 88.91 ± 0.39 11.46 ± 11.64 3.66 ± 0.52
DP-SGD 89.61 ± 1.12 21.98 ± 0.19 7.50 ± 0.42 3.58 ± 0.21 3.04 ± 0.23 28.17 ± 3.48 11.33 ± 1.17 5.38 ± 0.85 2.88 ± 0.35 2.24 ± 0.36 88.79 ± 1.02 88.74 ± 0.47 90.56 ± 0.77 13.23 ± 4.42 3.55 ± 0.06
RGP
36.65 ± 1.04 13.23 ± 0.78 10.17 ± 1.02 6.72 ± 0.09 6.37 ± 0.23 31.87 ± 2.62 21.24 ± 3.90 33.03 ± 7.12 34.06 ± 5.62 37.66 ± 8.97 90.30 ± 0.29 6.59 ± 0.95 3.86 ± 0.27
6.33 ± 2.24
4.78 ± 0.14
GEP
90.30 ± 0.29 90.25 ± 0.34 14.37 ± 1.83 2.67 ± 0.52 1.52 ± 0.02 86.22 ± 2.16 17.61 ± 1.35 4.36 ± 0.22 1.00 ± 0.04 0.46 ± 0.26
-
-
-
-
-
AdpAlloc 89.23 ± 0.81 18.79 ± 1.24 6.57 ± 0.20 3.59 ± 0.45 3.14 ± 0.29 24.26 ± 3.70 10.04 ± 2.13 4.91 ± 0.59 3.25 ± 0.57 2.48 ± 0.37 90.24 ± 0.22 89.00 ± 0.58 89.85 ± 1.04 6.44 ± 0.26
3.12 ± 0.04
AdpClip 88.17 ± 4.04 75.55 ± 11.05 7.79 ± 0.37 8.00 ± 0.30 8.10 ± 0.28 59.66 ± 2.95 7.46 ± 0.53 4.85 ± 0.40 4.17 ± 0.35 4.21 ± 0.41 88.92 ± 0.22 88.13 ± 0.24 89.06 ± 0.75 14.00 ± 2.44 4.78 ± 0.15
PATE
82.83 ± 3.94 71.81 ± 4.09 33.36 ± 12.33 11.89 ± 4.04 10.98 ± 2.78 90.86 ± 0.09 85.89 ± 6.08 30.74 ± 15.79 7.12 ± 3.66 10.38 ± 1.16 84.94 ± 3.47 76.17 ± 3.78 44.08 ± 23.93 32.15 ± 40.09 32.66 ± 39.74
Priv-kNN 61.22 ± 2.13 34.97 ± 3.37 33.13 ± 0.62 34.77 ± 0.94 33.80 ± 1.56 25.03 ± 5.76 9.70 ± 0.87 8.43 ± 0.63 9.30 ± 0.74 9.91 ± 1.25 49.05 ± 1.62 17.80 ± 2.17 17.16 ± 0.94 16.29 ± 0.39 14.74 ± 1.43
CIFAR-10
Hand-DP 90.06 ± 0.16 86.88 ± 4.04 48.67 ± 0.96 43.28 ± 0.74 44.14 ± 0.18 84.29 ± 2.26 58.34 ± 0.50 50.74 ± 0.58 41.95 ± 2.33 39.29 ± 3.11 90.03 ± 0.18 89.67 ± 0.07 89.86 ± 0.12 79.74 ± 7.71 37.58 ± 0.82
PrivSet
88.85 ± 0.65 87.78 ± 1.28 86.78 ± 0.71 88.83 ± 0.94 89.43 ± 0.78 89.56 ± 0.83 89.28 ± 0.37 89.45 ± 1.18 87.91 ± 2.03 85.43 ± 2.43 89.77 ± 0.29 88.14 ± 0.78 89.07 ± 0.38 90.04 ± 0.26 87.64 ± 2.81
DPGEN 90.16 ± 0.11 89.86 ± 0.09 89.66 ± 0.70 69.98 ± 2.28 76.98 ± 2.26 90.00 ± 0.35 90.00 ± 0.16 90.59 ± 1.39 79.51 ± 1.01 83.38 ± 4.20 90.52 ± 0.30 89.72 ± 0.21 89.47 ± 0.29 87.24 ± 3.07 88.86 ± 1.43
TanhAct 89.74 ± 0.64 69.95 ± 1.13 45.39 ± 0.92 32.93 ± 0.55 32.21 ± 0.21 82.55 ± 1.17 62.11 ± 0.33 55.52 ± 0.32 48.95 ± 1.17 49.28 ± 2.73 90.22 ± 0.00 90.07 ± 0.25 90.13 ± 0.10 64.46 ± 1.43 34.26 ± 0.28
FocalLoss 89.88 ± 0.08 88.17 ± 2.46 52.42 ± 0.47 38.55 ± 0.79 38.47 ± 0.90 84.36 ± 2.43 62.12 ± 0.53 52.06 ± 0.43 40.65 ± 2.12 39.00 ± 2.98 90.17 ± 0.26 89.75 ± 0.15 89.85 ± 0.23 66.36 ± 6.19 36.60 ± 0.33
DP-SGD 89.80 ± 0.30 89.13 ± 1.30 48.79 ± 0.24 40.03 ± 0.93 40.48 ± 0.86 81.92 ± 2.61 58.32 ± 0.44 49.57 ± 1.76 41.17 ± 2.67 38.66 ± 3.80 90.20 ± 0.56 89.38 ± 0.49 89.73 ± 0.09 89.81 ± 0.12 35.15 ± 0.35
RGP
90.15 ± 0.02 61.91 ± 1.32 58.52 ± 1.34 54.28 ± 0.68 54.41 ± 0.92 74.48 ± 0.54 65.24 ± 0.88 67.27 ± 2.00 66.38 ± 0.93 66.56 ± 0.82 90.16 ± 0.01 81.87 ± 4.19 53.66 ± 1.25 53.49 ± 0.09 54.37 ± 0.49
GEP
90.16 ± 0.00 90.16 ± 0.01 90.16 ± 0.00 35.11 ± 0.20 31.90 ± 0.24 88.68 ± 2.19 85.19 ± 0.20 46.72 ± 0.73 30.45 ± 0.36 26.64 ± 0.93
-
-
-
-
-
AdpAlloc 90.04 ± 0.30 89.89 ± 0.18 47.97 ± 0.57 38.49 ± 0.47 39.16 ± 0.88 80.04 ± 2.19 57.88 ± 0.86 48.86 ± 1.03 43.83 ± 1.52 42.22 ± 2.19 90.06 ± 0.05 89.57 ± 0.05 89.99 ± 0.08 51.42 ± 0.58 35.46 ± 0.42
AdpClip 89.71 ± 0.23 89.79 ± 0.26 64.50 ± 2.33 35.64 ± 0.82 34.12 ± 0.42 86.57 ± 1.43 64.08 ± 0.77 48.05 ± 1.15 37.17 ± 1.27 33.55 ± 1.98 89.70 ± 0.34 89.86 ± 0.68 90.21 ± 0.19 89.69 ± 0.01 44.47 ± 0.07
PATE
90.19 ± 1.30 91.70 ± 1.44 89.25 ± 0.53 83.30 ± 2.42 83.06 ± 0.54 88.34 ± 0.41 87.60 ± 1.13 85.99 ± 2.08 82.05 ± 1.18 83.50 ± 1.43 90.05 ± 1.23 91.60 ± 0.49 91.06 ± 1.11 89.92 ± 1.56 90.02 ± 2.79
Priv-kNN 89.52 ± 0.45 89.38 ± 0.12 88.94 ± 0.17 90.19 ± 0.05 90.29 ± 0.40 87.77 ± 1.70 81.35 ± 1.56 77.38 ± 1.27 74.96 ± 0.34 74.43 ± 1.27 89.85 ± 1.27 87.26 ± 2.55 85.42 ± 2.04 84.81 ± 1.32 84.41 ± 1.56
Table 4 :
4Overview of algorithms' tailored AUC in black-box MIA on different model architectures and privacy budgets. In every setting, we bold the value with the best performance (with the smallest value). The experimental results for GEP on InceptionNet and VGG are unavailable due to memory limits.SimpleCNN
ResNet
VGG
0.2
1
4
100
1000
0.2
1
4
100
1000
0.2
1
4
100
1000
CIFAR-10
Table 5 :
5Impact of per-sample clipping on model utility and defense to attacks. The table reports the accuracy and the AUC of models on CIFAR-10 with different privacy guarantees. Inf indicates normal SGD; Inf (Clip) denotes normal SGD with persample clipping.8
100
1000 Inf(clip)
Inf
SimpleCNN
ACC (%) 58.20 60.66 60.44
57.96
69.22
AUC
0.52
0.52
0.53
0.52
0.78
ResNet
ACC (%) 53.80 61.50 65.90
57.42
69.70
AUC
0.51
0.52
0.53
0.54
0.65
InceptionNet
ACC (%) 58.00 64.60 69.40
72.80
83.68
AUC
0.51
0.51
0.52
0.58
0.71
VGG
ACC (%) 10.36 10.02 64.66
67.72
71.36
AUC
0.50
0.50
0.52
0.59
0.78
7 Related Work
RelatedDifferential Privacy. Differential privacy (DP)[34, 36] is a widely used rigorous mathematical definition to formalize and measure privacy guarantees based on a parameter called privacy budget. It has been adopted for a number of data analysis tasks, such as synthetic dataset genera-tion [65, 66, 67, 68], marginal release [69], range query [70], and stream data analysis [71]. Some studies propose integrating DP with traditional machine learning algorithms, such as naive Bayes and Linear Support Vector Machine (SVM) [72, 73, 74]. Abadi et al. propose vanilla DP-SGD [6] as the first general DPML algorithm. Recent studies try to mitigate DP's impairment on utility by proposing new algorithms [9, 12, 31, 10] or relax DP definition for specific scenarios [75, 14, 76]. Membership Inference Attacks. The adversary in MIAs aims to infer whether a given data sample is used to train the target model. Currently, the MIA is one of the critical methods to assess the privacy risk of ML models [3, 39, 40, 77, 78, 79]. According to the accessibility to the target model, the MIA can be categorized into black-box and white-box attacks. Shokri et al. [3] propose the first black-box MIA against ML models. They propose to train multiple shadow models to simulate the behavior of the target model and use shadow models to generate the data used to train the attack model. Salem et al. [39] simplify their method by using one shadow dataset and one shadow model. Nasr et al. [40] first propose white-box MIAs, where the adversary knows the internal parameters of the target model. DPML Measurement. Several DPML measurement studies concentrate on different perspectives [16, 17, 18, 19]. Jayaraman et al. [16] analyzed the difference of privacy leakage of relaxed variants of differential privacy. They explore the difference in privacy leakage when using the same algorithm with different DP definitions. Iyengar et al. [17] evaluate several differentially private convex optimization algo-rithms. The work of Zhao et al. [18] and Jarin et al. [19]
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Table 6 :
6Detailed hyperparameter settings. Settings of DPGEN
and PrivSet are for classifier training. We follow the author's
setting for generated algorithms.
Learning Rate Batch Size
Epoch
Additional
vanilla DP-SGD
0.01
256
MNIST,FMNIST:60
SVHN,CIFAR-10:90
TanhAct
0.01
256
MNIST,FMNIST:60
SVHN,CIFAR-10:90
AdpAlloc
0.01
256
MNIST,FMNIST:60
ExpDecay
SVHN,CIFAR-10:90
k=0.01
AdpClip
0.01
256
MNIST,FMNIST:60
target unclipped quantile=0.7
clipbound learning rate=0.1
SVHN,CIFAR-10:90
max clipbound=10
min clipbound=0.05
FocalLoss
0.01
256
MNIST,FMNIST:60
weight decay=1e-4
SVHN,CIFAR-10:90
Handcrafted
0.01
256
MNIST,FMNIST:60
SVHN,CIFAR-10:90
GEP
0.1
256
MNIST,FMNIST:60
num groups=3
num bases=1000
SVHN,CIFAR-10:90
weight decay=2e-4
aux data size=2000
RGP
0.1
256
MNIST,FMNIST:60
width=1
rank=16
SVHN,CIFAR-10:90
weight decay=1e-4
PATE
0.001
200
500
n teacher=100
Priv-kNN
0.01
512
500
iteration=2
sample prob=0.15
DPGEN
0.01
1024
100
PrivSet
0.01
10
300
samples per class=10
LP-MST
0.01
256
200
stage=2
Table 7 :
7Shadow Testing Dataset is used to generate training data as non-members for attack models.Test accuracy of 5 model architectures on CIFAR-10
when given various privacy budgets.
0.2
0.4
1
4
100
1000
Inf
MLP
27.30 32.50 38.98 46.70 54.02 57.48 57.32
SimpleCNN
9.66
9.98 10.24 51.12 57.06 55.66 65.18
ResNet
14.78 28.04 42.02 49.36 60.80 64.58 72.16
InceptionNet 15.08 22.14 39.54 52.22 60.78 66.42 81.58
VGG
9.70 10.16 10.32 10.22 30.08 60.90 72.26
comprises 73257 training samples and 26032 test samples,
which are 32×32 RGB images. We trim the testset size to
10000 while keeping distribution consistent with the orig-
inal testset.
1. Target Training Dataset is regarded as private data and
member samples while evaluating the performance of
MIAs.
2. Target Testing Dataset is used to evaluate the utility
performance of the model. It is also used to evaluate the
performance of MIAs as non-member samples.
3. Shadow Training Dataset is used to train shadow mod-
els as auxiliary datasets of adversaries and then generate
training data as members for attack models.
4. C Details of Label DP Algorithms
• LP-MST Ghazi et al. [14] introduced RRWithPrior, a Ran-
domized Response (RR) [86]
Table 8 :
8Accuracy of RGP (w/o DP) and vanilla SGD. Other
settings keep the same as Table 3.
SimpleCNN
ResNet
InceptionNet
VGG
RGP(w/o DP) SGD RGP(w/o DP) SGD RGP(w/o DP) SGD RGP(w/o DP) SGD
MNIST
95.50
98.42
97.78
99.24
99.04
99.18
98.56
98.68
FMNIST
85.70
88.04
86.62
88.60
90.76
91.70
88.72
90.48
SVHN
73.70
87.69
90.58
93.84
93.09
94.90
88.14
89.77
CIFAR-10
50.40
69.22
59.94
68.16
75.74
83.68
68.32
71.36
Table 11reports the tailored AUC in white-box style on all model architectures, datasets, and privacy budget.Table 10reports the tailored AUC in blackbox on FMNIST and SVHN.
Table 9 :
9Overview of algorithms' utility loss on different model architectures, datasets, and privacy budget. For each privacy budget, we bold the value with the best performance (with the smallest value of utility loss). Supplement toTable 3.SimpleCNN
ResNet
InceptionNet
VGG
0.2
1
4
100
1000
0.2
1
4
100
1000
0.2
1
4
100
1000
0.2
1
4
100
1000
FMNIST
DPGEN
66.83 ± 1.90
67.80 ± 0.84
72.94 ± 0.77 43.30 ± 2.60 46.79 ± 2.73 75.84 ± 1.15 76.69 ± 2.18 73.98 ± 0.31 45.47 ± 3.09 47.87 ± 6.09 77.64 ± 2.13 77.92 ± 2.63 76.09 ± 1.60 44.21 ± 1.41 43.57 ± 0.23 74.17 ± 0.26 76.04 ± 1.32 71.35 ± 0.31
55.22 ± 6.61
48.50 ± 4.07
PrivSet
68.35 ± 3.18
62.69 ± 0.28
59.71 ± 4.98 49.09 ± 3.80 53.22 ± 6.59 86.93 ± 2.07 73.35 ± 8.89 52.70 ± 7.45 73.66 ± 11.15 49.31 ± 7.46 74.68 ± 15.51 60.46 ± 9.81 58.60 ± 2.53 67.81 ± 5.72 62.05 ± 7.41 73.91 ± 2.25 63.89 ± 4.96 85.38 ± 5.99
82.40 ± 9.16 75.13 ± 20.65
Hand-DP 71.89 ± 12.70 32.68 ± 1.89
17.55 ± 0.24 17.50 ± 0.25 17.12 ± 0.50 34.50 ± 1.06 25.48 ± 1.10 19.12 ± 0.31 15.24 ± 0.51 14.09 ± 0.36 49.09 ± 2.86 26.15 ± 1.12 19.99 ± 1.00 14.44 ± 0.73 11.53 ± 0.75 89.30 ± 0.07 89.65 ± 0.33 91.13 ± 0.40
17.91 ± 1.62
14.67 ± 1.31
TanhAct
84.79 ± 1.74
26.01 ± 1.07
12.85 ± 0.29 10.99 ± 0.32 11.00 ± 0.08 33.38 ± 0.99 25.50 ± 0.71 18.79 ± 0.76 14.00 ± 0.90 13.14 ± 0.85 64.14 ± 1.88 24.70 ± 0.67 20.01 ± 0.73 14.52 ± 0.48 12.42 ± 0.79 90.08 ± 0.09 89.81 ± 0.06 89.31 ± 0.19
19.38 ± 0.20
11.11 ± 0.04
FocalLoss 89.94 ± 0.47
38.54 ± 1.12
18.85 ± 0.04 14.44 ± 0.65 13.95 ± 0.43 43.65 ± 3.44 25.48 ± 0.61 19.46 ± 0.65 14.37 ± 0.20 12.88 ± 0.35 59.75 ± 6.38 26.32 ± 1.17 20.31 ± 1.31 14.48 ± 0.95 11.72 ± 0.30 79.11 ± 0.91 89.53 ± 0.47 87.89 ± 2.39
28.78 ± 9.79
14.35 ± 0.71
DP-SGD
77.79 ± 17.00 31.26 ± 1.65
17.24 ± 0.62 15.72 ± 0.56 15.83 ± 0.58 35.65 ± 2.99 24.92 ± 1.21 20.21 ± 0.77 15.76 ± 0.28 13.83 ± 0.82 53.24 ± 1.26 25.75 ± 0.84 20.15 ± 0.87 14.51 ± 1.33 12.40 ± 0.15 88.56 ± 0.19 89.34 ± 0.25 90.15 ± 0.14
29.09 ± 8.31
13.02 ± 1.54
AdpAlloc
54.58 ± 3.02
28.31 ± 1.91
16.03 ± 0.59 15.23 ± 0.67 15.45 ± 0.61 31.21 ± 0.90 23.88 ± 1.02 20.28 ± 0.19 15.63 ± 0.25 14.38 ± 0.05 41.08 ± 6.55 23.67 ± 0.35 19.88 ± 0.48 14.29 ± 0.24 12.68 ± 0.21 86.30 ± 1.76 90.32 ± 0.00 89.82 ± 0.24
17.14 ± 0.34
12.48 ± 0.06
AdpClip
78.18 ± 15.67 73.69 ± 11.55 20.30 ± 1.37 13.00 ± 0.31 13.26 ± 0.33 50.48 ± 3.76 24.08 ± 0.61 18.71 ± 0.49 12.65 ± 0.15 12.46 ± 0.53 70.79 ± 5.57 25.55 ± 0.19 19.02 ± 0.82 11.97 ± 0.58 11.81 ± 0.78 90.34 ± 0.37 90.13 ± 0.15 89.43 ± 0.03 62.83 ± 22.77 14.81 ± 0.57
GEP
89.85 ± 0.05
89.85 ± 0.05 68.11 ± 30.47 11.69 ± 0.46 10.35 ± 0.54 85.69 ± 2.27 33.40 ± 1.20 17.09 ± 0.99 10.60 ± 0.31 10.33 ± 0.20
-
-
-
-
-
-
-
-
-
-
RGP
44.80 ± 4.49
23.10 ± 0.76
23.04 ± 1.88 21.12 ± 1.65 20.67 ± 1.66 33.95 ± 1.86 30.97 ± 2.00 36.36 ± 2.44 37.00 ± 1.28 37.32 ± 1.08
28.00 ± 0.73 21.33 ± 0.33 21.42 ± 1.10 21.98 ± 0.65 23.52 ± 2.67 89.82 ± 0.05 23.08 ± 0.45 18.28 ± 0.46
20.10 ± 0.85
20.70 ± 0.42
PATE
81.62 ± 6.09
85.35 ± 4.39
51.37 ± 3.09 36.89 ± 1.88 38.07 ± 1.62 88.71 ± 3.48 81.72 ± 4.08 63.40 ± 8.27 42.57 ± 7.48 40.15 ± 5.16 84.24 ± 2.54 79.90 ± 3.35 62.70 ± 0.67 36.52 ± 1.90 37.80 ± 1.75 92.41 ± 1.76 86.05 ± 5.23 76.88 ± 15.73 65.12 ± 22.83 65.02 ± 21.25
Priv-kNN
72.31 ± 2.36
55.44 ± 2.34
51.94 ± 1.77 52.08 ± 0.75 50.97 ± 0.83 57.59 ± 3.89 46.23 ± 0.81 44.69 ± 0.54 44.55 ± 1.04 44.55 ± 1.20 55.54 ± 1.59 41.12 ± 1.26 39.95 ± 1.36 40.52 ± 0.86 40.12 ± 0.86 68.04 ± 1.31 47.17 ± 1.93 46.16 ± 0.65
45.90 ± 0.38
45.09 ± 0.49
SVHN
DPGEN
89.23 ± 0.13
92.20 ± 0.39
91.60 ± 1.03 51.49 ± 9.52 49.05 ± 4.32 88.80 ± 0.10 89.02 ± 0.77 85.70 ± 4.31 85.75 ± 5.11 84.56 ± 1.80 81.12 ± 3.22 86.86 ± 1.87 81.70 ± 1.76 65.34 ± 2.70 58.32 ± 3.10 90.93 ± 0.90 90.63 ± 0.84 89.91 ± 1.43 63.70 ± 19.35 56.55 ± 13.36
PrivSet
88.96 ± 4.87
88.36 ± 2.55
81.66 ± 1.12 79.83 ± 0.29 80.94 ± 1.15 91.14 ± 0.62 88.50 ± 0.74 87.49 ± 3.02 84.20 ± 2.91 85.08 ± 2.90 89.60 ± 1.67 88.21 ± 1.45 77.93 ± 1.06 78.17 ± 1.98 74.50 ± 2.11 90.23 ± 1.85 89.34 ± 3.79 92.26 ± 1.48
90.11 ± 3.31
87.44 ± 2.08
Hand-DP
83.78 ± 4.33
56.72 ± 2.80
23.46 ± 0.46 17.85 ± 0.20 16.77 ± 0.75 82.99 ± 0.76 36.15 ± 0.51 21.09 ± 1.58 11.23 ± 0.75
9.56 ± 1.07
80.24 ± 0.11 33.78 ± 0.96 16.57 ± 0.83 10.44 ± 0.28 7.71 ± 0.51 87.61 ± 3.42 83.85 ± 0.70 80.64 ± 0.18
80.24 ± 0.14
16.50 ± 0.56
TanhAct
88.96 ± 0.38
49.05 ± 1.51
22.64 ± 0.24 12.76 ± 0.16 12.11 ± 0.22 86.90 ± 0.30 80.12 ± 0.51 56.90 ± 0.35 40.32 ± 3.74 40.07 ± 5.48 80.82 ± 1.03 46.88 ± 0.32 20.66 ± 0.61 12.46 ± 0.66 9.35 ± 0.22 89.75 ± 0.13 90.83 ± 0.39 88.21 ± 1.12
61.78 ± 3.59
16.02 ± 0.68
FocalLoss 85.07 ± 3.75
74.53 ± 4.55
27.98 ± 0.45 17.12 ± 0.36 15.11 ± 0.27 81.79 ± 0.68 36.78 ± 3.60 22.59 ± 0.74 12.54 ± 0.97
9.80 ± 0.25
81.85 ± 0.41 38.60 ± 1.63 18.12 ± 0.52 10.53 ± 0.18 7.67 ± 0.33
90.38 ± 4.59 82.58 ± 1.02 80.60 ± 0.01
80.34 ± 0.14
17.56 ± 0.52
DP-SGD
85.22 ± 3.90
71.58 ± 4.43
23.23 ± 0.10 16.93 ± 0.80 15.46 ± 0.95 83.27 ± 1.93 34.64 ± 5.15 18.68 ± 1.01 11.47 ± 1.08 10.43 ± 1.56 80.25 ± 0.16 32.68 ± 2.33 15.40 ± 0.84 9.75 ± 0.16
8.58 ± 0.20 89.94 ± 0.77 83.92 ± 1.92 80.42 ± 0.28
80.38 ± 0.07
17.19 ± 0.13
AdpAlloc
84.76 ± 3.26
80.19 ± 0.13
23.95 ± 0.48 16.51 ± 0.46 15.83 ± 0.65 82.66 ± 0.68 33.80 ± 2.21 20.45 ± 0.81 12.23 ± 0.73 11.25 ± 0.73 80.19 ± 0.12 35.80 ± 1.66 16.02 ± 0.91 10.22 ± 0.38 8.31 ± 0.23 83.74 ± 6.73 80.77 ± 0.26 80.40 ± 0.05
27.93 ± 3.34
20.09 ± 1.53
AdpClip
89.67 ± 3.85
83.11 ± 1.84
44.26 ± 3.69 14.78 ± 0.27 14.86 ± 0.65 88.90 ± 0.57 38.17 ± 3.19 17.06 ± 0.66
9.27 ± 0.61
9.31 ± 0.81
82.00 ± 1.83 42.49 ± 2.77 15.79 ± 0.45 8.83 ± 0.83
8.38 ± 0.13 87.13 ± 4.53 90.06 ± 0.05 80.42 ± 0.11
80.20 ± 0.14
22.33 ± 0.73
GEP
89.25 ± 5.81
93.17 ± 0.01
55.13 ± 7.38 14.25 ± 0.83 11.93 ± 0.66 87.56 ± 1.05 89.73 ± 1.65 15.02 ± 0.24
7.77 ± 0.38
7.44 ± 0.57
-
-
-
-
-
-
-
-
-
-
RGP
93.17 ± 0.04
45.96 ± 2.54
37.55 ± 1.08 29.87 ± 1.53 28.75 ± 1.73 69.61 ± 1.38 49.92 ± 3.67 54.89 ± 2.85 57.00 ± 4.73 57.07 ± 5.13 78.54 ± 1.42 19.12 ± 0.88 17.20 ± 1.51 24.74 ± 3.14 26.37 ± 1.83 93.24 ± 0.04 81.11 ± 0.77 22.03 ± 0.26
27.76 ± 0.91
27.16 ± 1.08
PATE
90.64 ± 1.66
89.54 ± 1.08
82.13 ± 3.54 73.59 ± 4.85 70.62 ± 2.16 90.10 ± 0.98 87.74 ± 3.17 85.94 ± 3.54 73.20 ± 2.04 74.73 ± 0.54 88.11 ± 3.20 90.13 ± 1.10 79.08 ± 1.78 58.29 ± 6.01 63.46 ± 2.76 89.05 ± 4.66 88.61 ± 2.47 79.95 ± 0.94
74.80 ± 7.42
76.07 ± 5.64
Priv-kNN
87.78 ± 0.71
77.79 ± 1.02
70.58 ± 0.65 70.21 ± 1.42 69.29 ± 0.55 81.30 ± 1.82 44.26 ± 0.67 40.86 ± 1.87 39.84 ± 1.14 41.74 ± 2.01
68.28 ± 5.75 48.09 ± 0.69 45.75 ± 2.42 44.43 ± 1.83 44.55 ± 1.06 81.14 ± 1.03 46.04 ± 0.37 41.76 ± 3.15
42.19 ± 1.55
40.58 ± 0.65
Table 10 :
10Overview of algorithms' tailored AUC in black-box on on FMNIST and SVHN..
Table 11 :
11Overview of algorithms' tailored AUC in white-box style on all model architectures, datasets, and privacy budget .SimpleCNN
ResNet
InceptionNet
VGG
0.2
1
4
100
1000
0.2
1
4
100
1000
0.2
1
4
100
1000
0.2
1
4
100
1000
FMNIST
Table 12 :
12SimpleNet architecture and details of BasicBlock.Layer Type Architecture
BasicBlock
filters=16
BasicBlock
filters=32
BasicBlock
filters=64
Flatten
Relu FC
500 units
FC
10 units
BasicBlock
filters
Conv2D
filters, kernel size=3, padding=1
Activation
ReLU
MaxPooling
kernel size=2, stride=2
Table 13 :
13ResNet architecture.Layer Type
Architecture
Input Layer
filters=16
ResBlock 1
filters=16
ResBlock 1
filters=16
ResBlock 1
filters=16
ResBlock 2
filters=32, stride=2
ResBlock 2
filters=32, stride=2
ResBlock 2
filters=32, stride=2
ResBlock 2
filters=64, stride=2
ResBlock 2
filters=64, stride=2
ResBlock 2
filters=64, stride=2
AdaptiveAvgPool2D
output size=(1,1)
FC
10 units
Table 14 :
14Details of ResBlock for ResNet. (Shortcut perform identity mapping, and their outputs are added to the outputs of the stacked layers [20])Input Layer
filters
Conv2D
filters, kernel size=3
GroupNorm num groups=4, num channels=filters, affine=False
Activation
ReLU
ResBlock 1
filters
Conv2D
filters, kernel size=3, stride=1
GroupNorm
num groups=4, num channels=filters
Activation
ReLU
Conv2D
filters, kernel size=3, stride=1
GroupNorm
num groups=4, num channels=filters
Shortcut
Activation
ReLU
ResBlock 2
filters, stride
Conv2D
filters, kernel size=3, stride
GroupNorm
num groups=4, num channels=filters
Activation
ReLU
Conv2D
filters, kernel size=3, stride=1
GroupNorm
num groups=4, num channels=filters
AvgPool2D
kernel size=1, stride
GroupNorm
num groups=4, num channels=filters
Shortcut
Activation
ReLU
Table 15 :
15InceptionNet architecture.Layer Type
Architecture
InceptionBlock
filters=32, kernel size=3, stride=1
InceptionBlock
filters=32, kernel size=3, stride=1
MaxPool2D
kernel size=2, stride=1
InceptionA
InceptionB
InceptionC
InceptionD
InceptionE
AdaptiveAvgPool2d
output size=(1,1)
Dropout
p=0.5
FC
10 units
Table 16 :
16VGG architecture and the details of the VGGBlock.Layer Type
Architecture
VGGBlock
filters=64
MaxPool2D kernel size=2, stride=2
VGGBlock
filters=128
MaxPool2D kernel size=2, stride=2
VGGBlock
filters=256
VGGBlock
filters=256
MaxPool2D kernel size=2, stride=2
VGGBlock
filters=512
VGGBlock
filters=512
MaxPool2D kernel size=2, stride=2
VGGBlock
filters=512
VGGBlock
filters=512
MaxPool2D kernel size=2, stride=2
Flatten
FC
4096 units
Activation
ReLU
Dropout
p=0.5
FC
4096 units
Activation
ReLU
Dropout
p=0.5
FC
10 units
VGGBlock
filters
Conv2D
filters, kernel size=3, padding=1
Activation
ReLU
Table 17 :
17Details of InceptionBlock for InceptionNet. InceptionBlock filters=32, kernel size=5, padding=2 InceptionBlock filters=48, kernel size=3, padding=1 InceptionBlock filters=16, kernel size=1 InceptionBlock filters=48, kernel size=3, padding=1 Concat InceptionB InceptionBlock filters=96, kernel size=3, stride=2 InceptionBlock filters=32, kernel size=1 MaxPool2D kernel size=3, stride=2 InceptionBlock filters=48, kernel size=3, padding=1 InceptionBlock filters=48, kernel size=3, stride=2 Concat InceptionC InceptionBlock filters=48, kernel size=1 InceptionBlock filters=48, kernel size=1 AvgPool2D kernel size=3, stride=1, padding=1 InceptionBlock filters=48, kernel size=7, padding=3 InceptionBlock filters=48, kernel size=1 InceptionBlock filters=48, kernel size=7, padding=3 Concat InceptionD InceptionBlock filters=48, kernel size=1 InceptionBlock filters=96, kernel size=1 MaxPool2D kernal size=3, stride=2 InceptionBlock filters=96, kernel size=3, stride=2 InceptionBlock filters=96, kernel size=7, padding=3 InceptionBlock filters=96, kernel size=3, stride=2 Concat InceptionE InceptionBlock filters=80, kernel size=1 InceptionBlock filters=96, kernel size=1 InceptionBlock filters=112, kernel size=1 AvgPool2D kernel size=3,stride=1,padding=1 InceptionBlock filters=96, kernel size=3, padding=1 InceptionBlock filters=96, kernel size=3, padding=1 InceptionBlock filters=48, kernel size=1 InceptionBlock filters=96, kernel size=3, padding=1 ConcatInceptionBlock
filters, kernel size, padding
Conv2D
filters, kernel size, padding
GroupNorm
num groups=4, num channels=filters
Activation
ReLU
InceptionA
InceptionBlock
filters=32, kernel size=1
InceptionBlock filters=24, kernel size=1
InceptionBlock filters=32, kernel size=1
AvgPool2D
kernel size=3,stride=1,padding=1
A survey on deep learning based face recognition. Computer vision and image understanding. Guodong Guo, Na Zhang, 189102805Guodong Guo and Na Zhang. A survey on deep learn- ing based face recognition. Computer vision and image understanding, 189:102805, 2019.
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Fedgraphnn: A federated learning benchmark system for graph neural networks. Chaoyang He, Keshav Balasubramanian, Emir Ceyani, Carl Yang, Han Xie, Lichao Sun, Lifang He, Liangwei Yang, S Philip, Yu Yu, Rong, ICLR 2021 Workshop on Distributed and Private Machine Learning (DPML). 2021Chaoyang He, Keshav Balasubramanian, Emir Ceyani, Carl Yang, Han Xie, Lichao Sun, Lifang He, Liang- wei Yang, Philip S Yu, Yu Rong, et al. Fedgraphnn: A federated learning benchmark system for graph neural networks. In ICLR 2021 Workshop on Distributed and Private Machine Learning (DPML), 2021.
| {'fraction_non_alphanumeric': 0.09044641306235887, 'fraction_numerical': 0.14275664408546118, 'mean_word_length': 3.2739792130660725, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 4, 'https://': 1, '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': "Differential privacy (DP), as a rigorous mathematical definition quantifying privacy leakage, has become a wellaccepted standard for privacy protection. Combined with powerful machine learning techniques, differentially private machine learning (DPML) is increasingly important. As the most classic DPML algorithm, DP-SGD incurs a significant loss of utility, which hinders DPML's deployment in practice. Many studies have recently proposed improved algorithms based on DP-SGD to mitigate utility loss. However, these studies are isolated and cannot comprehensively measure the performance of improvements proposed in algorithms. More importantly, there is a lack of comprehensive research to compare improvements in these DPML algorithms across utility, defensive capabilities, and generalizability.We fill this gap by performing a holistic measurement of improved DPML algorithms on utility and defense capability against membership inference attacks (MIAs) on image classification tasks. We first present a taxonomy of where improvements are located in the machine learning life cycle. Based on our taxonomy, we jointly perform an extensive measurement study of the improved DPML algorithms, over twelve algorithms, four model architectures, four datasets, two attacks, and various privacy budget configurations. We also cover state-of-the-art label differential privacy (Label DP) algorithms in the evaluation. According to our empirical results, DP can effectively defend against MIAs, and sensitivity-bounding techniques such as per-sample gradient clipping play an important role in defense. We also explore some improvements that can maintain model utility and defend against MIAs more effectively. Experiments show that Label DP algorithms achieve less utility loss but are fragile to MIAs. Machine learning practitioners may benefit from these evaluations to select appropriate algorithms. To support our evaluation, we implement a modular re-usable software, DPMLBench, 1 which enables sensitive data owners to deploy DPML algorithms and serves as a benchmark tool for researchers and practitioners.", 'arxivid': '2305.05900', 'author': ['Chengkun Wei \nZhejiang University\n\n', 'Minghu Zhao \nZhejiang University\n\n', 'Zhikun Zhang \nStanford University\n\n', 'Min Chen \nCISPA Helmholtz Center for Information Security 4 DBAPPSecurity\n\n', 'Wenlong Meng \nZhejiang University\n\n', 'Bo Liu ', 'Yuan Fan \nZhejiang University\n\n', 'Wenzhi Chen \nZhejiang University\n\n'], 'authoraffiliation': ['Zhejiang University\n', 'Zhejiang University\n', 'Stanford University\n', 'CISPA Helmholtz Center for Information Security 4 DBAPPSecurity\n', 'Zhejiang University\n', 'Zhejiang University\n', 'Zhejiang University\n'], 'corpusid': 258588373, 'doi': '10.48550/arxiv.2305.05900', 'github_urls': [], 'n_tokens_mistral': 50538, 'n_tokens_neox': 37896, 'n_words': 19547, 'pdfsha': 'a19834ebe67c36829b42ca301f6a343bedafd2f5', 'pdfurls': ['https://export.arxiv.org/pdf/2305.05900v1.pdf'], 'title': ['DPMLBench: Holistic Evaluation of Differentially Private Machine Learning', 'DPMLBench: Holistic Evaluation of Differentially Private Machine Learning'], 'venue': ['the 30th ACM SIGSAC Conference on Computer and Communications Security']} |
arxiv |
THE MATRIX MODEL FOR DESSINS D'ENFANTS
18 Jun 2014
Jan Ambjørn
Leonid Chekhov
THE MATRIX MODEL FOR DESSINS D'ENFANTS
18 Jun 2014arXiv:1404.4240v2 [math.AG]Belyi functiontopological recursiontau functionMiwa transform AMS classification: 05A1514H7015B52
We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over 0, 1, and ∞ (Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by one of the authors (L.Ch.) and K.Palamarchuk. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev-Petviashvili (KP) hierarchy τ -function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich-Penner model introduced by the authors and Yu. Makeenko. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times.
Introduction
In general, Hurwitz numbers pertain to combinatorial classes of ramified mappings f : CP 1 → Σ g of the complex projective line onto a Riemann surface of genus g. Commonly, single and double Hurwitz numbers correspond to the cases in which ramification profiles (defined by the corresponding Young tableauxes λ or λ and µ) are respectively given at one (∞) or two (∞ and 1) distinct points whereas we assume the existence of m other distinct ramification points with only simple ramifications.
Generating functions for Hurwitz numbers have been considered for long in mathematical physics. Notably, Okounkov and Pandharipande [29] showed that the exponential of the generating function for double Hurwitz numbers is a tau-function of the Kadomtsev-Petviashvili (KP) hierarchy. The same result was obtained by A. Yu. Orlov and Shcherbin [30], [31] using the Schur function technique and, in a more general setting, by Goulden and Jackson [21] using Plucker relations.
Orlov and Shcherbin [30] also addressed the case of the generating function for the case of Grothendieck dessins d'enfants where we have only three ramification points with multiple ramifications and the ramification profile is fixed at one or two of these points. In this case, they also obtained that the exponentials of the corresponding generating functions are the tau functions of the KP hierarchy.
On the other hand, Hurwitz numbers manifest properties intrinsic for conformal theories including sets of Virasoro constraints and closely related loop equations. That simple Hurwitz numbers satisfy the topological recursion-the technique originated in matrix models-was conjectures in [7] and proved in [8]. [26]. In a nice recent paper [32] Zograf provided recursion relations for the generating function of Grothendieck's dessins d'enfants enumerating the Belyi pairs (C, f ), where C is a smooth algebraic curve and f a meromorphic function f : C → CP 1 ramified only over the points 0, 1, ∞ ∈ CP 1 .
We recall some mathematical results relating Belyi pairs to Galois groups and begin with Theorem 1.1. (Belyi, [6]) A smooth complex algebraic curve C is defined over the field of algebraic numbers Q if and only if it exists a nonconstant meromorphic function f on C (f : C → CP 1 ) ramified only over the points 0, 1, ∞ ∈ CP 1 .
For a Belyi pair (C, f ) let g be the genus of C and d the degree of f . If we take the inverse image f −1 ([0, 1]) ⊂ C of the real line segment [0, 1] ∈ CP 1 we obtain a connected bipartite fat graph with d edges with vertices being preimages of 0 and 1 and with the cyclic ordering of edges entering a vertex coming from the orientation of the curve C. This led Grothendieck to formulating the following lemma: Lemma 1.2. (Grothendieck, [22]) There is a one-to-one correspondence between the isomorphism classes of Belyi pairs and connected bipartite fat graphs.
We define a Grothendieck dessin d'enfant to be a connected bipartite fat graph representing a Belyi pair.
It is well known that we can naturally extend the dessin f −1 ([0, 1]) ⊂ C corresponding to a Belyi pair (C, f ) to a bipartite triangulation of the curve C. For this, we cut the complex plane along the (real) line containing 0, 1, ∞ coloring upper half plane white and lower half plane gray. This defines the partition of C into white and grey triangles such that white triangles has common edges only with grey triangles. We then consider a dual graph in which edges are of three types (pre-images of the three edges shown in Fig. 1): the type of an edge depend on
which segment-f −1 ([0, 1]) ⊂ C, f −1 ([1, ∞ + ]) ⊂ C, or f −1 ([∞ − , 0]) ⊂ C-it intersects (∞ ±
indicate the directions of approaching the point of infinity along the real axis in CP 1 ). Each face of the dual partition then contains a preimage of exactly one of the points 0, 1, ∞, so they are of three sorts (bordered by solid, dotted, or dashed lines in the figure). We call such a graph a Belyi fat graph.
The type of ramification at infinity is determined by the set of solid-line bounded faces of a Belyi fat graph: the order of branching is r for a 2r-gon, so we introduce the generating function that distinguishes between different types of branching at infinity. We let n 1 , n 2 , n 3 denote the numbers of respective solid-, dotted-, and dashed-line cycles (faces) and let m r denote the number of solid-line cycles of length 2r in a Belyi fat graph We are interested in the following counting problem: we are going to calculate the generating function
(1.1) F {t m }, β, γ; N = Γ 1 |Aut Γ| N 2−2g β n 2 γ n 3 n 1 i=1 t r i ,
where N, β, γ, and t r are formal independent parameters and the sum ranges all (connected) Belyi fat graphs. Often a factor α n 1 is also added; it can however be adsorbed into the times t r by scaling t r → αt r for all r. The structure of the paper is as follows. In Sec. 2, we show that generating function (1.1) is the free energy of a special matrix model. We demonstrate that this model is the twologarithm matrix model of [17], and it therefore belongs to the class of generalized Kontsevich models (GKM) [24]. In Sec. 3, we present the solution of this model from paper [17] in which it was reduced, upon a special transformation of times, to a Hermitian one-matrix model with a general potential. In Sec. 4, we present the direct solution of the original generating function in terms of the Hermitian one-matrix model without appealing to the external field model thus again establishing the equivalence between the two models and describing the corresponding topological recursion. In Sec. 5, we construct the matrix model for clean Belyi morphisms (those having ramifications only of type (2, 2, . . . , 2) over 1) and show that the corresponding generating function is the original Kontsevich-Penner model of [15]. This model is also equivalent [16] to the Hermitian one-matrix model with a general potential and to the BGW model of [27]. Finally, in Sec. 6, we combine the techniques of Secs. 2, 3, and 4 establishing that the generating function for the two-profile Belyi morphisms (with the given ramifications at two points, ∞ and 1) is again given by the GKM integral thus being a tau function of the KP hierarchy (that is, it satisfies the bilinear Hirota relations). We conclude with the discussion of our results.
Throughout the entire text we disregard all multipliers not depending on external fields; all equalities in the paper must therefore be understood modulo such factors.
The model
In our conventions the indices i, i 1 , i 2 , etc. take positive integer values between 1 and αN, the indices j, j 1 , etc. take positive integer values between 1 and βN, and the indices k, k 1 , etc. take positive integer values between 1 and γN. We introduce three complex-valued rectangular matrices R k,i , G i,j , and B j,k and one diagonal matrix (the external field) Λ i 1 ,i 2 = λ i 1 δ i 1 ,i 2 . The action is given by the integral in which we can perform the Gaussian integration w.r.t. G, G thus obtaining
DR DR := γN k=1 αN i=1 dRe R k,i dIm R k,i .(2.5) DR DR e −N tr(RR) det δ i 1 ,i 2 − (Λ RRΛ) i 1 ,i 2 −βN .
After the change of variables R → RΛ this integral becomes
(2.6) αN i=1 |λ i | −2γN DR DR e −N tr RR[ΛΛ] −1 det δ i 1 ,i 2 − (RR) i 1 ,i 2 −βN .
For definiteness, let γ ≥ α. A general rectangular matrix R can then be reduced to the form
R = U † M V , where U ∈ U(αN), V ∈ U(γN)/U((γ − α(|m i 2 | 2 − |m i 1 | 2 ) 2 αN i=1 |m i | 2(γ−α)N ,
Introducing the new variables x i = |m i | 2 ranging from zero to infinity, we reduce the integral in (2.6) to the αN-fold integral w.r.t. x i and to the integration w.r.t. the unitary group:
αN i=1 |λ i | −2γN ∞ 0 dx 1 . . . dx αN DUe −N i 1 ,i 2 x i 1 U i 1 ,i 2 |λ i 2 | −2 U † i 2 ,i 1 × × ∆(x) 2 αN i=1 x (γ−α)N i (1 − x i ) −βN . (2.10)
The integral over DU is given by the Itzykson-Zuber-Mehta formula (we write it having in mind that we subsequently integrate it over variables x i with a totally symmetric measure),
DU e −N i 1 ,i 2 x i 1 U i 1 ,i 2 |λ i 2 | −2 U † i 2 ,i 1 = e −N i x i |λ i | −2 ∆(x i )∆(|λ i | −2 ) ,
so the final formula for the generating function reads
(2.11) αN i=1 |λ i | −2γN ∆(|λ| −2 ) ∞ 0 dx 1 . . . dx αN ∆(x)e N i [−x i |λ i | −2 +(γ−α) log x i −β log(1−x i )] .
The integral (2.11) is equivalent to the matrix-model integral
(2.12) αN i=1 |λ i | −2γN αN ×αN DH ≥0 e N tr[−HΛ −2 +(γ−α) log H−β log(1−H)] ,
where the integration goes over Hermitian (αN × αN)-matrices with positive eigenvalues. We thus obtain the following statement.
Lemma 2.1. The generating function for Grothendieck dessins d'enfants (Belyi fat graphs (1.1)) is the matrix-model integral (2.12).
The integral (2.12) belongs to the class of generalized Kontsevich models (GKM) [24]; in terms of variables ξ i = 1/|λ i | 2 it can be calculated as the ratio of determinants of (αN ×αN)-matrices,
∂ i 1 −1 f (ξ i 2 ) ∂ξ i 1 −1 i 2 /∆(ξ), where f (ξ) = ∞ 0 dxe −N xξ x (γ−α)N (1 − x) −βN ,
and as such is a tau-function of the Kadomtsev-Petviashvili (KP) hierarchy in times t n =
i ξ −n i = i |λ i | 2n (cf. (2.
3)) i.e., we come to the following theorem proved by Zograf [32] by purely combinatorial means with the using of the cut-and-joint operator. The integral (2.12) was studied by one of the authors and Palamarchuk [17] in relation to exploring possible explicit solutions of matrix models with external fields. It was called the two-logarithm model there and it was proved that this integral admits Virasoro constraints that, upon a proper change of times, become the Virasoro constraints of the matrix model introduced in [15] (the term Kontsevich-Penner model was coined there), which, in turn, is equivalent [16] to a Hermitian one-matrix model with the potential related to the external-field variables ξ i via the Miwa transformation. As such, this integral must also satisfy the equations of the Toda chain hierarchy.
Remark 2.
3. An important remark concerning integral (2.12) is that its asymptotic behavior as N → ∞ is different depending on whether γ − α ≃ O(1) or γ − α ≃ O(1/N). In the first case, we have an infinite repulsive potential at the origin and an eigenvalue distribution is confined within an interval [x ′ − , x ′ + ] (see below) with 0 < x ′ − < x ′ + . The 1/N-expansion then is "insensitive" to the hard edge at the origin, and we can assume that we integrate over the whole real axis (the difference between the restricted and nonrestricted integrations is then
exponentially small in N). If γ = α or γ − α ∼ O(1/N), representation (2.12) still remains valid, but in this case the eigenvalue support is [0, x ′ + ]
, so it reaches the hard edge x = 0 at the origin. We then again have a topological expansion (about 1/N-expansion in matrix models with hard edges, see, e.g., review [11]) but with the differential ydx finite at
x = 0 (y ∼ 1/ √ x as x → 0 and y ∼ x − x ′ + as x → x ′ + )
. The asymptotic expansions of integral (2.12) are therefore different in the corresponding regimes and do not admit an analytical transition as γ → α.
Remark 2.4. In Sec. 4, we present a simpler, straightforward way of proving that generating function (1.1) for general Belyi morphisms is indeed a Hermitian one-matrix model free energy. However, the external field technique of this and next sections will be instrumental when proving a general correspondence between the generating functions for clean (Sec. 5) and two-profile (Sec. 6) Belyi morphisms and free energies of the corresponding generalized Kontsevich models.
The two-logarithm matrix model
In this section, we present the results of [17] adapted to the notation of integral (2.12).
3.1. Constraint equations for integral (2.12). We first perform the variable change
(3.1)Ñ = αN,Λ = Λ −2 /(2α),H = 2H − 1 α = β/α,β = 1 − γ/α.
in (2.12). Disregarding here and hereafter factors not depending on λ's, the integral then takes the form
(3.2)Ñ i=1 |λ i | γN e −Ñ |λ i | Ñ ×Ñ DH ≥0 e −Ñ tr[HΛ+α log(1−H)+β log(1+H)] :=Ñ i=1 |λ i | γN e −Ñ |λ i | Z[λ],
where we let Z[λ] denote the integral (2.12) without the normalization factor.
The Schwinger-Dyson equations for the integral (3.2) follow from the identity (here all the indices range from 1 to αN)
(3.3) 1 N 3 ∂ ∂Λ jk ∂ ∂Λ li − 1 N Ñ ×Ñ DH ∂ ∂H ij e −Ñ tr[HΛ+α log(1−H)+β log(1+H)] = 0.
In terms of the eigenvaluesλ i of the matrixΛ, the correspondingÑ equations read (3.4)
− 1 N 2λ i 1 ∂ 2 ∂λ 2 i 1 − 1 N 2 i 2 =i 1λ i 2 λ i 2 −λ i 1 ∂ ∂λ i 2 − ∂ ∂λ i 1 +α +β − 2 N ∂ ∂λ i 1 +β −α +λ i 1 Z[λ] = 0,
We can equivalently write the constraint equations (3.4) in terms of the times
(3.5) t n = 1 n i 1 λ n i , n ≥ 1.
They then becomes the set of Virasoro constraints 1
(3.6) V k Z {t n } = 0, k ≥ 0, where V k [t] := − ∞ m=1 mt m ∂ ∂t m+k − k m=1 ∂ ∂t m ∂ ∂t k−m −Ñ(α −β + 1)(1 − δ k,0 − δ k,−1 ) ∂ ∂t k + 2Ñ(1 − δ k,−1 ) + δ k,−1 t 1 ∂ ∂t k+1 +Ñ 2α (β − 1)δ k,0 , k = −1, 0, 1, . . . . (3.7)
(Here, for the future use, we have also introduced the operator V −1 .)
The operators V k enjoy the Virasoro algebra
(3.8) [V k , V l ] = (l − k)V k+l , k, l ≥ −1.
3.2.
Equivalence to the Hermitian one-matrix model. In [17] it was shown that the two-logarithm model is equivalent to the Kontsevich-Penner model [15], which in turn was known [16], [24] to be equivalent to a Hermitian one-matrix model. In this paper, we skip the intermediate step and demonstrate the equivalence between (2.12) and a Hermitian one-matrix model defined as an integral
(3.9) Z 1MM {ξ m }, M := M ×M DY e −V (Y ) , V (Y ) = ∞ m=1 ξ m tr Y m .
It is well-known that this integral satisfies the set of Virasoro constraints uniformly written in the form (3.10)
L n Z 1MM {ξ m }, M = n m=0 ∂ 2 ∂ξ m ∂ξ n−m + ∞ m=1 mξ m ∂ ∂ξ n+m Z 1MM {ξ m }, M = 0, n ≥ −1,
where we have used a convenient notation
∂ ∂ξ 0 Z 1MM {ξ m }, M = −MZ 1MM {ξ m }, M .
In order to establish the correspondence it is necessary to shift the original variableλ,
(3.11) µ i =λ i − ρ, ρ ∈ C,
introducing an auxiliary parameter ρ. We also introduce the new times
(3.12) τ n := 1 nÑ i=1 1 µ n i , n ≥ 1,
and the new normalizing factor
(3.13) N [µ] :=Ñ i=1 µÑ (β−1) i eÑ µ i
The following set of constraints was found in [17]:
Lemma 3.1. (see [17]) The normalized integral Z[λ]/N [µ] whereλ i = µ i + ρ satisfies the set of Virasoro constraints L k Z[λ]/N [µ] = 0, k = −1, 0, 1, . . . ,
in times (3.12) with
L k = − ∞ m=1+δ k,−1 m(τ m − 2Ñδ m,1 ) ∂ ∂τ m+k − k−1 m=1 ∂ 2 ∂τ m ∂τ k−m + 2Ñα KP (1 − δ k,0 − δ k,1 ) ∂ ∂τ k −2ϕÑ m=1+δ k,−1 1 (−ρ) m ∂ ∂τ k+m − (Ñ α KP ) 2 δ k,0 +Ñ α KP τ 1 − 2Ñ − 2ϕÑ ρ δ k,−1 ,(3.
14)
where α KP =β − 1 and ϕ = −(α +β − 1)/2. Remark 3.2. In order to derive constraints (3.14) the following trick was used in [17]: constraint equations (3.4) after shift (3.11) were written in the form
∞ k=1 µ −k i L k Z[λ] = 0, where L k = V k+1 [τ ] + ρV k [τ ] + ρÑ (α +β − 1) (1 − δ k,0 − δ k,−1 ) ∂ ∂τ k − (β − 1)Ñδ k,0 +ρ(β − 1)Ñ(τ 1 − 2Ñ )δ k,−1 , k ≥ −1,
were differential operators in (shifted) times τ s and where we let V s [τ ] denote operators (3.7) upon the substitution t → τ . The "proper" Virasoro operators L k (3.14) were finally obtained upon the upper-triangular transformation
L k = ∞ s=0 (−1) s ρ s+1 L k+s , k ≥ −1.
We see that in order to perform all these replacements we have to keep ρ nonzero and finite.
Lemma 3.3. (see [17]) Upon the substitution
(3.15) ξ n = τ n + 1 n 2ϕÑ (−ρ) n − 2Ñδ n,1 , M =Ñα KP
the Virasoro constraints (3.14) become the Virasoro constraints (3.10) of the Hermitian onematrix model. Because these conditions determine the corresponding integrals unambiguously, these two models are equivalent.
In terms of the original variables, we have the following lemma.
= αN i=1 1 2α − ρ|λ i | 2 −γN e αN 1 2α|λ i | 2 −ρ × ×Z 1MM ξ m = τ m + 1 m (γ − β)N (−ρ) m − 2αNδ n,1 , M = −γN (3.16) with τ m = 1 m αN j=1 1 µ m j where µ i + ρ = 1/(2α|λ i | 2 ). Here Z 1MM {ξ m }, M is matrix integral (3.9).
In the next section we demonstrate that this statement enables us to write explicit formulas for terms of the genus expansion of F provided we know the answer for the free energy of matrix model (3.9) either in terms of momentums [3] or in terms of the topological recursion technique of [20], [13], [14], [1].
Remark 3.5. The shift of variables (3.11) is a convenient technical tool that was used in [17] for passing to the full half-Virasoro constraint algebra that includes also the operator L −1 . If |γ − α| O(1/N) we have a hard edge at the origin, which is specific for the complex matrix model of [5] or the BGW model of [27], and we shall lose the L −1 Virasoro operator. 2 We reconstruct the L −1 -operator in the model with logarithmic potential for the price of unfreezing all times of the hierarchy. And, as we demonstrate in the next section, the final answers for genus expansion terms do not depend on the auxiliary parameter ρ.
3.3. The genus expansion. An extensive literature is devoted to solving the one-matrix model (3.9) in the topological (genus) expansion; its free energy F admits a representation F = ∞ h=0 M 2−2h F h , which can be interpreted as a semiclassical expansion of a (quasi)stationary statistical theory. As such, in the large-M limit, we observe a stationary distribution of eigenvalues described by a spectral curve of the model. In the present paper, as in [17], we assume that this stationary distribution spans a single interval, and we therefore have a one-cut solution based on a spectral curve that is just a double cover of the complex plane with two branching points, x + and x − (a sphere). These two points are determined by the constraint equations for the so-called master loop equation [25] (3.17)
C D dw 2πi V ′ (w) (w − x + )(w − x − ) = 0, C D dw 2πi wV ′ (w) (w − x + )(w − x − ) = 2M,
where the integration contour encircles the eigenvalue domain (the interval [x − , x + ] in this case) and not other singularities (including possible singularities of V ′ (w)). After the Miwa time transformation (3.15) we obtain for V ′ (w) the expression
(3.18) V ′ (w) = −2αN − αN i=1 1 w − µ i − (γ − β)N 1 w + ρ
and we assume that all µ i and −ρ are situated outside the integration contour. We can then take the integrals in (3.17) by residues at µ i , −ρ, and infinity. For the first equation we obtain
−2αN + αN i=1 1 (µ i − x + )(µ i − x − ) + (γ − β)N 1 (p + x + )(p + x − ) = 0
and shifting the branching points
x + + ρ = x ′ + , x − + ρ = x ′ −
and recalling that µ i + ρ =λ i we obtain the constraint equation solely in terms ofλ i :
(3.19) − 2αN + αN i=1 1 (λ i − x ′ + )(λ i − x ′ − ) + (γ − β)N 1 x ′ + x ′ − = 0
For the second constraint equation we obtain
−αN(x ′ + + x ′ − − 2ρ) + αN i=1λ i − ρ (λ i − x ′ + )(λ i − x ′ − ) − αN − (γ − β)N + (γ − β)N −ρ x ′ + x ′ − = −2γN
and the term linear in ρ is just the first constraint equation and thus vanishes. So, the second constraint equation becomes
(3.20) (γ + β − α)N − αN(x ′ + + x ′ − ) + αN i=1λ i (λ i − x ′ + )(λ i − x ′ − ) = 0.
We see that, as expected, all the dependence on ρ disappears from constraint equations (3. 19) and (3.20).
Remark 3.6. Equations (3.19) and (3.20) exactly coincide with the respective first and second constraint equations in Eq. (2.14) of [17] upon the substitution
(3.21) λ →λ, N → αN β − α → 1 − γ/α − β/α, c → (β − γ) 2 /4α 2 , b/a → −x ′ + − x ′ − , c/a → x ′ − x ′ + .
The answer for F 0 (formula (2.16) in [17]) obtained from these constraint equations therefore coincides (up to the normalization factor αN i=1 |λ i | γN e −αN |λ i | ) with the genus zero contribution to generating function (1.1).
3.3.1.
Genus-zero term. It follows from Remark 3.6 that the genus-zero term F 0 of our generating function (1.1) upon the substitutions (3.21) and (3.1) coincides with F 0 found in [17] with the added normalization term αN i=1 γN logλ i − αNλ i . In terms of variables x ′ ± ,λ the corresponding expression reads
F 0 = 1 4 (β 2 N 2 + γ 2 N 2 ) log (x ′ + − x ′ − ) 2 +N 2 (α − β − γ) |β − γ| log x ′ + + x ′ − − 2 x ′ + x ′ − x ′ + + x ′ − + 2 x ′ + x ′ − + x ′ + + x ′ − 2 +N 2 α 2 8 (x ′ + + x ′ − ) 2 + α|β − γ| x ′ + x ′ − − (β − γ) 2 4 log[x ′ + x ′ − ] +N αN i=1 β + γ 2 log |λ i | + g(λ i ) −λ i + α − β − γ 2 log λ i − x ′ + + x ′ − 2 + g(λ i ) − |β − γ| 4 log g(λ i ) −λ i (x ′ + +x ′ − ) 2 √ x ′ + x ′ − + x ′ + x ′ − g(λ i ) +λ i (x ′ + +x ′ − ) 2 √ x ′ + x ′ − − x ′ + x ′ − − 1 4 αN i 1 ,i 2 =1 log g(λ i 1 )g(λ i 2 ) +λ i 1λ i 2 −λ i 1 +λ i 2 2 (x ′ + + x ′ − ) + x ′ + x ′ − (3.22)
where we have introduced the notation g(λ i ) :
= (λ i − x ′ + )(λ i − x ′ − ).M r = C D dw 2πi V ′ (w) (w − x + ) r+1/2 (w − x − ) 1/2 , J r = C D dw 2πi V ′ (w) (w − x + ) 1/2 (w − x − ) r+1/2 , r ≥ 1.
Using representation (3.18), we obtain for the moments the following expressions (3.24)
M r = αN i=1 1 (λ i − x ′ + ) r+1/2 (λ i − x ′ − ) 1/2 + (γ − β)N (−1) r (x ′ + ) r+1/2 (x ′ − ) 1/2 J r = αN i=1 1 (λ i − x ′ + ) 1/2 (λ i − x ′ − ) r+1/2 + (γ − β)N (−1) r (x ′ + ) 1/2 (x ′ − ) r+1/2 r ≥ 1.r 1 . . . r m ; q 1 . . . q l |r q p h M r 1 · · · M rm J q 1 · · · J q l M r 1 J q 1 |x ′ + − x ′ − | p , h > 1,
and [4] (3.26)
F 1 = − 1 24 log M 1 J 1 |x ′ + − x ′ − | 4 .
Here r 1 . . . r m ; q 1 . . . q l |r q p h are finite (for a fixed h) sets of rational numbers given by the topological recursion technique for the standard Hermitian one-matrix model (see [13]). They are subject to restrictions:
m + l − r − q = 2 − 2h, m s=1 (r 1 − 1) + l s=1 (q s − 1) + p = 4h − 4, p ≥ h − 1.
Using topological recursion we can effectively calculate the numbers r 1 . . . r m ; q 1 . . . q l |r q p h . The quantity |x ′ + − x ′ − |, which is often denoted by d, is the length of the interval of eigenvalue support. Formulas (3.25), (3.26), and (3.24) thus describe generating function (1.1) in all orders of the genus expansion.
Spectral curve and topological recursion
In this section, we directly derive the spectral curve without appealing to a matrix model with external fields. For this, we shrink all solid-line cycles assigning just the original times t r to the obtained 2r-valent vertices of the field B, B. The generating function (1.1) is then described by the matrix-model integral over rectangular (γN × βN)-matrices B: which, using the Jacobian from Appendix A under assumption that β > γ, can be reduced to the γN-fold integral over positive x k :
(4.2) Z[t] = ∞ 0 dx 1 . . . dx γN [∆(x)] 2 γN k=1 x (β−γ)N k e −N ∞ r=1 γN k=1 1 r (δ r,1 −tr)x r k .
This integral is again a Hermitian one-matrix model with a logarithmic term in the potential:
(4.3) Z[t] = γN ×γN DX ≥0 e −N tr ∞ r=1 1 r (δ r,1 −tr)X r −(β−γ) log X ,
We have thus obtained another representation of generating function (1.1). Because we have reduced the original problem to a mere Hermitian one-matrix model integral, we can directly apply a standard topological recursion procedure [13] (see [10] where it was generalized to the case of rational functions V ′ (x)). We let
(4.4) U ′ (x) := N ∞ r=1 (δ r,1 − t r )x r−1
denote the polynomial part of the potential with times t r with the shifted first time. The hyperelliptic spectral curve is a sphere with two branching points x ′ + and x ′ − whose positions are determined by the standard constraints (3.17) in which
(4.5) V ′ (x) = U ′ (x) − N(β − γ) x , M = γN.
Constraints (3.17) then become
(4.6) C D dw 2πi U ′ (w) (w − x ′ + )(w − x ′ − ) = N(β − γ) x ′ + x ′ − , C D dw 2πi wU ′ (w) (w − x ′ + )(w − x ′ − ) = N(β + γ),
i.e., precisely constraints (3.19) and (3.20) after the inverse Miwa transformation. 3 The y-variable of the topological recursion is given by the integral over the contour that encircles the eigenvalue support and the point x,
(4.7) y(x) := C [x ′ − ,x ′ + ] ∪{x} dw 2πi V ′ (w) (x − x ′ + )(x − x ′ − ) (w − x) (w − x ′ + )(w − x ′ − )
, which can be evaluated by residues at infinity and at w = 0 (due to the presence of a pole term in V ′ (w)) The result reads
(4.8) y(x) = res ∞ U ′ (w) (w − x) (w − x ′ + )(w − x ′ − ) + N(β − γ) x ′ + x ′ − (x − x ′ + )(x − x ′ − )
The genus expansion for h ≥ 1 has the same form as in Lemma (3.7) with the moments given by the standard integrals taken by residues at infinity and at w = 0:
(4.9) M r = res w=∞ U ′ (w) (w − x ′ + ) r+1/2 (w − x ′ − ) 1/2 + (γ − β)N (−1) r (x ′ + ) r+1/2 (x ′ − ) 1/2 J r = res w=∞ U ′ (w) (w − x ′ + ) 1/2 (w − x ′ − ) r+1/2 + (γ − β)N (−1) r (x ′ + ) 1/2 (x ′ − ) r+1/2 r ≥ 1.
The term F 0 has the general form [9] (for the number of eigenvalues equal t 0 N)
(4.10) F 0 = − 1 2 C [x ′ − ,x ′ + ] y(x)V (x) − ζt 0 ,
where ζ is the Lagrange multiplier most conveniently obtained as the limit of the integral
(4.11) ζ = lim Λ→+∞ Λ x ′ + y(x)dx − V (Λ) − t 0 log Λ .
Generating functional for clean Belyi morphisms
5.1. The model. A clean Belyi morphism is a special class of Belyi pairs (C, f ) that have profile (2, 2, . . . , 2, 1, 1, . . . , 1) over the branch point 1 ∈ CP 1 . This means that all dotted cycles (in Fig. 1) have either lengths 2 (no ramification) or 4 (simple ramification). In [19] the authors demonstrated that the generating function for ramifications of sort (2, 2, . . . , 2) satisfies the topological recursion relations with the spectral curve (x = z + z −1 ; y = z).
In this section, we demonstrate that the matrix model corresponding to clean Belyi morphisms is just the Kontsevich-Penner model [15], which is in turn equivalent [16] to the Hermitian one-matrix model with a general potential.
We thus have to calculate the generating function (1.1) in which the sum ranges over only clean Belyi morphisms. In terms of the diagrammatic technique of Sec. 2 this means that we count only dotted cycles of lengths 2 and 4. Counting cycles of length 2 reduces to a mere changing of the normalization of the R R -propagators:
Λ Λ β Λ Λ β Λ Λ β + + + · · · R R R R R R
so that the propagator becomes
R R ∼ 1 N δ i 1 ,i 2 δ k 1 ,k 2 1 − β|λ i 1 | 2
and the corresponding quadratic form gets an external field addition:
(5.1) − N tr[RR(1 − β|Λ| 2 )].
The new interaction vertex arises from the dotted cycles of length four:
Λ Λ Λ Λ β ∼ 1 2 Nβ tr[RRΛΛ RRΛΛ]
where the factor 1/2 takes into account the symmetry of the four-cycle. We therefore have that the generating function F is the logarithm of the integral
(5.2) DR DR e N tr[−RR(1−β|Λ| 2 )+ 1 2 βRR|Λ| 2 RR|Λ| 2 ] ,
where we integrate over rectangular complex (γN × αN)-matrices R. We first rescale the integration variable R → RΛ, which results in the integral
(5.3) αN i=1 |λ i | −2γN DR DR e N tr[−RR(|Λ| −2 −β)+ 1 2 βRRRR] .
Performing now the same chain of transformations as in Sec. 2, we obtain eventually that integral (5.3) is equivalent to the Hermitian one-matrix model integral 4). This matrix-model integral is the (original) Kontsevich-Penner matrix model [15], [16].
(5.4) αN i=1 |λ i | −2γN
Remark 5.2. If we demand the ramification profile at the point 1 to be just (2, 2, . . . , 2) (no dotted two-cycles are allowed), then in order to obtain the corresponding generating function we must merely replace Λ −2 − β by Λ −2 in (5.4).
From now on, for simplicity, we restrict ourselves to the case of ramification profile (2, 2, . . . , 2) at the point 1.
5.2.
Solving integral (5.4). That the Kontsevich-Penner matrix model integral (5.4) is equivalent to the Hermitian one-matrix model integral (3.9) is well known. This equivalence was established using the Virasoro constraints in [16] or using explicit determinant relations in [24]. We recall here the logic of [24].
We begin with the standard eigenvalue representation for integral (3.9),
(5.5) dy 1 . . . dy M [∆(y)] 2 e − ∞ k=1 M i=1 ξ k y k i
in which we again perform the Miwa change of variables with the Gaussian shift,
(5.6) ξ k = 1 k N j=1 1 µ k j + 1 2 δ k,2 .
Summing up the terms in the exponential into logarithms, we transform integral (5.5) to the form
dy 1 . . . dy M [∆(y)] 2 M i=1 N j=1 (µ j − y i ) N j=1 µ −M j e − 1 2 M i=1 h 2 i .
We now use that ∆(y) M
H M (µ 1 ) H M (µ 2 ) . . . H M (µ N ) H M +1 (µ 1 ) H M +1 (µ 2 ) . . . H M +1 (µ N ) . . . . . . . . . . . . H M +N −1 (µ 1 ) H M +N −1 (µ 2 ) . . . H M +N −1 (µ N )
On the other hand, we obtain the same ratio of determinants multiplied by e − 1 2 j µ 2 j if we consider the N-fold integral
(5.8) dx 1 . . . dx N ∆(x) ∆(µ) N j=1 x M j e N j=1 (x j µ j + 1 2 x 2 j )
because dx x s e xµ+ 1 2 x 2 = e − 1 2 µ 2 H s (µ). Expression (5.8) is nothing but the Kontsevich-Penner integral, so we obtain the relation between two matrix integrals of different sizes:
(5.9) N ×N DX e tr[Xµ+ 1 2 X 2 +M log X] = M j=1 µ M j e − 1 2 µ 2 j M ×M DY e − ∞ k=1 ξ k tr Y k , ξ k = 1 k N j=1 1 µ k j + 1 2 δ k,2 .
After a simple algebra, we come to the following lemma.
(5.10) Z[t; γ, β] = αN i=1 |λ i | −2γN M ×M DY e − ∞ k=1 t k k (−1) k tr Y k − N 2β tr Y 2 , t k = αN i=1 λ 2k i , M = (γ − α)N.
Because this integral is also equivalent to Kontsevich-Penner matrix model (5.4) (with the external field term Λ −2 instead of Λ −2 − β), it also belongs to the GKM class thus being a tau function of the KP hierarchy.
Remark 5.4. Note again that the above correspondence is valid only in the 1/N asymptotic expansion and only when γ − α ≃ O(1). If γ − α O(1/N) the above correspondence fails because in this case we must take into account that we integrate in formula (5.4) over positive definite matrices, contrary to formula (5.9) in which no restriction on integration domain is assumed. So, again, the case γ = α is special and must be treated separately.
A general case of two-profile Belyi morphisms
Combining the techniques of Secs. 2 and 4 we now address the most general case of Belyi morphisms with the given profiles at two branching points: infinity and 1. We take these profiles into account in two different ways: at infinity we, as in Sec. 4, introduce the times t m responsible for the profile whereas the times at 1 will be taken into account by introducing, as in Sec. 2, the external field Λ with where the times t s are given by (6.1).
Performing the same operation as in (4.1)-(4.3), we obtain that integral (6.3) is equal to the integral over Hermitian positive definite (γN × γN)-matrix X with the external matrix field Λ = |Λ| −2 :
(6.4) Z[t, τ ] = γN k=1 |λ k | −2βN γN ×γN DX ≥0 e N tr −X|Λ| −2 + ∞ m=1 tm m X m +(β−γ) log X ,
Integral (6.4) is again a GKM integral [24]; after integration over eigenvalues x k of the matrix X it takes the form of the ratio of two determinants,
(6.5) Z[t, τ ] = γN k=1 |λ k | −2βN ∂ k 1 −1 ∂λ k 1 −1 k 2 f (λ k 2 ) γN k 1 ,k 2 =1 ∆(λ) , where (6.6) f (λ) = ∞ 0 x N (β−γ) e −N xλ+N ∞ m=1 tm m x m .
Because any GKM integral (in the proper normalization) is a τ -function of the KP hierarchy, and for a model with the logarithmic term in the potential it was demonstrated in [27], we immediately obtain the following theorem.
Theorem 6.2. The exponential e F [{t},{t},γ;N ] of generating function (6.2) modulo the normalization factor γN k=1 |λ k | −2βN is a τ -function of the KP hierarchy (that is, it satisfies the bilinear Hirota relations) in times t s given by (6.1).
Conclusion
We have proved that generating functions for numbers of three different types of Belyi morphisms are free energies of special matrix models all of which are in the GKM class thus being tau functions of the KP hierarchy. Besides this, it is interesting to establish other relations between, say, generating function (1.1) for clean Belyi morphisms and the free energy of the Kontsevich-Penner matrix model, which is known (see [12], [28], [18]) to be related to the numbers of integer points in moduli spaces M g,n of curves of genus g with n holes with fixed (integer) perimeters; the very same model is also related [12] by a canonical transformation to two copies of the Kontsevich matrix model expressed in times related to the discretization of the moduli spaces M g,n . It is tempting to find possible relations between these discretizations, cut-and-join operators of [32], and Hodge integrals of [23].
Of course, the possibility of using GKM techniques when studying enumeration problems for Belyi morphisms deserves more detailed studies; we consider this note a first step in exploring this perspective field of knowledge.
It is also interesting to clarify the role of cut-and-join operators of [23] and [32] in the matrixmodel context. After this text was completed, an interesting paper [2] extending the formalism of cut-and-join operators to the case of generalized Hurwitz numbers has appeared.
Appendix A Deriving the Jacobian of transformation (2.8)
The invariant measure DU DV in the vicinity of the unity becomes DH DH DP DP . For dR i,k we then obtain (A.1) dR i,k = dm i δ i,k + idH i,k m k + im i dH i,k , k ≤ αN m i dP i,k−αN , k > αN .
The elements dm i appear only for i = k with the unit factor, so we have to calculate only "non-diagonal" differentials DR DR. For i < k ≤ αN we have:
(A.2) dR i,k = idH i,k m k + im i dH i,k , dR k,i = idH * i,k m i + im k dH * i,k , dR k,i = −idH i,k m i − im k dH i,k , dR i,k = −idH * i,k m k − im i dH * i,k .
Combining the columns in these relations, we obtain
(A.3) dR i,k ∧ dR k,i = dH i,k ∧ dH i,k [m k m k − m i m i ], dR k,i ∧ dR i,k = dH * i,k ∧ dH * i,k [m i m i − m k m k ], 1 ≤ i < k ≤ αN,
and we obtain that
(A.4) αN ∧ i,k=1 dR i,k ∧ dR k,i = DH ∧ DH ∧ αN i=1 dm i ∧ dm i 1≤i<k≤αN |m i | 2 − |m k | 2 2 .
For the remaining part we merely obtain from (A.1) that (A.5) ∧ i=1,...,αN k=αN+1,...,γN dR i,k ∧ dR k,i = DP ∧ DP αN i=1 |m i | 2(γ−α)N , so we finally obtain formula (2.9) for the Jacobian of transformation (2.8).
Figure 1 .
1The Belyi graph Γ 1 corresponding to the Belyi pair (CP 1 , id); ∞ ± indicate directions of approaching the infinite point in CP 1 . By Λ, Λ we indicate the insertions of the external field in the matrix-model formalism of Sec. 2. For example, this graph contributes the term N 2 βγ tr(ΛΛ).
F
[{t r }, β, γ; N] := DR DR DB DB DG DG e N tr(−BB−RR−GG+RΛGB+B G Λ R) . The free energy F [{t r }, β, γ; N] is given by the sum over all connected bipartite three-valent fat graphs Γ weighted by (2.2) 1 |Aut Γ| N 2−2g β n 2 γ n 3 r t mr r r m r = n 1 where n 1,2,3 are the respective numbers of solid-, dotted-, and dashed-line cycles in Γ, i | 2r are the times of the model, and m r is the number of solid-line cycles of length 2r in Γ. Measures of integration are the standard Haar measures; for instance,
In the vicinity of the unities of the unitary groups, we can write U = e iǫH and V = e iǫQ with the Hermitian (αN × αN)-matrix H and Hermitian (γN × γN)-matrix Q of the form(2.7) Q = H P P † 0 ,in which H is another Hermitian (αN × αN)-matrix and P is the general complex (αN × (γ − α)N)-matrix. The Jacobian of the transformation(2.8) DR DR = Jac DU DV i dm i dm ican then be easily calculated (see Appendix A)
Theorem 2 . 2 .
22The generating function for Belyi fat graphs (1.1) is the tau-function of the KP hierarchy in times (2.3).
Lemma 3. 4 .
4The generating function F [{t r }, β, γ; N] (1.1) for the Belyi fat graphs is given by the exact formula e F [{tr },β,γ;N ]
It is easy to see that in the domain of largeλ i , the expansion in (3.22) contains only negative powers ofλ: the linear and the logarithmic inλ i terms vanish in this domain. 3.3.2. Higher genus expressions. All higher genus corrections to the Hermitian one-matrix model can be written in terms of moments [3] M r , J r of the potential: (3.23)
DB
DB e −N tr[BB]+N ∞ r=1 1 r tr tr[(BB) r ] ,
Lemma 4. 1 .
1Generating function (1.1) can be presented as a Hermitian one-matrix model integral (4.3) with a logarithmic term in the potential.
DH
≥0 e N tr[−H(Λ −2 −β)+(γ−α) log H+ 1 2 βH 2 ] . Lemma 5.1. The generating function for clean Belyi fat graphs ((1.1) with ramification profiles(2, . . . , 2, 1, . . . , 1) at the point 1)is the matrix-model integral (5.
µ j − y i ) = ∆(y, µ)/∆(µ), where ∆(y, µ) is the Vandermonde determinant of the set of variables y i and µ j , write each of the determinants ∆(y, µ) and ∆(y) as determinants of the Hermitian polynomials H s (x), where s ranges from 0 to M + N − 1 and x are either y i or µ j in the first determinant and s ranges from 0 to M − 1 and x are y i in the second determinant. Because the Hermitian polynomials are orthogonal with the measure e − 1 2 x 2 , we can integrate out all the y-variables; the remaining expression will be the determinant of the (N × N)-matrix H M +j 1 −1 (µ j 2 ) , j 1 , j 2 = 1, . . . , N, and the original integral (5.
Lemma 5. 3 .
3The generating function (1.1) for the clean Belyi morphisms with the ramification profile (2, 2, . . . , 2) at the point 1 is given by the following Hermitian one-matrix model integral for γ − α ≃ O(1):
k | 2s .We then have the following statementLemma 6.1. The generating function (6.2) F [{t 1 , t 2 , . . . }, {t 1 , t 2 , . . . }, β; N] morphisms in which we have two sets of ramification profiles: {t r 1 , . . . , t rn 1 } at infinity and {t s 1 , . . . , t sn 3 } at 1 is given by the integral over complex rectangular (βN × γN)-matrices B, B: (6.3) Z[t, t] := e F [{t},{t},β;N ] = γN ×βN DB DB e −N tr[BB]+N ∞ m=1 1 m tm tr[(BB ΛΛ) m ] ,
The authors were reported by M. Kazarian that the same constraints can be derived by pure combinatorial means [M. Kazarian, P. Zograf, paper in preparation].
The authors thank A. Mironov for this comment.
The term (β + γ) in the r.h.s. of the second equation is not a misprint.
AcknowledgmentsThe authors thank Maxim Kazarian, Andrei Mironov, and Petr Zograf for the useful discussion.The authors acknowledge support from the ERC Advance
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| {'fraction_non_alphanumeric': 0.09401184934405417, 'fraction_numerical': 0.04289039356749894, 'mean_word_length': 3.38495082575617, 'pattern_counts': {'":': 0, '<': 7, '<?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': 63, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "We present the matrix models that are the generating functions for branched covers of the complex projective line ramified over 0, 1, and ∞ (Grotendieck's dessins d'enfants) of fixed genus, degree, and the ramification profile at infinity. For general ramifications at other points, the model is the two-logarithm matrix model with the external field studied previously by one of the authors (L.Ch.) and K.Palamarchuk. It lies in the class of the generalised Kontsevich models (GKM) thus being the Kadomtsev-Petviashvili (KP) hierarchy τ -function and, upon the shift of times, this model is equivalent to a Hermitian one-matrix model with a general potential whose coefficients are related to the KP times by a Miwa-type transformation. The original model therefore enjoys a topological recursion and can be solved in terms of shifted moments of the standard Hermitian one-matrix model at all genera of the topological expansion. We also derive the matrix model for clean Belyi morphisms, which turns out to be the Kontsevich-Penner model introduced by the authors and Yu. Makeenko. Its partition function is also a KP hierarchy tau function, and this model is in turn equivalent to a Hermitian one-matrix model with a general potential. Finally we prove that the generating function for general two-profile Belyi morphisms is a GKM thus proving that it is also a KP hierarchy tau function in proper times.", 'arxivid': '1404.4240', 'author': ['Jan Ambjørn ', 'Leonid Chekhov '], 'authoraffiliation': [], 'corpusid': 50662402, 'doi': '10.4171/aihpd/10', 'github_urls': [], 'n_tokens_mistral': 17665, 'n_tokens_neox': 15576, 'n_words': 8877, 'pdfsha': '93a3189b09e359a3444c4a6916e2dcfaafc31ab5', 'pdfurls': ['https://arxiv.org/pdf/1404.4240v2.pdf'], 'title': ["THE MATRIX MODEL FOR DESSINS D'ENFANTS", "THE MATRIX MODEL FOR DESSINS D'ENFANTS"], 'venue': []} |
arxiv |
Image2GIF: Generating Cinemagraphs using Recurrent Deep Q-Networks
Yipin Zhou
University of North Carolina at Chapel Hill
Yale Song
Yahoo Research
Tamara L Berg
University of North Carolina at Chapel Hill
Image2GIF: Generating Cinemagraphs using Recurrent Deep Q-Networks
Given a still photograph, one can imagine how dynamic objects might move against a static background. This idea has been actualized in the form of cinemagraphs, where the motion of particular objects within a still image is repeated, giving the viewer a sense of animation. In this paper, we learn computational models that can generate cinemagraph sequences automatically given a single image. To generate cinemagraphs, we explore combining generative models with a recurrent neural network and deep Q-networks to enhance the power of sequence generation. To enable and evaluate these models we make use of two datasets, one synthetically generated and the other containing real video generated cinemagraphs. Both qualitative and quantitative evaluations demonstrate the effectiveness of our models on the synthetic and real datasets.
Introduction
Based on our life-long observations of the natural world, humans have the ability to reason about visual appearances of static and dynamic objects. In particular, given an image, one can easily picture which objects will move and how they might move in the future. For instance, given an image showing a can pouring liquid into a cup, as illustrated in Fig. 1 (left), one can imagine how the liquid must be flowing due to gravity and what that might look like. Or, given an image showing a woman with billowing hair (right), we can imagine how her hair might wave in the breeze.
Cinemagraphs are still photographs in which a repeated movement of one or more objects occur, often served as an animated GIF to produce a repeating effect. Often these objects are natural entities, e.g. water flowing in a fountain, plants blowing in the breeze against a still background, and the steps of an escalator ascending. In this way, cinema- * This work was done while the author was an intern at Yahoo Research. Figure 1. Example images where one can easily imagine dynamic motion -the coke will flow, the hair will wave. graphs can be seen as highlighting and illustrating the motion of specific dynamic objects in static scenes. Because of this, we posit that understanding and generating cinemagraphs is useful for providing insight into the world -for understanding what objects can move and how they moveand for evaluating how well computational models learn to represent the dynamic nature of objects.
In this work, our goal is to learn a computational model to reason about visual appearances of static and dynamic objects. Specifically, we train a generative model to animate part of the input photograph and output a cinemagraph sequence. Recent generation works have explored the tasks of generating novel images [5,6,3,24,33,26,37,1,9,39,2,11] and future video frames [29,25,17,38,35,10,15,30,4,32], but this work is the first to explore the generation of cinemagraphs from a single image, which include the features of both images (static) and videos (dynamic). Cinemagraph generation from a single image is very challenging because the model needs to figure out both where to animate (localization) and how to animate (generation).
With both dynamic and static characteristics, cinemagraphs highlight particular objects within an image, bestowing a sense of feeling or emphasis to a static image. As a result, they have become quite popular on the Internet (querying cinemagraph on Google yields 1,960,000 results). However, one drawback is that creating a cinemagraph requires tedious manual work: people use videos to create a cinemagraph using commercial softwares, fixing part of frames and animating others. Thus, one useful practical benefit of this work is to automate the process of generating a cinemagraph from a single image. Other benefits include learning about what objects can move in a scene and how they move. This can be accomplished by developing models to generate cinemagraphs and then evaluating how well their predictions match the reality.
We explore different model architectures stacked in a recurrent structure to recursively generate cinemagraph sequences. Additionally, we incorporate Deep Q-networks, a deep reinforcement learning algorithm (Sec. 3), to improve performance by brining in stochasticity to the model. To enable evaluation of these models, we use two new datasets: one is synthetically generated, the other is cinemagraph data we collected from the Internet (Sec. 4).Finally, we provide both quantitative and qualitative evaluation of the models on our synthetic and real datasets (Sec. 5).
Our contributions include: 1) We introduce a new task of generating cinemagraphs from a single image, 2) We release two new datasets for our task, 3) We explore and evaluation various recurrent generative model structures combined with deep Q-net for cinemagraph generation.
Related Work
Image generation: Applying generative models on natural images has attracted a great deal of attention in recent years. Gregor et al. [6] recurrently generate different areas of a single image using an attention mechanism with variational autoencoders. Goodfellow et al. [5] proposed generative adversarial networks (GANs) that greatly improve image generation quality. GANs consist of a generative model and a discriminative model, trained jointly for enhancing the realism of generated images. Radford et al. [24] explored combining deep convolutional neural networks in GANs to further improve the image quality. Pathak et al. [23] proposed a network to generate contents of an arbitrary image region conditioned on its surroundings. Finally, Zhao et al. [37] proposed energy-based GANs that consider the discriminative model as an energy function. Following these great successes, we also explore the use of GANs for cinamagraph sequence generation. Video frame generation: As a step beyond creating realistic images, video generation requires the model to incorporate temporal information into the generation process. Therefore, video generation models usually generate frames conditioned on the previous frame (or several previous frames), rather than generating from noise, as is usually done in image generation tasks. Some related approaches [29,25] train generative models either to reconstruct input video frames or to generate the next few consecutive frames, learning representations in an unsupervised framework. Mathieu et al. [17] combines a mean square error (MSE) loss with an adversarial loss to produce high-quality generation results. Vondrick et al. [31] proposed a GAN with a spatio-temporal convolutional architecture to learn a scene's foreground and background simultaneously. Zhou et al. [38] incorporated a recurrent neural network with a generative model to generate frames showing future object states in timelapse videos. Inspired by their work, we apply an LSTM [8] to encode temporal information during cinemagraph generation. Deep reinforcement learning: Reinforcement Learning (RL) has been applied to many applications [18,12,14]. Recently, works combining RL (especially model-free RL) with deep neural networks have attracted extensive attention [20,16,40,19,22]. Model-free RL algorithms can be divided into two types: Q-learning and policy gradient learning. Deep Q-learning (DQL) predicts which action to take at each time step to maximize future rewards, while policy gradient methods directly optimize a policy of expected reward using gradient descent. For example, Mnih et al. [20] use DQL to control an agent to play ATARI games (manipulating the joystick), while Gu et al. [7] proposed a DQL method with a continuous action space. Silver et al. [28], Lillicrap et al. [16], Mnih et al. [19], and Schulman et al. [27] make use of an actor-critic or an asynchronous variant of the actor-critic algorithm based on deterministic policy gradient that can solve the tasks with continuous action space, e.g., cartpole swing-up, 3D locomotion, or other robotics tasks.
On the Atari task, Mnih et al. [20] utilized a Deep Qnetwork (DQN) to select an action a, which they apply to the state s in order to achieve maximal award for each time step. Here, a (e.g. joystick operation), s (game screen) and award (game score) can be easily defined. Moreover the action space is discrete (e.g. agents can only move up and down). Later work [7] applies a DQN to a continuous action space for controlling robotic arms. Those works have clear definitions for action space (e.g., angles, positions). Previous success of Deep Q-learning witnessed the potential of DQN for sequential decision-making. In this work, we apply a DQN in a sequence generation task, which differs with previous works in that our action space consists of abstract concepts that help with the sequence decision making. In other words, we select actions to apply to the current image to further lower the generative loss (maximize award).
Methods
Our goal is to generate a cinemagraph with n frames Y = {y 1 , ..., y n } given a static image x as input. This requires building models that can generate depictions of future states of moving objects over time. Given an image, we expect the model to learn both what part of the image should be animated (what objects can move and where they are in the image) and how to create an animation for their motion. In this work, we explore two frameworks for the genera- tion task. In the first framework, we stack an autoencoderlike structure with a recurrent neural network layer to recursively generate future frames. The second framework adds a Deep Q-network to help the model make better decisions sequentially while generating future frames.
Recurrent Model
In this scenario, we recursively generate frames of a cinemagraph, one by one, from a single static image. We do this by applying an autoencoder-style convolutional neural network stacked with an LSTM layer [8], recursively generating future frames based on a hidden variable z (illustrated in Fig 2). An LSTM is ideal for our purpose due to its recurrent architecture and the ability to cope with temporal information of relatively long sequences.
Inspired by work on recursively generating future frames for timelapse videos of objects [38], we design our network with a similar structure and define a loss function with two tasks. The first is a mean square error (MSE) loss, that is, pixel-wise mean square error between generated outputs and the ground truth frames:
loss mse = |Y − G(x)| 2(1)
where x is the input image (the first frame in a cinemagraph sequence), G(.) is the output of the generative model, and Y is the ground truth cinemagraph sequence. The second part of the loss function is an adversarial loss. We define an autoencoder as the generator and introduce an additional binary CNN classifier as the discriminator. The discriminator classifies a given input image as real or fake (i.e. generated). These two components, generator and discriminator, can be seen as adversaries as they operate in competition with one another, and forms a generative adversarial network (GAN). Several previous genera- tion works [5,24,3,21,36] have used GANs for generating high quality images.
The adversarial loss is represented as:
loss adv = − log(D[G(x)])(2)
where D[.] is the output from the discriminator. This loss encourages the generated frames look like real images. Our dual-task loss function is then represented as
loss = loss mse + λ adv * loss adv(3)
where λ adv is a hyper-parameter that controls the impact of adversarial loss. The recurrent model partially solve our sequential generation task. However, for cinemagraph generation, which is sightly different from video frame generation, we need a more expressive model with high stochasticity because the model needs to reason about the notion of foreground (moving) and background (fixed). Next, we discuss one possible way to achieve this by using Deep Q-network [20].
Recurrent Deep Q-Network
Recent work [20] has applied deep Q-learning to train a model to play Atari games, where the model makes decisions for each time step regarding how to adjust a joystick. Similarly, our cinemagraph generation task is a sequential decision making process. During generation, the model must decide which action to apply to generate the next frame.
Inspired by the success of deep Q-learning, we incorporate a Deep Q-network (DQN), which is a convolutional neural network trained with Q-learning, into our framework. For DQN, the input is a state s (e.g., game screenshots in Atari games) and the output is a Q-vector. The dimension of the Q-vector is equal to the number of possible actions (e.g. up and down). The value q(s, a) for each dimension is defined as the discounted cumulative maximum future expected reward of taking action a from state s.
q(s, a) = max π E[r t + σr t+1 + σ 2 r t+2 ...|s t , a t = s, a, π](4)
where r t is a reward at time step t, σ is a discount factor, and π is a policy mapping input states to actions. For each time step, the policy (what action to take when in state s) would then pick the action that maximizes the expected future reward in a greedy manner.
In this work, we apply DQN in a supervised generation task to make decisions about what "action" to take to generate the next frame. We first quantize a continuous action space into a discrete space (a Q-vector with limited dimensions). We then make use of the decoding structure of our generative model to decode how a "quantized action" will affect the state to generate the next frame. We evaluate how sensitive our model is to the size of the discretized space in Section 5. 6 Specifically, we propose a generation model cooperating with a DQN. The generator structure is the same as the recurrent model described in Sec. 3.1. We introduce a DQN whose input is the previous generated frame from the generator (or the input image for the first time step). For the output, we quantize the continuous action space into a discrete space by defining the output Q-vector as an N dimension vector to decide what "action" to take for a given state. We then encode it to a one-hot vector by assigning the maximal value of the Q-vector to 1 and others to 0. For each time step, we compute a different Q-vector based on the previous generation and concatenate it with the latent variable from the LSTM layer (as illustrated in Fig. 3). We let the concatenated vector pass through the decoding part to generate the frame showing the next state.
As we show in Sec. 5, adding the DQN to our recurrent generative model helps us achieve superior results -in terms of both the sequence generation aspect (Sec. 5.3) and the "action" prediction aspect, i.e., how the foreground object will move in the next step (Sec.5.5).
Datasets
We collected two datasets: synthetic and real. The synthetic data allows us to evaluate our models with a large amount of data in a controlled setting, and to objectively measure performances with clearly defined ground truth labels. The real-data, on the other hand, allows us to test whether our models can indeed tackle the real-world problem of generating cinemagraphs from a single image.
Synthetic Dataset
To emulate a cinemagraph-like data, we collect 10 cluttered texture images and randomly pick multiple random offsets between [0.90, 1.1] for 3 channels as the fixed background. We then draw a randomly sized rectangle filled with a random color to one of the texture images as the "foreground object." The foreground object will move in one of 6 moving patterns: "I" pattern, "O" pattern, "L" pattern, "8" pattern, rotate (counter clockwise), and scale (horizontally and vertically). Fig. 4 shows example frames from the synthetic sequences.
Cinemagraph Dataset
To show the ability of our models generating cinemagraphs from natural images, we collect a dataset of real cinemagraphs from the Giphy website. We crawl GIFs with the tag "cinemagraph" and manually annotate the data with our predefined category names, e.g., water flowing, water pouring, fire, and candle light. We then select those categories with more than 200 samples. The resulting dataset contains 2 categories of cinemagraphs, depicting water flowing and fire. The number of cinemagraphs in the water flowing and fire categories are 926 and 350, respectively, with a total of 1,276 cinemagraphs.
A cinemagraph with long duration usually contains a great deal of movement replication. To alleviate this, we cut the cinemagraphs to be no more than 1 second. For long cinemagraphs that are more than 3 seconds long, we sample two to three 1-second clips with a maximum margin. This Figure 5. Example frames of "Water flowing" and "fire" categories from our real cinemagraph dataset. We show the cropped patches on the up-right or left-down corner to indicate the details. We observe along the columns that the corresponding objects are indeed moving. helps us augment the data for training, since the movements at different time steps may look different. We make sure to assign clips from the same cinemagraph to the same training/testing split. Some example frames are shown in Fig.5. We observe that frames contain both static and dynamic parts depicting water flowing and fire moving. Since the two moving types are difficult to see in still image frames, we crop and enlarge some key patches to show details of moving objects. 1
Experiments
Implementation Details
For the generator of both the RNN (Sec 3.1) and the RNN+DQN (Sec 3.2) models, inputs and outputs at each time step are of dimension 64x64x3, and the encoding and the decoding parts include 4 convolution/deconvolution layers with a kernel of size 5x5 and a stride of 2. The decoding part of each time step shares the same weights, and the number of feature maps for the layers are 64, 28, 256, 512/512, 256, 128, 64 respectively. The size of the hidden variable z is 64 for synthetic data and 512 for real cinemagraph data generation. After each layer, we use ReLU as the activation function; for the last layer we apply a tanh activation function.
For the RNN+DQN model, the architecture of DQN is the same as the encoding part of the generator, except we do not apply the tanh activation on the output layer. The output Q-vector is a 64 dimensional vector for synthetic data task, and 512 dimension for the real cinemagraph genera-tion task.
We implement all models using Tensorflow. For all experiments, we use an ADAM optimizer [13] with a learning rate of 0.0002, and a batch size of 64. We set the weight for the adversarial loss λ adv to 0.005 for synthetic data and to 0.05 for real data experiments (equation 3). We raise the λ adv in real data experiments in order to enhance the realistic appearance of natural images.
Synthetic Data Results
We evaluate the generation performance of the RNN model (Sec 3.1), RNN+DQN model (Sec 3.2), and a baseline generation method from Mathieu et al. [17] on 6 categories of synthetic data. For each category, we generate 100K sequences (95K for training and 5K for testing). Our proposed RNN and RNN+DQN models receive one input frame and recurrently generate the future frames. For each synthetic category, we train the model for 6K iterations. We note that Mathieu et al. [17] is stateless, in that it takes the previous output as the input for the next time step to generate the sequence. For their model we experiment with several parameters settings, and find that applying the default setting described in their paper works the best. We also train their model for 6K iterations.
For the synthetic data, we sample frames of sequences of each category. If the length of sampled sequences in one category is different, we pad the short sequences with a black image of the same size. The maximum length for 6 categories are: I Pattern : 27; O pattern : 21; L Pattern : 26; Eight Pattern : 21; Rotate pattern : 21; Scale pattern : 17. Fig.6 qualitative results. We observe that the baseline method [17] has a difficult time modeling the temporal movement of the foreground objects. We believe this is because it generates the future frame conditioned on only the previous (several) inputs while the sequence is relatively long. The superior quality of results of our RNN and RNN+DQN models suggest that they successfully learned the moving patterns of different categories. We can also see that the RNN+DQN model produces slightly more visually appealing results (the foreground objects have sharper boundary and more accurate shapes) than RNN method.
We also quantitatively evaluate the methods. For the evaluation metrics, we compute the Peak Signal to Noise Ratio (PSNR) [34] and Structural Similarity Index (SSIM) [34] between a generated sequence and the ground truth sequence. Table 1 shows that the RNN+DQN model works better than the RNN model. This is because, although both can generate a sharp background, RNN+DQN generates more accurate foreground objects. These two methods both significantly outperform the baseline method [17], which does not apply an RNN structure to incorporate temporal information.
Real Data Results
We evaluate our RNN and RNN+DQN models, as well as a baseline [17] method, on "water flowing" and "fire" categories of our real cinemagraph dataset. For each cinemagraph, we sample 6 frames (1 input and 5 generation outputs). We split the training and testing sets with a ratio of 0.85 : 0.15. We augment the training data by cropping different regions from frames (random location fixed across time) and flipping frames left-right across time. For all three methods and categories, we train the models for 13,500 iterations.
We first evaluate the method qualitatively. Fig.7 shows input frames and generated frames under each method. We refer to our project website for detailed examples of generated cinemagraphs. The baseline method [17] achieves decent results because the length of the generation sequences is shorter than the synthetic data. However, similar to the case with synthetic data, background and foreground is still not well distinguished.
Overall, our models learn the correct motions for each object. The results suggest that our RNN and RNN+DQN models better differentiate and animate the foreground motions of water flowing and fire flickering, while keeping background fixed. 2 For the "fire" category, the resulting animated fires sometimes wave less strongly than the ground truth due to the limited data and high variance.
For a quantitative evaluation, we compute PSNR and SSIM values between generated results and the ground truth sequences. Table 2 shows the results. RNN+DQN, which generates more visually appealing results, also achieves the best quantitative performance compared to the other two methods.
Constant Baseline
To further show that the proposed method is able to capture the (foreground) moving objects while making the background static, we provide a simple "constant" baseline, i.e., new frames are simply copied from the first frame. To capture the temporal dynamics, we compute PSNR/SSIM on frame-wise difference images (two consecutive frames) rather than on the original frames. For the constant baseline, frame-wise difference results in all-zero images because there is no movement over time. Table 3 shows that RNN+DQN method achieves better performance than the baseline, which suggests the capability of our method discriminating foreground objects from backgrounds.
Deep Q-Network
Deep Q-network (DQN) helps improve the results by bringing in more stochasticity into our model, improving sequential decision making and thus the overall visual quality of generated cinemagraphs. Here, we show the ability of DQN to help decide what objects in a scene should move and how they should move.
We conduct an experiment on synthetic dataset to evaluate the model's ability to predict the location of moving objects (find which object to move and how to move). Specif- Table 6. Human evaluation results. Column 2-3 shows how often humans preferred one method over the others. Column 4-5 shows how often humans were "fooled" into believing that the generated results are real cinemagraphs.
ically, we pre-compute the segmentation maps (segmenting the moving object and the background) of the synthetic data. During testing, we segment out the object from the ground truth data and run a sliding window on the generated frames to get the resulting segmentation maps. And we compute the average Euclidean distance between the object centers of ground truth and generated frames. Table 4 shows the results. The results suggest that adding DQN indeed improves the prediction quality on how objects move in future frames. We do not evaluate Mathieu et al. [17] because it often fails to generate moving objects, and thus the sliding window technique cannot be applied.
Sensitivity Analysis
We conduct an ablation study to show how sensitive the results are to the size of z (bottleneck of the auto-encoder) and the Q-vector (output of DQN). Specifically, we evaluate the RNN+DQN model with small (4-dim) and large (2048dim) sizes of z and Q-vector, and compare them with the original results we presented earlier (64 for synthetic data and 512 for real data) Table 5 reports average PSNR/SSIM scores of small and large z/Q-vector on both synthetic and real datasets. We no-tice that utilizing the small bottleneck and/or Q-vector significantly harm the results due to the severe information loss and over quantization (except for applying small Q-vector on synthetic dataset because the action space of partial synthetic categories is relatively small).
Human Evaluation
There is a universal problem in evaluating video prediction methods: There might be multiple correct predicted results. There might be multiple cinemagraphs that share the first frame but look different from each other, all of which could be considered reasonable to a human observer. Therefore, we design two human experiments to judge the quality of real cinemagraph generated by different methods.
In the first experiment, we show human subjects three cinemagraphs generated from different methods (RNN, RNN+DQN, Mathieu et al. [17]) and ask them to choose one that looks the most superior. We used Amazon Mechanical Turk (Mturk) for this study. Results are shown in Table 6 (Col 2-3), where numbers indicate how often humans selected results from each method as the best cinemagraph. We note that RNN+DQN method achieves the best results under both categories.
In the second experiment, we show human subjects a single cinemagraph (either real or generated) at a time and ask them whether it looks realistic. We measure how often humans choose the generated ones as real and report the results in Table 6 (Col 4-5), which shows that our RNN+DQN is the most successful at this task.
Conclusion
In this paper, we explore a challenging task of generating a cinemagraph from a single image. We propose a method that combines recurrent generative models with a deep Q-network to learn what regions should move and how they should move over time. On both synthetic and real datasets, we evaluate qualitative and quantitative results of the proposed methods and show improved performance over a baseline method of Mathieu et al. [17].
Future work includes expanding the natural cinemagraph dataset to additional categories with increased numbers of examples. We also plan to analyze the learned models to better understand what the models are learning about image content semantics and motions of objects in the world.
Figure 2 .
2Recurrent generation architecture: Input to the network is an image, and the architecture consists of an autoencoder CNN stacked with an LSTM layer that recurrently generates future frames.
Figure 3 .
3Recurrent generator with a deep Q-network. The Qvector is concatenated with the latent state from an LSTM layer at each time step. Note that we show the concatenation only in the first time step to reduce clutter.
Figure 4 .
4We show frames of six categories sub-sampled from synthetic generated sequences. (a) "I" Pattern frames; (b) "O" Pattern frames; (c) "L" Pattern frames; (d) "Eight" Pattern frames; (e) "Rotate" Pattern; and (f) "Scale" Pattern frames.
Figure 6 .
6Qualitative results on synthetic data. We show an input frame along with results from a) Mathieu et al.[17]; (b) our RNN model; (c) our RNN+DQN model. (d) is the ground truth. We show 7 frames uniformly sampled from each sequence.
Figure 7 .
7Qualitative results on the real cinemagraph data. For each sample, we show the input image and 5 generated frames generated by method: (a) Mathieu et al. [17]; (b) RNN method; (c) RNN+DQN method; and (d) ground truth.
Table 1. PSNR and SSIM scores of Mathieu et al.[17], our RNN and RNN+DQN models on synthetic data across 6 categories.I
O
L
Eight
Rotate
Scale
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
PSNR
SSIM
Mathieu et al. [17] 19.9801 0.7917 21.1132 0.8134
9.9721 0.0631 19.9538 0.8004 20.7091 0.7820 14.4876 0.4502
RNN
24.2417 0.8652 26.4261 0.8879 27.2542 0.8641 26.5649 0.8861 27.2592 0.9101 25.2220 0.7611
RNN+DQN
25.4979 0.8672 27.9527 0.9210 27.3036 0.8782 27.5839 0.9161 27.3559 0.9115 25.5889 0.7483
RNN+DQN 27.9192/0.5576 31.0523/0.6868 27.4687/0.6167 31.7454/0.6941 30.5351/0.6186 30.5754/0.6081 29.8827/0.6303 29.4949/0.5873 29.4030/0.5565 29.4490/0.5719I
O
L
Eight
Rotate
Scale
Average
Water
Fire
Average
Constant
27.1174/0.5541 27.9468/0.6270 26.7426/0.6097 28.9732/0.6361 29.4275/0.5893 31.0325/0.6459 28.5400/0.6104 29.3681/0.5768 28.7743/0.5434 29.0712/0.5601
Table 3 .
3PSNR/SSIM scores of synthetic and real data and their averages on Constant baseline and RNN+DQN. (Col 2-8: synthetic data; Col 9-11: real data.)Table 4. Average Euclidean distance between centers of the foreground object from the ground truth and from generated frames. Small Q-vec25.8371 / 0.8556 17.9049 / 0.3974 Large Q-vec 26.1532 / 0.8585 19.6085 / 0.4885 Table 5. The average PSNR/SSIM scores with different dimensions of z and Q vectors.I
O
L
Eight
Rotate
Scale
RNN
2.9278 2.2839 2.6734 2.4735 2.0150 2.3675
RNN+DQN 2.6440 2.1125 2.6285 2.3181 1.9931 2.1842
Synthetic
Real
Original
26.8805 / 0.8737 19.8437 / 0.5015
Small z
21.3788 / 0.7794 13.7626 / 0.1364
Large z
26.2362 / 0.8670 18.4141 / 0.4111
Water flowing
Fire
Water flowing
Fire
Mathieu [17]
30.00%
26.42%
11.43%
20.75%
RNN
32.86%
35.84%
27.86%
26.41%
RNN+DQN
37.14%
37.74%
35.00%
33.96%
We include examples of animated cinemagraphs in the demo video in the project webpage.
We note that, in the resulting frames shown in the paper, the foreground motion is hard to observe due to small image size and the nature of these small motions. We show videos of the generated cinemagraphs in our project webpage to better illustrate the results.
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arxiv |
Relativity of motion in vacuum
arXiv:quant-ph/9801071v1 30 Jan 1998
Marc-Thierry Jaekel
Laboratoire de Physique Théorique de l'ENS *
24 rue LhomondF75231, Cedex 05Paris
Astrid Lambrecht
Serge Reynaud
Laboratoire du CNRS associéà l'Ecole Normale Supérieure età l'Université Paris-Sud † Laboratoire de l'Ecole Normale Supérieure et de l'Univer
Relativity of motion in vacuum
arXiv:quant-ph/9801071v1 30 Jan 1998
b Laboratoire Kastler Brossel † , UPMC case 74, Jussieu, F75252 Paris Cedex 05 (Contribution for Vacuum, eds E. Gunzig and S. Diner)The principle of relativity is one of the main general laws of physics. Since it applies in particular to motion in empty space it is related to the symmetries of this space. Thoughts about this subject within the framework of classical physics have led to the theory of relativity [1]. The emergence of quantum theory has then profoundly altered our conception of empty space by forcing us to consider vacuum as the realm of quantum field fluctuations. The notion of quantum vacuum and the existence of vacuum fluctuations unavoidably lead us to reconsider the question of relativity of motion. The present article is devoted to this aim with a main line which can be formulated as follows: "The principle of relativity of motion is directly related to symmetries of quantum vacuum". Keeping close to this statement, we discuss the controversial relation between vacuum and motion. We introduce the question of relativity of motion in its historical development before coming to the results obtained more recently.Movement in space cannot be defined otherwise than movement in vacuum. This assertion was formulated by Leucippus and Democritus more than 2000 years ago. In the context of that time, it has to be understood as a logical necessity rather than a physical statement in the modern sense. The existence of movement is a matter of evidence which forces us to conceive a space in which movement takes place [2]. The absence of resistance to motion is a key condition in this respect and in fact constitutes the main argument for the definition of the concept of vacuum. At the same time vacuum is a positive physical reality which clearly differs from nothingness. It is in particular the reference with respect to which movement has to be identified. These properties are certainly paradoxical at first sight. This paradox has raised numerous discussions and it has conserved its whole pertinence ever since the birth of modern physics until today.After a long eclipse dominated by Aristotelian concepts, the question of relativity of motion was brought anew to the fore by Galileo [3]. His central argument, "Movement is like nothing", not only signifies that a motion with uniform velocity is indistinguishable from rest. It also implies that a motion with uniform velocity has no
The principle of relativity is one of the main general laws of physics. Since it applies in particular to motion in empty space it is related to the symmetries of this space. Thoughts about this subject within the framework of classical physics have led to the theory of relativity [1]. The emergence of quantum theory has then profoundly altered our conception of empty space by forcing us to consider vacuum as the realm of quantum field fluctuations. The notion of quantum vacuum and the existence of vacuum fluctuations unavoidably lead us to reconsider the question of relativity of motion. The present article is devoted to this aim with a main line which can be formulated as follows: "The principle of relativity of motion is directly related to symmetries of quantum vacuum". Keeping close to this statement, we discuss the controversial relation between vacuum and motion. We introduce the question of relativity of motion in its historical development before coming to the results obtained more recently.
Movement in space cannot be defined otherwise than movement in vacuum. This assertion was formulated by Leucippus and Democritus more than 2000 years ago. In the context of that time, it has to be understood as a logical necessity rather than a physical statement in the modern sense. The existence of movement is a matter of evidence which forces us to conceive a space in which movement takes place [2]. The absence of resistance to motion is a key condition in this respect and in fact constitutes the main argument for the definition of the concept of vacuum. At the same time vacuum is a positive physical reality which clearly differs from nothingness. It is in particular the reference with respect to which movement has to be identified. These properties are certainly paradoxical at first sight. This paradox has raised numerous discussions and it has conserved its whole pertinence ever since the birth of modern physics until today.
After a long eclipse dominated by Aristotelian concepts, the question of relativity of motion was brought anew to the fore by Galileo [3]. His central argument, "Movement is like nothing", not only signifies that a motion with uniform velocity is indistinguishable from rest. It also implies that a motion with uniform velocity has no * Laboratoire du CNRS associéà l'Ecole Normale Supérieure età l'Université Paris-Sud † Laboratoire de l'Ecole Normale Supérieure et de l'Université Pierre et Marie Curie associé au CNRS observable effect when it is composed with a second motion. This is the deep meaning of the famous discussion of motion in a moving boat used by Galileo in the Dialogo to emphasize the property of relativity [4]. Of course this property is rigorously true only when the resistance of air to motion can be ignored. Newton stressed this fact in the Principia in order to identify the empty space in which motion takes place with the vacuum which had just been the subject of the first experiments of Torricelli, Pascal, von Guericke and Boyle. As we know today, the Newtonian laws describing inertia or gravity already contain the Galilean principle of relativity, as it was named by Einstein, although Newton himself had argued for the absolute character of his space reference. The theory of relativity has freed space from this absolute character and one of the reasons for that was precisely that it was contradictory with Galilean relativity. After the advent of this theory, the relation between motion and space can be made explicit as follows: the expression of the laws of motion are symmetry properties of space or, equivalently, the symmetries of empty space imply the laws of motion.
The formulation of the theory of relativity has largely been built upon the symmetries of Maxwell's equations. Indeed the empty space of classical physics is a reference for writing not only the laws of mechanics but also the propagation of the electromagnetic field. However, the development of classical physics was based on the idealization that space can be thought as being absolutely empty. This classical idealization could not be maintained, not even as a limiting case, when it was realized that space is always filled with freely propagating radiation fields after the birth of statistical mechanics and then of quantum mechanics. Black body radiation is present at every non-zero temperature and it undoubtedly produces real mechanical effects. It exerts a pressure onto the reflecting boundaries of a surrounding cavity and produces a friction force when reflecting surfaces are moving. Those two effects are analogous to the pressure exerted by air molecules on the walls of a container as was made clear by Einstein's theory of Brownian motion [5]. It is precisely for explaining the properties of black body radiation that Planck introduced the first quantum law in 1900. At zero temperature this law describes a region of space limited by a cavity and entirely emptied out of radiation. In order to approach a practical realization of vacuum, it is not sufficient to remove all matter from this enclosure since one also has to lower the temperature down to zero to eliminate thermal radiation.
Later on in 1912, Planck was led to modify his law in order to improve its agreement with known results in the classical regime reached at high temperatures. The modified law predicts that fluctuations remain present at zero temperature, from which Nernst deduced in 1916 that space is permanently filled with an electromagnetic field propagating with the speed of light [6]. The existence of these zero-point fluctuations or vacuum fluctuations, which correspond to half of the energy of one photon per electromagnetic field mode, was confirmed by the full quantum theory developed after 1925 [7]. Fluctuations thus appear as an inescapable consequence of Heisenberg's inequalities. To sum up this lesson of quantum theory, one may say that it establishes a necessary identity between the potentiality of movement and the presence of movement. As soon as space allows movement, which is of course an essential feature of space, then movement exists and in particular subsists at zero temperature. In the same way, as soon as space allows field propagation, which is equally a fundamental property of space, then fields are present. Quantum physics forbids the absence of movement as well as the absence of propagating fields. The ground state of a mechanical oscillator is defined as the state where its mechanical energy is minimal. In the same way, quantum vacuum is defined as the field state where the energy of field fluctuations is minimal. Then naturally the question arises whether the presence of fluctuations in quantum vacuum is compatible with the principle of relativity of motion. It is this discussion which we will sum up in a non-technical manner in this article. More technical arguments as well as a list of references can be found elsewhere [8].
Some preliminary remarks should still be mentioned. Most of the time the discussions devoted to Heisenberg's inequalities stress the limits they impose on the precision of a measurement. Insisting on the notions of uncertainty or even of indeterminacy however presents the serious disadvantage that it fixes the ambition of physicists with respect to a classical description of physical phenomena, although quantum theory has been developed precisely in order to remove the deficiencies of this description. The successes of quantum theoretical description of natural phenomena plead for a quite renewed point of view. Quantum fluctuations are intrinsic properties of physical quantities which display their inherent quantum nature. Today's knowledge allows for a theoretical understanding and also for an experimental study of these properties. In certain experiments one can even manipulate the quantum fluctuations in order to reduce the noise they produce on a particular signal [9].
The fluctuations of the electromagnetic field which remain in the vacuum state have well known observable effects [10]. An isolated atom in vacuum interacts only with vacuum fluctuations and this interaction is responsible for the spontaneous emission processes during which the atom changes its internal state by falling from a higher energy level to a lower one. When fallen in the ground state, the state of lowest energy where it can no longer emit photons, the atom is still coupled to vacuum fluctuations and this coupling results in measurable effects like the Lamb shift of the atomic absorption frequencies. Two atoms in vacuum are coupled to the field fluctuations which produce an attractive force between them, the so-called Van der Waals force. This force plays a very important role in physical and chemical processes and its quantum theoretical interpretation has been studied since the first years of quantum theory. While studying this problem, Casimir discovered in 1948 that there exists also a force between two mirrors placed in quantum vacuum [11]. The vacuum fluctuations are modified by the cavity formed by the mirrors and their energy depends on their relative distance. Hence vacuum exerts a force which mutually attracts the mirrors to each other. This Casimir force, as it was called later on, depends only on the distance and on two fundamental constants, the speed of light c and the Planck constanth. This is a remarkably universal feature in particular because the Casimir force is independent of the electronic charge in contrast to the Van der Waals forces. We have already noticed that any pragmatic definition of vacuum necessarily involves a region of space limited by some enclosure. The Casimir effect signifies in fact that the energy of vacuum depends on the configuration of this cavity from which it follows that its boundaries experience forces arising from radiation pressure of vacuum fluctuations. Although the Casimir force is relatively small it has been observed in several experiments [12].
Vacuum fluctuations thus play a central role in the modern description of the structure of matter. They explain numerous novel effects which have been observed without any ambiguity. Their existence is directly associated with a fundamental property of quantum physics, namely the representation of any physical quantity by an observable defined with the help of non-commuting mathematical objects. Their status nevertheless continues to raise intricate questions. One reason for this situation is the obvious fact that their representations are often incompatible with the intuitions inherited from classical physics. More fundamental reasons are related to the serious difficulties which have remained unsolved for a long time and, for some of them, still remain unsolved at the interface between the physics of vacuum fluctuations and the laws of mechanics or gravitation.
The archetype of these difficulties is the relation of vacuum energy to gravitation. The total vacuum energy, that is to say the energy summed over all field modes in their vacuum state, takes an infinite value. This implies that vacuum energy does not contribute in a standard way to gravitation since the universe would have a very different appearance otherwise. One simple way to deal with this problem is to set the vacuum energy equal to zero and, therefore, to use it as a reference for all other energies [13]. In contrast to a widely spread opinion this prescription does not allow to ignore the gravitational effects of vacuum. Indeed vacuum energy is modified in a space curved by the gravitational field. Furthermore, even if the mean value of the vacuum energy does not contribute to gravitation its variations necessarily contribute to it. Yet, the spatial distribution of vacuum energy changes continuously as it is composed of field fluctuations which propagate with the speed of light. The energy density of vacuum shows fluctuations which manifest themselves as fluctuations of space-time curvature. It is possible to discuss some of these questions in analogy with the standard formalism of quantum field theory (a discussion and references can be found in [14]). However a final answer to these questions will not be reached until a satisfactory quantum description of gravitation is available.
As already stated in the introduction, from the mere point of view of logics movement in space must be understood as movement in vacuum, so that the presence of field fluctuations in quantum vacuum forces us to reconsider the notion of motion itself. This question has been discussed at length in connection with attempts to obtain a consistent description of motion for elementary quantum objects like the electron [15]. It has been learned from classical electrodynamics that the expression of the force acting on a moving charged particle contains a contribution, known as the Abraham-Lorentz force, describing the reaction to motion entailed by the electromagnetic field emitted by the particle. The modern quantum theory tells us that the radiation reaction force is directly related to the vacuum field fluctuations through the fluctuation-dissipation relations, which are the quantum generalizations of the classical results of Brownian motion theory [16]. The occurence of a dissipative force induced by motion of the electron and its direct association with vacuum fluctuations forbid to obtain any consistent description of atoms within the classical framework. In the quantum theory, this crisis was only solved at a very high price since any mechanical description of movement was abandoned for elementary quantum objects [17]. Before discussing this point in more detail in the next paragraphs, let us notice that this was not the end of the story. Vacuum field fluctuations were also shown to modify the inertial mass of a point-like scatterer and, furthermore, to lead to an infinite mass correction. A renormalization prescription was designed, which states that the real mass of the particle is finite, as the result of adding the infinite positive correction to a bare mass which is itself infinitely negative. However, other difficulties emerge as outcomes of this approach. In particular, the mechanical response of the particle to an applied force shows instability and violates causality [18]. In fact, the renormalization procedure used to keep the particle mass finite in spite of an infinite correction is incompatible with a causal mechanical description [19].
These difficulties have led most of the theoretical physicists to adopt a pragmatic point of view which has proved itself useful in the microscopic domain. Any mechanical description is then given up in this domain where physical descriptions are instead built on quantum field theory. Taken seriously, this approach forces physicists to renounce to general principles of mechanics at the elementary level of microscopic physics whereas the mechanical description of nature has a validity restricted to the macroscopic domain [20]. This a priori separation between microscopic and macroscopic domains has now lost the pragmatical pertinence it had when the argument was formulated, due to the progress towards highly sensitive measurements approaching the quantum level of sensitivity for macroscopic objects [21]. From a more fundamental point of view, this solution cannot be satisfactory since it rejects the mechanical questions which concern in particular inertia and gravitation outside the framework of quantum theory. Although the problems of inertia and gravitation differ, they are directly related to each other. In the domain of gravitation also, vacuum field fluctuations modify the laws of motion by the introduction of dissipative reaction terms which lead to stability problems [22]. Both cases point at difficulties generated by the mixture of classical and quantum theoretical descriptions, namely Einstein or Newton equations on one hand and quantum field theory on the other hand. It is known more generally that any description built on a too crude mixing between classical and quantum formalisms is necessarily plagued with inconsistencies [23].
The general problem of quantum field theory with moving boundaries has emerged as an outcome of the thoughts about possible approaches to this problem [24]. This subject may be considered as an extension of the study of Casimir force, precisely of vacuum radiation pressure on reflecting boundaries, to the case of moving boundaries. At the same time, it is directly related to the questions raised by quantum descriptions of motion, inertia or gravitation. As it will become apparent in the following, this problem has been given formulations well adapted to the present theoretical framework of quantum physics and has led to satisfactory solutions of some of the general difficulties mentioned above.
Let us first consider the simple case of a mirror moving with a uniform velocity. Clearly there exist two very different situations depending on whether motion takes place in vacuum or in a thermal field. The principle of relativity of motion applies to the first situation but not to the second one. Should vacuum exert a force on a mirror with a uniform velocity, the reaction of vacuum would distinguish between inertial motion and rest. As expected, quantum theory predicts the friction force to vanish in this case so that the principle of relativity of uniform motion is valid in quantum vacuum. Besides, quantum theory gives an interesting interpretation of this property according to which vacuum fluctuations appear exactly identical to an inertial observer and to an observer at rest. The invariance of vacuum under Lorentz transformations is an essential condition for the principle of relativity of motion and it establishes a precise relation between this principle and a symmetry of vacuum.
We now come to the general case of arbitrary motion in vacuum. In 1976, Fulling and Davies showed that a perfectly reflecting mirror moving in vacuum emits radi-ation as soon as the mirror has a non uniform acceleration [25]. Meanwhile, the motion tends to be reduced to a uniform acceleration by the reaction of vacuum. The change of mechanical energy associated with damping is dissipated in form of emitted radiation and the reaction force follows from the associated momentum exchange. This motional force is directly related to fluctuations of the vacuum radiation pressure as they may be evaluated for a mirror at rest, in full consistency with the quantum fluctuation-dissipation relations [26,27]. When the equations of motion in vacuum are modified in order to take the motional force into account, it turns out that mirrors possess causal mechanical response functions and hence stable motions in vacuum. It has to be noted that the expression of the reaction force coincides with the Abraham-Lorentz derivative in the limit of a perfectly reflecting mirror which thus raises the same problems as in the case of the electron. This difficulty is solved by a careful description of the reflection properties of the mirror, accounting in particular for the fact that any real mirror is certainly transparent at high frequencies.
The motion of a real mirror in vacuum is described in a consistent manner once these points are satisfactorily dealt with [28]. Furthermore, a full treatment of the coupling between the mirror's motion and the field scattering shows that the vacuum fluctuations of the quantum field are eventually transmitted to the position of the mirror, even if the latter was originally considered as classical. In any realistic situation, the mechanical effect of vacuum upon the mirror may be considered as vanishingly small and the mirror is found in this limit to obey the standard Schrödinger equation [29]. These results prove that the objections against the possibility of giving a consistent quantum representation of motion in vacuum can be bypassed. They also show that the solution involves subtle correlations between the descriptions of motion in vacuum and of vacuum itself. For instance, the fluctuations associated with motion in vacuum can in no way be treated as uncorrelated with the fields fluctuations of vacuum in which motion takes place.
The existence of motional effects also questions the principle of relativity of motion applied to arbitrary motions in vacuum. As the reaction of vacuum vanishes for uniform velocities it does not raise any problem to the theory of special relativity. In contrast a non-uniformly accelerated motion produces observable effects, namely the resistance of vacuum against motion and the emission of radiation by the moving mirror into vacuum. Quantum theory thus allows us to sketch the outline of a framework where the questions raised in the introduction find consistent answers. Space is not empty since vacuum fluctuations are always present. These fluctuations effectively represent a reference for the definition of motion because they give rise to real dissipative effects in the general case of an arbitrary motion. At the same time, they do not damp uniform motions since quantum vacuum obeys Lorentz invariance. The physical properties of quantum vacuum are thus consistent with logical requirements of ancient atomists as well as with the Galilean principle of relativity of motion.
In this context, it would be extremely interesting to obtain experimental evidence for motional dissipative effects. Although these effects are extremely small for any motion which could be achieved in practice for a single mirror, an experimental observation could turn out to be conceivable with a cavity, instead of a single mirror, oscillating in vacuum. Due to a resonance enhancement of the emission of motional radiation the experimental figures become indeed much more promising in this case [30].
The radiation reaction force calculated by Fulling and Davies for a single mirror in vacuum is proportional to the Abraham-Lorentz derivative associated with its motion. Hence, it vanishes exactly for a uniformly accelerated motion. One question then arises, which naturally generalizes the one already met when considering uniform motion. How do vacuum fluctuations appear for an observer with uniform acceleration? The absence of reaction of vacuum against uniformly accelerated motion would suit well the invariance of vacuum fluctuations under a group of transformations corresponding to observers with uniform acceleration as well as uniform velocity. This question has raised and still raises numerous controversial discussions between physicists [31]. To understand this controversy it is necessary to describe what is often called the Unruh effect which predicts that vacuum fluctuations should appear as thermal fluctuations in a uniformly accelerated reference frame [32]. The effective temperature, which is proportional to Planck constant and to acceleration, is analogous to the Hawking temperature of the field radiated by the surface of a black hole [33]. This analogy plays an important role in the arguments pleading in favor of the physical reality of the Unruh effect but it can be questioned as both situations are clearly different within the framework of general relativity. The curvature of space-time is not involved in a change of reference frame whereas it plays a central role in the description of a black hole.
Concerning the problem of relativity of motion, the Unruh effect shows serious drawbacks. If vacuum is really different for accelerated and inertial observers, this certainly makes it difficult, if not impossible, to describe inertia in a consistent quantum formalism. In particular difficulties should arise when analysing situations which involve composed motions. For example a motion with uniform velocity in an accelerated frame should give rise to a vacuum reaction like uniform velocity in an ordinary thermal field whereas a uniformly accelerated motion in a frame obtained from a Lorentz transformation would be free from dissipation. Contradictions are also met concerning energy conservation for an accelerated detector. Furthermore, the interpretation of detection processes appear to depend on the reference frame. As far as this point is concerned different results have been obtained depending on the model used for the detector [31]. For those models involving detectors which click when accelerated in vacuum it can even happen that the detection process is found to be related now to absorption then to emission of a photon. More generally it is impossible to preserve the usual understanding of the principle of relativity because the notions of vacuum or photon number are not the same for different observers [34].
These difficulties have their origin in a too rapid identification of uniformly accelerated reference frames with their Rindler representation. This particular representation of accelerated frames favors a criterion of rigidity in the transformation of solid bodies. A consequence of this choice is that Maxwell equations are modified in the change of reference frame. The Unruh effect essentially tells us that the definition of vacuum is also altered in a Rindler transformation and that the accelerated vacuum obtained after the transformation appears as a thermal field. But the Rindler representation of accelerated reference frames is not the only possible one. In fact an infinite number of representations exist when one only imposes the condition that a given point with uniform acceleration is brought to rest. Furthermore, the theory of general relativity does not provide any manner to privilegiate one particular representation of accelerated frames since it convincingly argues that physical results cannot depend on the choice of a particular map of spacetime.
In quantum theory in contrast, motion must be understood as taking place in a space filled with vacuum fluctuations. This feature implies to attach a particular importance to those transformations to accelerated frames which leave vacuum invariant. This leads to favor conformal coordinate transformations which have been known for a long time to leave Maxwell equations invariant [35]. These tranformations represent a natural extension of Lorentz transformations which also include uniformly accelerated frames [36]. An essential property of conformal transformations is to preserve the definition of vacuum fluctuations [37] and, more generally, the definition of particle number [38]. Hence there is no more difference between the viewpoints of accelerated and of inertial observers as far as their perception of vacuum is concerned, provided that accelerated observers are defined through conformal transformations. The fact that vacuum exerts no force on a uniformly accelerated object, already discussed for electrons and for mirrors, is in fact a direct consequence of this symmetry property of quantum vacuum. In this sense, the principle of relativity which was known for uniform velocities now has a domain of application extended to uniform accelerations.
In the conformal representation of accelerated frames, the Unruh effect disappears as well as the paradoxes it creates. A deep reason for the difficulties mentioned above is that Rindler transformations do not form a group. In fact they do not even compose well with Poincaré transformations. But the discussion of the principle of relativity is mainly based on group properties obeyed by the composition of motions and these argu-ments do not hold for Rindler transformations. The situation is much more favorable for conformal coordinate transformations which form a group including the Poincaré transformations besides the conformal changes to accelerated frames. In particular, the composition of a uniformly accelerated motion with an inertial motion results in another uniformly accelerated motion. Associated with preservation of Maxwell equations and of electromagnetic vacuum, these group properties amply justify the use of conformal transformations to represent uniformly accelerated reference frames. The conformal representation is the natural choice when relativistic properties are interpreted as invariance properties rather than mere form-invariance relations [39].
Furthermore, a clear understanding of the notion of quantum particles requires that the notion of vacuum has been understood before. This point has been thoroughly discussed in the present paper for what concerns the principle of relativity of motion. It must equally be taken into account when considering the extension of the equivalence principle to the quantum domain. The thoughts presented here plead for bringing out the symmetries of vacuum, the quantum version of empty space, in the forefront of primary questions. As it has been shown here, this should ensure compatibility of the conception of motion with the principle of relativity. It has also been shown elsewhere that this approach fits perfectly well the description of localization in space-time built upon symmetries of quantum fields. Precisely, conformal symmetry allows to define quantum observables associated with positions in space-time and to obtain their transformations under Lorentz transformations and transformations to accelerated frames [40].
It has often been noted that the serious problems arising in the mechanical description of electrons is directly associated with the fact that they are treated as pointlike structures although such a treatment certainly contradicts their quantum nature. More generally, the classical conception of space as a set of points upon which classical relativity is built with the help of differential geometry is challenged by the quantum nature of physical observables. The design of new conceptions of motion and localization in space could therefore reveal itself a preliminary step to the solution of yet unsolved problems lying at the interface between quantum theory and gravitation.
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. M T Jaekel, S Reynaud, Phys. Rev. Lett. 762407EuroPhys. Lett.. Foundations of Physics to appearJaekel M. T. and Reynaud S. 1996 Phys. Rev. Lett. 76 2407; Phys. Lett. A220 10; 1997 EuroPhys. Lett. 38 1; 1998 Foundations of Physics to appear
| {'fraction_non_alphanumeric': 0.028786075572746206, 'fraction_numerical': 0.021719726271942873, 'mean_word_length': 4.568548943807815, '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': 'b Laboratoire Kastler Brossel † , UPMC case 74, Jussieu, F75252 Paris Cedex 05 (Contribution for Vacuum, eds E. Gunzig and S. Diner)The principle of relativity is one of the main general laws of physics. Since it applies in particular to motion in empty space it is related to the symmetries of this space. Thoughts about this subject within the framework of classical physics have led to the theory of relativity [1]. The emergence of quantum theory has then profoundly altered our conception of empty space by forcing us to consider vacuum as the realm of quantum field fluctuations. The notion of quantum vacuum and the existence of vacuum fluctuations unavoidably lead us to reconsider the question of relativity of motion. The present article is devoted to this aim with a main line which can be formulated as follows: "The principle of relativity of motion is directly related to symmetries of quantum vacuum". Keeping close to this statement, we discuss the controversial relation between vacuum and motion. We introduce the question of relativity of motion in its historical development before coming to the results obtained more recently.Movement in space cannot be defined otherwise than movement in vacuum. This assertion was formulated by Leucippus and Democritus more than 2000 years ago. In the context of that time, it has to be understood as a logical necessity rather than a physical statement in the modern sense. The existence of movement is a matter of evidence which forces us to conceive a space in which movement takes place [2]. The absence of resistance to motion is a key condition in this respect and in fact constitutes the main argument for the definition of the concept of vacuum. At the same time vacuum is a positive physical reality which clearly differs from nothingness. It is in particular the reference with respect to which movement has to be identified. These properties are certainly paradoxical at first sight. This paradox has raised numerous discussions and it has conserved its whole pertinence ever since the birth of modern physics until today.After a long eclipse dominated by Aristotelian concepts, the question of relativity of motion was brought anew to the fore by Galileo [3]. His central argument, "Movement is like nothing", not only signifies that a motion with uniform velocity is indistinguishable from rest. It also implies that a motion with uniform velocity has no', 'arxivid': 'quant-ph/9801071', 'author': ["Marc-Thierry Jaekel \nLaboratoire de Physique Théorique de l'ENS *\n24 rue LhomondF75231, Cedex 05Paris\n", 'Astrid Lambrecht ', 'Serge Reynaud ', "\nLaboratoire du CNRS associéà l'Ecole Normale Supérieure età l'Université Paris-Sud † Laboratoire de l'Ecole Normale Supérieure et de l'Univer\n\n"], 'authoraffiliation': ["Laboratoire de Physique Théorique de l'ENS *\n24 rue LhomondF75231, Cedex 05Paris", "Laboratoire du CNRS associéà l'Ecole Normale Supérieure età l'Université Paris-Sud † Laboratoire de l'Ecole Normale Supérieure et de l'Univer\n"], 'corpusid': 118245220, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 10047, 'n_tokens_neox': 8570, 'n_words': 6411, 'pdfsha': 'c0535a1d74e340013cc483c8dd1ad50b42d93c4f', 'pdfurls': ['https://arxiv.org/pdf/quant-ph/9801071v1.pdf'], 'title': ['Relativity of motion in vacuum', 'Relativity of motion in vacuum'], 'venue': []} |
arxiv |
Learning More Discriminative Local Descriptors for Few-shot Learning
Qijun Song
Siyun Zhou
Liwei Xu
Learning More Discriminative Local Descriptors for Few-shot Learning
Few-shot learning for image classification comes up as a hot topic in computer vision, which aims at fast learning from a limited number of labeled images and generalize over the new tasks. In this paper, motivated by the idea of Fisher Score, we propose a Discriminative Local Descriptors Attention (DLDA) model that adaptively selects the representative local descriptors and does not introduce any additional parameters, while most of the existing local descriptors based methods utilize the neural networks that inevitably involve the tedious parameter tuning. Moreover, we modify the traditional k-NN classification model by adjusting the weights of the k nearest neighbors according to their distances from the query point. Experiments on four benchmark datasets show that our method not only achieves higher accuracy compared with the state-of-art approaches for few-shot learning, but also possesses lower sensitivity to the choices of k.
Introduction
Image classification is an important research area in computer vision. Much of the related works rely on collecting and labeling a large amount of data, which is often very difficult and expensive. In addition, such image classification mechanisms are quite different from the human discrimination of images that enables to recognize kinds of targets given only a single image from some certain class [9]. These call for not merely an adequate reduction in the sample size for learning, but also the ability of imitating the intelligent human behavior in image discrimination. In this context, there is increasing concern about the few-shot learning, which can be generally categorized into three types: meta-learning based methods [1,4,15,18,19], data augmentation methods [3,14,17,21,24,26,30] and metric-learning based methods [7,10,12,22,26,29,32]. The meta-learning methods aim to provide a model for adjusting the parameters, and thus it can quickly learn new tasks based on the acquired knowledge, while the data augmentation methods pay more attention to the the limited number of the available data in the few-shot learning. And the metric-based learning methods first map the original images to a high-dimensional semantic space, and then compute the distance between the samples in the support set and the ones in the query set to do the classification tasks.
In this work, we concentrate on the metric-based learning methods. The literatures [7,22,26] all use the image-level features for classification, and the only difference is the metric adopted. However, due to the small sample size of each class, the image-level features based methods may not be effective. On this account, the Deep Nearest Neighbor Neural Network (DN4) model [12] uses the local descriptors of the image to learn an image-to-class metric and consequently, the model is more capable of catching the discriminative information of the image. Recently, Zheng et al. [32] apply the bi-directional local alignment methods to the DN4 model and attain better performance. Note that in these methods, not all local descriptors are well-valued in the classification task. For this point, the work [10] uses the Convolutional Neural Networks (CNNs) to generate weights for all local descriptors such that the representative local features can be emphasized. However, there still exist two major problems to be solved: most of this type methods are based on (i) the CNNs for highlighting the representative local descriptors, which introduces extra model parameters, and thus increases the model complexity and the cost for parameter adjustment; (ii) the k-NN or variants thereof for classification, which is usually sensitive to k.
To address the above two limitations, in this paper, we propose a Discriminative Local Descriptor Attention (DLDA) model and an improved k-NN based classification model. As can be seen from Fig. 1, all images are first put into a feature embedding model, and then move to the DLDA model. Correspondingly, the DLDA model produces an attention graph for each image, where the discriminative local descriptors are fully valued. This not only stresses the representative local features of each class, but also weakens the effect of noise in the classification. The proposed DLDA model is non-parametric, and does not change the size of the feature map. In the last stage, the images get into the improved k-NN based classification model, where the k-NN algorithm finds the nearest k neighbors for each local descriptor in the query set. Then we assign greater weights to those k neighbors that are more concentrated, and thus the impact of k can be reduced. Figure 1: A flow chart of the proposed few-shot learning method for the 5-way 1-shot classification. The learning framework is composed of three models: (i) a feature embedding model implemented on a CNN to extract the local descriptors of images; (ii) a discriminative local descriptors attention model to highlight the representative local features of each class of the images in the support set; (iii) an improved k-NN based classification model to measure the similarity between the images in the query set and each class in the support set The rest of this paper is organized as follows. An overview of the related works in few-shot learning is provided in Section 2. The details of the proposed method are presented in Section 3, with extensive numerical experiments following in Section 4. The conclusions of this work is finally given in Section 5.
Related work
In the following, we will provide a relatively comprehensive review of existing works on the few-shot learning, which can be classified primarily into three categories as mentioned before. Meta-learning based methods. Meta-learning [20], also known as "learning to learn" [25], draws on the previous experience to help understand the new tasks. Finn et al. [1] propose a Model-Agnostic Meta-Learning (MAML) algorithm to obtain a universal initialization strategy such that the model can converge within just a few number of iterations when faced with a new task. However, Jamal et al. [4] hold the view that such uniformly-initialized model of the meta-learner can not be well adapted to the new task due to the differences among tasks and in this regard, they design several Task-Agnostic Meta-Learning (TAML) algorithms to prevent the model from over-executing on some certain tasks. For the same purpose of getting a better initialization, Ravi et al. [18] present a Long Short Term Memory (LSTM) based meta-learner model to optimize another neural network classifier. Similarly, by taking the advantage of an LSTM-based meta-learner to have access to external memory, Santoro et al. [19] develop a memory-enhanced neural network. Although the meta-learning type approaches do spur the development of few-shot learning, it remains a tricky issue in training the models that are of complex memory network structures [15]. Data augmentation methods. Data augmentation can effectively alleviate the matter of the limited number of samples in few-shot learning through constructing new samples based on the old ones. Various geometric transformations can be taken to realize data augmentation, such as flipping [17], rotating [26], etc. Meanwhile, another technique called Random Image Cropping and Patching (RICAP) [24] randomly crops four images and patches them to generate new training images, which owns the advantages similar as the label smoothing. In [30], Zhang et al. propose the mixup method that trains a neural network using the new samples formed by the convex combinations of the samples and their labels. By the use of the Delta encoder, the work [21] is conducted to the extraction of transferable intra-class deformations between the training samples of the same class, and hence the goal of sample synthesis for an unseen class with only few provided samples can be achieved. Besides, Mehrotra et al. [14] apply the generative adversarial networks [3] into the few shot learning for data augmentation, and provide the generating adversarial residual pairwise networks for the one-shot learning problem. Metric-learning based methods. Metric-learning approaches are intended for choosing a metric of similarity which computes the distance between the samples in the query set and each class in the support set. They can be roughly divided into two categories: (i) image-level features based methods; (ii) local descriptor features based methods. Among the image-level features based methods, Siamese neural network [7] extracts the features from two given images and takes the weighted L 1 -norm to measure the distance between two feature vectors, while the matching network [26] makes good use of LSTM to enhance the network and improve the learning ability. Besides, the prototypical network [22] is another popular approach based on the image-level features, where the classification is implemented by calculating the Euclidean distance of the class prototypes in the learned embedding space. Among the local descriptor features based methods, as mentioned earlier, Li et al. [12] propose a DN4 model to learn an image-to-class measurement. Later, Zhang et al. [29] investigate the structural distance between local feature representations by using the Earth Movement Distance (EMD) to acquire the image correlation. Most recently, a Bi-Directional Local Alignment (BDLA) method is presented in [32], which designs a convex combination of the bidirectional distances between the query point and the classes of the support set for classification module. In addition, to reduce the influence of noises and obtain more representative local descriptors, Li et al. [10] develop a More Attentional Deep Nearest Neighbor Neural Network (MADN4), where the convolutional block attention module is employed for the local descriptor extraction.
The proposed method
Problem statement
In the standard few-shot learning, we are given three datasets:
(i) a training set D = (x i , y i ) N i=1
with samples x i (i = 1, 2, · · · , N ) and corresponding labels y i (i = 1, 2, · · · , N ); (ii) a support set
S = (x i , y i ) M K i=1
, where M represents the number of classes contained and K is the number of samples in each class; (iii) a query set Q, which is comprised of the samples sharing the same classes as the set S but all unlabeled. And the label space of D and the one of S are disjoint, thus the sets satisfy Q ∩ D = S ∩ D = ∅. The goal of few-shot learning is to classify the samples in the query set Q according to the support set S, and this problem is referred to as "M -way K-shot classification". Since the number of the samples in S is limited, we use the abundant samples in the training set D to learn the transferable knowledge such that the classification performance of the model on Q can be improved.
Following the episode training mechanism in [26], it is an effective way to make full use of the training set D. To achieve this, we construct multiple episodes to train our model. In each episode, we randomly select M classes of samples and K samples belonging to each class from the training set D, which form the training support set D S , and take the remaining samples of these M classes in D as the training query set D Q . Once all training episodes have been completed, we use the fully trained model to classify the query set Q according to the support set S in the testing stage.
Framework of the proposed method
As can be seen in Fig. 1, our few-shot learning method mainly consists of three parts: a feature embedding model, a discriminative local descriptors attention model, and a classification model. Following common practice, the feature embedding model is composed of a CNN, which is used for feature extraction of images in the support set and the query set. To emphasize discriminative local descriptors in the image, we introduce an additional attention mechanism model as an intermediate processing, and to improve the classification performance, we incorporate a modified k-NN into the last stage, i.e., the classification model, which calculates the similarity scores of the query image with each class of images in the support set, and then predicts the query image to be belonging to the class with the highest score.
Feature embedding model
Given an image X, we input it to the feature embedding model, which can be formulated as a mapping F θ with the neural network parameter θ. Then, we get the corresponding output F θ (X) ∈ R h×w×d , where h and w represent the height and width, respectively, and d denotes the number of channels. Note that F θ (X) can be written into the following matrix form
F θ (X) = [x 1 , . . . , x r ] ∈ R d×r ,(3.1)
where r = hw, and x i ∈ R d is called the i-th (i = 1, · · · , r) local descriptor (the red block with dotted lines in Fig. 1) of the feature map of image X. And then we perform a normalization step on each column of F θ (X).
Discriminative local descriptors attention model
In some previous studies [12,29], all local descriptors are treated fairly and some underlying representative information of the image may thus be ignored, as illustrated in Fig. 2. To address this, the work [10] generates the specified weights for local descriptors using CNNs, which however introduces additional parameters that usually requires careful tuning, and may even worsen the overfitting issue in few-shot learning. At this point, we propose a discriminative local descriptors attention model Φ, which not only underscores the importance of the discriminative local descriptors in the support set, but also avoids the involvement of the extra parameters. The proposed DLDA model is inspired by the Fisher Score approach [11], where for each local descriptor, the ratio of the intra-class similarity to the inter-class one obtained by k-NN is taken as the weight. Now we take the 5-way 1-shot problem as an example. Given five images X i (i = 1, · · · , 5) belonging to five classes that are selected from the support set, one has the output F θ (X i ) = [x i1 , · · · , x ir ] ∈ R d×r of the feature embedding model F θ . For each local descriptor x ij (j = 1, · · · , r), we find its k nearest neighbors, denoted by x m ij (m = 1, · · · , k), and then return the corresponding cosine similarities. Based on that, the intra-class and inter-class similarities are characterized by the following formulas Then, the weight w ij of x ij in the DLDA model is defined as
intra-class (x ij ) = k m=1 cos x ij , x m ij ,(3.w ij = intra-class (x ij ) inter-class (x ij ) ,(3.4)
which finally leads to the weighted feature map of the image X i aŝ
F θ (X i ) = [w i1 x i1 , · · · , w ir x ir ] [x i1 , · · · ,x ir ] ∈ R d×r . (3.5)
For the 5-way 5-shot case where there are five images per class, we first compute an average feature map of the five ones of each class, and then follow the same steps as the 5-way 1-shot case.
In addition, the DLDA model is only performed on the images in the support set, but not on the ones in the query set. It is also worth noting that the DLDA model is a non-parametric model, which may facilitate the overfitting problem to some extent.
An improved k-NN based classification model
The final stage is for classification, and one can choose any appropriate technique to deal with it. One popular choice is the k-NN approach [12,10], which however implicitly assumes that the k nearest neighbors are of equal importance in the classification decision regardless of their distances from the query point. To remedy this, we slightly modify the PNN method [28] for the final classification, where different weights are assigned based on the distance from the query point. The details of the proposed modified k-NN based classification model are described as follows.
Given an image Y in the query set, we denote the corresponding output from the feature embedding model as F θ (Y ) = [y 1 , . . . , y r ] ∈ R d×r . For each descriptor y s (s = 1, · · · , r), we find its k nearest neighborsx 1 ij , · · · ,x k ij in class i, and compute the corresponding cosine similarity as cos(y s ,x 1 ij ), · · · , cos(y s ,x k ij ). According to the basic idea that a larger cosine similarity means a smaller distance, and therefore a greater weight should be assigned, we then give the formula of the weights for the k nearest neighbors of y s as follows w sn = cos(y s ,x n ij ) k p=1 cos(y s ,x p ij )
, n = 1, · · · , k,
where cos(y s ,x n ij ) is assumed to be positive for all 1 ≤ n ≤ k. Such assumption is natural and reasonable since the parameter k is often set to a small integer, e.g., 1, 3, 5.
Finally, the similarity between image Y and class i is defined as
Similarity(Y, class i) = r s=1 k n=1
w sn cos(y s ,x n ij ), (3.7) which is sum of the rk weighted similarities between r descriptors and their k nearest neighbors.
Less sensitivity to k can be expected
As in the traditional k-NN, all k neighbors are treated equally, and thus the differences in the distances of these k neighbors from the query point are neglected. With this in mind, our proposed improved k-NN assigns a greater weight to the neighbor that is closer to the query point, or to say, attaches greater importance to the neighbor in the final classification.
For illustrative purposes, we consider the simplest k = 2 case. Suppose that the two nearest neighbors of the query point y are x 1 and x 2 , and the relation cos(y, x 1 ) > cos(y, x 2 ) holds. In the traditional k-NN, the percentages of the scores of x 1 and x 2 can be expressed as
x 1 :
cos(y, x 1 ) cos(y, x 1 ) + cos(y, x 2 ) , x 2 : cos(y, x 2 ) cos(y, x 1 ) + cos(y, x 2 )
, (3.8) while in our proposed improved k-NN, the percentages can be written as
x 1 : cos 2 (y, x 1 ) cos 2 (y, x 1 ) + cos 2 (y, x 2 ) , x 2 : cos 2 (y, x 2 ) cos 2 (y, x 1 ) + cos 2 (y, x 2 )
.
(3.9)
Comparing (3.8) and (3.9), we have cos 2 (y, x 1 ) cos 2 (y, x 1 ) + cos 2 (y, x 2 ) − cos(y, x 1 ) cos(y, x 1 ) + cos(y, x 2 ) = cos 2 (y, x 1 )[cos(y, x 1 ) + cos(y, x 2 )] − cos(y, x 1 )[cos 2 (y, x 1 ) + cos 2 (y, x 2 )] [cos 2 (y, x 1 ) + cos 2 (y, x 2 )][cos(y, x 1 ) + cos(y, x 2 )] = cos(y, x 1 ) cos(y, x 2 )[cos(y, x 1 ) − cos(y, x 2 )] [cos 2 (y, x 1 ) + cos 2 (y, x 2 )][cos(y, x 1 ) + cos(y, x 2 )] > 0, which implies that the proposed improved k-NN will enhance the importance of the nearest neighbor and relegate the farthest neighbor to lower importance, and consequently, the final classification will depend more on the nearest neighbor. More specifically, we may encounter two possible situations as shown in Fig. 3 and Fig. 4. When the cosine similarity between x 1 and y is close to the one between x 2 and y, the traditional k-NN and the improved one are nearly equivalent, but when the cosine similarity between x 1 and y is significantly larger than the one between x 2 and y, the improved k-NN may have marked effect on the insensitivity to the choice of k as the farther point x 2 is considered less informative. Therefore from the discussions above, the improved k-NN can behave more stably with respect to different choices of k (k > 1) than the traditional k-NN.
Experimental settings
Network architecture
For the sake of fairness, we adopt the same network structure of other few-shot learning methods that will be used for comparison in our experiment, e.g., [12,32]. To be specific, we use four convolutional blocks to construct the feature embedding model, and each convolutional block contains one convolutional layer with 64 filters of size 3 × 3, one batch normalization layer, and one Leaky ReLU layer. In addition, a 2×2 pooling layer is added to the first two convolutional blocks. In general, this embedding network is referred to as Conv4.
Implementation details
The experiments are implemented by PyTorch [16]. Both the 5-way 1-shot and 5-way 5-shot classifications are considered in the experiments. At the training stage, we construct 600,000 episodes randomly from the training part of the MiniImageNet dataset and 300,000 episodes from the one of the three fine-grained datasets. In each episode, for the 1-shot and 5-shot settings, we choose 1 and 5 support images, 15 and 10 query images from each class, respectively. Take the 5-way 1-shot setting as an example, we have 5 support images and 75 query images in each episode. The Adam optimizer [6] with a cross-entropy loss is used for training. The learning rate is initialized to 0.001, and will be reduced by half every 100,000 episodes. At the testing stage, we construct 600 episodes randomly from the testing part of each dataset. The testing process will be repeated 5 times and the average top-1 accuracy along with the 95% confidence interval will be presented in the results.
Baselines
To prove the feasibility and superiority of our proposed few-shot learning method, for the MiniImageNet dataset, we make comparisons with the following twelve methods: MAML [1], TAML [4], Meta-Learner LSTM [18], MetaGAN [31], GNN [2], TPN-semi [13], Relation Net [23], Matching Net [26], Prototypical Net [22], DN4 [12] and BDLA [32], MADN4 [10]. And for the three fine-grained datasets, we make comparisons with the following five methods: Matching Net [26], Prototypical Net [22], DN4 [12], BDLA [32] and GNN [2].
Results
Comparison on the MiniImageNet dataset
The results on MiniImageNet are given in Table 1. Compared with the classical metric-based methods, in the case of 5-way 1-shot, the proposed DLDA model with k = 1 gains 2.37%, 9.25% and 3.39% improvements over Relation Net [23], Matching Net [26] and Prototypical Net [22], respectively. And for the more recent DN4, our DLDA model improves the accuracy by 1.57% and 0.74% in the 5-way 1-shot and 5-way 5-shot settings, respectively, which suggests that the DLDA model is more able to emphasize the discriminative local features. Moreover, the combination of the DLDA model and the improved k-NN algorithm further enhances the classification accuracy and outperforms all of the previous methods. Note additionally that though the proposed method has similar results as the ones of MADN4 [10], our method does not introduce any new parameters, which thus avoids the tedious parameter tuning and may be beneficial to alleviate the problem of overfitting to some degree.
Comparison on fine-grained datasets
One distinguishing feature of fine-grained datasets is the small inter-class variation but the large intraclass variation, which hence makes the classification much more challenging. As can be seen from Table 2, in the case of 5-way 1-shot, the DLDA improves the accuracy by 4.03%, 1.02%, 8.28% over DN4 in the Stanford Dogs, Stanford Cars and CUB-200, respectively. And the one with the improved k-NN is ahead of DN4 under both of the 1-shot and 5-shot cases on three fine-grained datasets. Particularly, on the Standford Dogs dataset, the accuracy is enhanced by 3.67% and 6.65% in the 1-shot and 5-shot classifications, respectively. Overall, although the DLDA with/without the improved k-NN lags behind the BDLA in the 1-shot case on the Standford Cars dataset, it is safe to say that our method is the winner among all competitors.
Results on the sensitivity to k
Observe from the previous studies on k-NN involved few-shot learning methods, their numerical performance is usually dependent on the choice of k, which requires a large number of experiments such that it is set appropriately. Specifically, from Table 3 we can find that in the case of 5-way 1-shot, the accuracy of the DN4 model drops from 52.35% at k = 1 to 50.31% at k = 7, with a fluctuation of 2.04%, and the one of the BDLA model reaches its peak 52.36% at k = 3 but slides to 45.94% at k = 7, with a fluctuation of 6.42%, while the fluctuation of our proposed DLDA model added with an improved k-NN is a much smaller 0.39%. To address this issue, we provide a visual relationship between the selection of k and the corresponding accuracy in Fig. 5 and Fig. 6, which shows that our method is apparently less sensitive to the choices of k over DN4 and BDLA, both in the cases of 5-way 1-shot and 5-way 5-shot.
Results on cross-domain classification
The cross-domain classification is to generalize the model that is pre-trained on the source domain to the target domain, which appears to be an important indicator of measuring the ability of handling the differences among multiple sources in few-shot learning. In our experiments, considering that the types of the images in the MiniImageNet dataset differ significantly from the ones in the three fine-grained datasets, we use the former dataset as the source domain to train the model and use the latter three datasets as the target domain to test the model. Moreover, we select three models, Prototypical Net [22], DN4 [12] and BDLA [32] for comparison. It can be obviously seen from Table 4 that our method is the best performer under the domain shift. This result indicates that aside from an improved classification performance and higher robustness to the parameter k, our method has a better cross-domain generalization capability over other competitors, which may be because the DLDA model together with the modified k-NN makes the local descriptors extracted by the embedded network more discriminative and transferable.
Conclusions
In this paper, we focus on the few-shot image classification, and develop a new method for this problem, which includes a Discriminative Local Descriptors Attention (DLDA) model and an improved k-NN based classification model. Inspired by Fisher Score, the DLDA model gives a weight to each local descriptor in the support set for highlighting the representative local features before the image-to-class classification. Based on the idea that the value of the information can be partly reflected by the distance, in the final classification, the improved k-NN model assigns larger weights to those local descriptors that are closer to the query point. Extensive experimental results on the benchmark datasets illustrate that, compared with the state-of-the-art few-shot learning methods, the proposed method obtains a higher accuracy and a lower sensitivity to the parameter k, especially on the MiniImageNet dataset. In addition, it also shows to be more capable of dealing with the situation of domain shift.
2 )Figure 2 :
22Consider the 2-way 1-shot case. The two support images are selected from the CUB-200 dataset, which clearly belong to two different breeds of birds. The local features highlighted in the yellow boxes of the two images are quite similar, and thus the query image is hard to be correctly classified according to this type of local features. On the contrary, the local features in the red boxes are more conducive to distinguishing the query image among kinds of images, which naturally play a much more important role in the classification task inter-class (x ij ) =
Figure 3 :Figure 4 :
34The case of k = 2: cos(y, x 1 ) is slightly larger than cos(y, x 2 ) The case of k = 2: cos(y, x 1 ) is much larger than cos(y,x 2 )MiniImageNet[26]: the dataset is composed of 60,000 images selected from the ImageNet dataset, with 100 classes and 600 images per class. Each image is of size 84 × 84. Following the splitting approach in[18], we take 64 classes for training,16 classes for validation and the remaining 20 classes for testing. CUB-200 [27]: the dataset covers 200 bird species and the number of images included in each class varies. We take 130 classes for training, 20 classes for validation and the remaining 50 classes for testing. This dataset is the most widely used benchmark for fine-grained image classification. StanfordDogs [5]: the dataset consists of 120 breeds of dogs with a total of 20,580 images. We take 70 classes for training, 20 classes for validation and the remaining 30 classes for testing. StanfordCars [8]: the dataset contains 16,185 car images of 196 classes, and the classes are mainly derived according to the brand, the model, and the year of manufacture. We take 130 classes for training, 17 classes for validation and the remaining 49 classes for testing. The images in the three fine-grained image classification datasets, i.e., CUB-200, StanfordDogs, and StanfordCars, are resized uniformly to 84×84, which are consistent with the ones in the MiniImageNet dataset.
Figure 5
5Figure 5: 5-way 1-shot classification on the Mini-ImageNet dataset Figure 6: 5-way 5-shot classification on the Mini-ImageNet dataset
Table 1 :
1Average accuracy with 95% confidence intervals on the MiniImageNet datasetModel
Embedding
5-Way Accuracy (%)
1-shot
5-shot
MAML[1]
Conv4-32
48.70 ± 1.84
63.11 ± 0.92
TAML[4]
Conv4
51.73 ± 1.88
66.05 ± 0.85
Meta-Learner LSTM[18]
Conv4-32
43.44 ± 0.77
60.60 ± 0.71
MetaGAN[31]
Conv4-32
52.71 ± 0.64
68.63 ± 0.67
GNN[2]
Conv4-64
49.02 ± 0.98
63.50 ± 0.84
TPN-semi[13]
Conv4-64
52.78 ± 0.27
66.42 ± 0.21
Relation Net[23]
Conv4-64
50.44 ± 0.82
65.32 ± 0.70
Matching Net[26]
Conv4-64
43.56 ± 0.84
55.31 ± 0.73
Prototypical Net[22]
Conv4-64
49.42 ± 0.78
68.20 ± 0.66
DN4[12]
Conv4-64
51.24 ± 0.74
71.02 ± 0.64
BDLA[32]
Conv4-64
52.97 ± 0.35
71.31 ± 0.68
MADN4[10]
Conv4-64
53.20 ± 0.52
71.66 ± 0.47
DLDA(k = 1)
Conv4-64
52.81 ± 0.79
71.76 ± 0.66
DLDA+Improved k-NN(k = 3)
Conv4-64
53.20 ± 0.82
71.76 ± 0.47
Table 2 :
2Average accuracy with 95% confidence intervals on the fine-grained datasetsModel
Embed.
5-Way Accuracy (%)
Stanford Dogs
Stanford Cars
CUB-200
1-shot
5-shot
1-shot
5-shot
1-shot
5-shot
Matching Net[26]
Conv4-64
35.80
± 0.99
47.50
± 1.03
34.80
± 0.98
44.70
± 1.03
45.30
± 1.03
59.50
± 1.01
Prototypical Net[22]
Conv4-64
37.59
± 1.00
48.19
± 1.03
40.90
± 1.01
52.93
± 1.03
37.36
± 1.00
45.28
± 1.03
GNN[2]
Conv4-64
46.98
± 0.98
62.27
± 0.95
55.85
± 0.97
71.25
± 0.89
51.83
± 0.98
63.69
± 0.94
DN4(k = 1)[12]
Conv4-64
45.41
± 0.76
63.51
± 0.62
59.84
± 0.80
88.65
± 0.44
46.84
± 0.81
74.92
± 0.62
BDLA[32]
Conv4-64
48.53
± 0.87
70.07
± 0.70
64.41
± 0.84
89.04
± 0.45
50.59
± 0.97
75.36
± 0.72
DLDA(k = 1)
Conv4-64 49.44
± 0.85
69.36
± 0.69
60.86
± 0.82
89.50
± 0.41
55.12
± 0.86
74.46
± 0.65
DLDA+Improved k-NN(k = 3) Conv4-64
49.08
± 0.83
70.16
± 0.67
60.04
± 0.83
89.62
± 0.42
54.53
± 0.85
75.85
± 0.68
Table 3 :
3Average accuracy with different k (k = 1, 3, 5, 7) on the MiniImageNet datasetModel
Table 4 :
4Average accuracy with 95% confidence intervals on the fine-grained datasets using the model trained on the MiniImageNet datasetDataset
Prototypical Net
DN4
BDLA
DLDA+
Improved k-NN
Stanford Dogs
1-shot
33.11 ± 0.64
36.32 ± 0.68
35.55 ± 0.66
37.10 ± 0.70
5-shot
45.94 ± 0.65
53.43 ± 0.71
52.64 ± 0.69
53.99 ± 0.70
Stanford Cars
1-shot
29.10 ± 0.75
30.77 ± 0.57
30.62 ± 0.58
31.48 ± 0.56
5-shot
38.12 ± 0.60
46.93 ± 0.62
45.99 ± 0.61
49.63 ± 0.66
CUB-200
1-shot
39.39 ± 0.68
39.89 ± 0.73
40.40 ± 0.76
41.36 ± 0.74
5-shot
56.06 ± 0.66
59.03 ± 0.71
58.23 ± 0.72
60.02 ± 0.71
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| {'fraction_non_alphanumeric': 0.05972904927466731, 'fraction_numerical': 0.039443711785157654, 'mean_word_length': 4.027848101265823, 'pattern_counts': {'":': 0, '<': 0, '<?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': 3, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'Few-shot learning for image classification comes up as a hot topic in computer vision, which aims at fast learning from a limited number of labeled images and generalize over the new tasks. In this paper, motivated by the idea of Fisher Score, we propose a Discriminative Local Descriptors Attention (DLDA) model that adaptively selects the representative local descriptors and does not introduce any additional parameters, while most of the existing local descriptors based methods utilize the neural networks that inevitably involve the tedious parameter tuning. Moreover, we modify the traditional k-NN classification model by adjusting the weights of the k nearest neighbors according to their distances from the query point. Experiments on four benchmark datasets show that our method not only achieves higher accuracy compared with the state-of-art approaches for few-shot learning, but also possesses lower sensitivity to the choices of k.', 'arxivid': '2305.08721', 'author': ['Qijun Song ', 'Siyun Zhou ', 'Liwei Xu '], 'authoraffiliation': [], 'corpusid': 258685822, 'doi': '10.48550/arxiv.2305.08721', 'github_urls': [], 'n_tokens_mistral': 12846, 'n_tokens_neox': 11122, 'n_words': 6739, 'pdfsha': 'fd7b00d048217d215a8312b38ea7eb35efe89cee', 'pdfurls': ['https://export.arxiv.org/pdf/2305.08721v1.pdf'], 'title': ['Learning More Discriminative Local Descriptors for Few-shot Learning', 'Learning More Discriminative Local Descriptors for Few-shot Learning'], 'venue': []} |
arxiv |
Measuring Self-Supervised Representation Quality for Downstream Classification using Discriminative Features
Neha Kalibhat
Department of Computer Science
University of Maryland
College Park 2 Meta AI
Kanika Narang
Department of Computer Science
University of Maryland
College Park 2 Meta AI
Hamed Firooz
Department of Computer Science
University of Maryland
College Park 2 Meta AI
Maziar Sanjabi
Department of Computer Science
University of Maryland
College Park 2 Meta AI
Soheil Feizi
Department of Computer Science
University of Maryland
College Park 2 Meta AI
Measuring Self-Supervised Representation Quality for Downstream Classification using Discriminative Features
* Correspondence to [email protected]
Self-supervised learning has shown impressive results in downstream classification tasks. However, there is limited work in understanding their failure modes and interpreting their learned representations. In this paper, we study the representation space of state-of-the-art selfsupervised models including SimCLR, SwaV, MoCo, BYOL, DINO, SimSiam, VICReg and Barlow Twins. Without the use of class label information, we discover discriminative features that correspond to unique physical attributes in images, present mostly in correctly-classified representations. Using these features, we can compress the representation space by up to 40% without significantly affecting linear classification performance. We then propose Self-Supervised Representation Quality Score (or Q-Score), a model-agnostic, unsupervised score that can reliably predict if a given sample is likely to be mis-classified during linear evaluation, achieving AUPRC of 91.45 on ImageNet-100 and 78.78 on ImageNet-1K. Q-Score can also be used as a regularization term on any pre-trained self-supervised model to remedy low-quality representations. Fine-tuning with Q-Score regularization can boost the linear classification performance of state-of-the-art self-supervised models by up to 5.8% on ImageNet-100 and 3.7% on ImageNet-1K compared to their baselines. Finally, using gradient heatmaps and Salient ImageNet masks, we define a metric to quantify the interpretability of each representation. We show that discriminative features are strongly correlated to core attributes and enhancing these features through Q-score regularization makes representations more interpretable across all self-supervised models.
Introduction
Self-supervised models [13,11,14,26,15,8,31,12,3,49] learn to extract useful representations from data without relying on human supervision, and perform comparably to supervised models in downstream classification tasks. Pre-training these models, however, is extremely time-consuming and resource-intensive. It is therefore crucial that the learned representations are of high quality such that they are explainable and generalizable. However, in practice, these representations are often quite noisy and uninterpretable, causing difficulties in understanding and debugging their failure modes [30,29,23].
In this paper, our goal is to study the representation space of pre-trained self-supervised models such as Sim-CLR [13], SwaV [11], MoCo [14], BYOL [26], SimSiam [15], DINO [12], VICReg [3] and Barlow Twins [49] and discover their informative features in an unsupervised manner. We observe that representations are mostly sparse, containing a small number of highly activating features ( Figure 2). These features can strongly activate a small, moderate or large number of samples in the population. We refer to the moderate category of features as discriminative features.
We observe some intriguing properties of discriminative features: (i) Although discovered without any class label information, they can be strongly correlated to a particular class or group of classes (See Figure 1); (ii) They highlight useful/informative attributes in the activating samples which are often related to the ground truth of those samples; (iii) There are more such features in correctly classified representations than mis-classified representations (as shown in Figure 4) and finally, iv) Representations can be compressed by up to 40% using discriminative features without significantly affecting linear evaluation performance.
Building on these observations, we propose a modelagnostic, unsupervised, sample-wise Self-Supervised Representation Quality Score (Q-Score). A high Q-Score for a sample implies that its representation contains highly activating discriminative coordinates which is a favorable Figure 1. Discriminative Features in Self-Supervised Models: We plot the percentage of highly activating samples for each feature in the SimCLR [13] (ResNet-50 [27]) representation space. The features that show very low or very high percentage activations are nondiscriminative as they likely correspond to very uncommon (lower tail) or very general attributes (upper tail). The features that activate a moderate number of samples (middle portion) are called discriminative features. As shown in the gradient heatmaps, these features encode important physical attributes shared among specific classes. These features play a key role in assessing the quality of self-supervised representations for downstream linear classification tasks.
representation property. We empirically observe that Q-Score can be used as a zero-shot predictor in distinguishing between correct and incorrect classifications for any selfsupervised model achieving AUPRC of 91.45 on ImageNet-100 and 78.78 AUPRC on ImageNet-1K.
We next apply Q-Score as a regularizer and fine-tune pre-trained self-supervised models to improve low-quality representations. Across all baselines, Q-Score regularization improves the linear classification accuracy. On BYOL, we observe the highest improvement of 5.8% on ImageNet-100 and 3.7% on ImageNet-1K. The representations, after regularization, show increased activations for discriminative features ( Figure 4) due to which several previously mis-classified samples get correctly classified with higher confidence.
Finally, we define a metric for quantifying representation interpretability by computing mean Intersection over Union (mIoU) with Salient ImageNet [40] masks used as ground truth. We show that, across all baselines, the discriminative features of self-supervised representations are strongly correlated to core features of Salient ImageNet. We can therefore interpret these features better by potentially correlating their meanings with the worker annotations provided for core features in Salient ImageNet. We also observe that mis-classified representations in our baselines show relatively lower IoU scores with core features compared to correct classifications. However, after Q-score regularization, we observe that the interpretability of mis-classified representations also improves via discriminative features.
We summarize our contributions as follows:
• We study the representation space of self-supervised models and discover discriminative features which often have unique physical meaning and mostly exist in correctly-classified representations. Although these features are discovered without any label information, they do show strong correlation to class labels.
• We introduce Self-Supervised Representation Quality Score (Q-Score), a model-agnostic, unsupervised score, to measure the quality of each learned representation. We empirically observe that the higher the Q-Score, the more likely that the sample will be correctly classified, achieving an AUPRC of up to 91.45 on ImageNet-100 and 78.78 on ImageNet-1K.
• We apply Q-Score as a light-weight regularizer to the self-supervised loss and show that, by improving the quality of low-score samples, we can improve downstream classification accuracy by up to 5.8% on ImageNet-100 and 3.7% on ImageNet-1K. We also define an interpretability metric to show that using the Q-Score regularization produces more explainable representations.
Related Work
Unsupervised methods for classification has been a longstanding area of research, generally involving the use of clustering techniques [5,20,47,4,9,10,28]. Selfsupervised learning, is a recent approach to learn without human supervision by training models to prepare their own labels for each example [5,20,45,19] usually with the help of a contrastive loss. Contrastive learning [1,43,2] usually uses a temperature-controlled cross-entropy loss between positive pairs of similar samples and negative pairs of dissimilar samples. Positive pairs are usually considered as multiple transformations (views) [41] of a given sample using stochastic data augmentation. Through this approach, several state-of-the-art self-supervised techniques [13,11,15,26,14,31] have produced representations that show linear classification accuracy comparable to that of supervised approaches.
Understanding these learned representations is relatively less explored. [30], observes that self-supervised representations collapse to a lower dimensional space instead of the entire embedding space. Other methods [44,46], propose to separate the representation space into variant and invariant information so that augmentations are not task-specific. [25] observes representations across layers of the encoder and compare it to supervised setups. Clustering-based or prototypical-based methods have also been proposed where the representation space is collapsed into a low-rank space [21,32]. [6] uses an RCDM model to understand representation invariance to augmentations. [24] proposes a score based on the rank of all post-projector embeddings that can be used to judge and compare various self-supervised models.
In this work, we focus more on studying representations across correct and incorrect classifications in downstream classification tasks and understanding their properties (without using any labels). We investigate the connection between these unsupervised properties in the representation space and mis-classifications. Unlike [24] which requires computing rank over the entire dataset, our analysis leads to the development of an unsupervised sample-wise quality score which can be used as a regularizer and effectively improve downstream classification performance.
Understanding Representations and their Failure Modes
Let us consider a pre-trained self-supervised model with a ResNet [27] backbone encoder f (.). Given an input sample, x i ∈ R n its representation is denoted by, f (x i ) = h i ∈ R r , where r is the size of the representation space. Our goal is to study this representation space and identify the properties which are essential for better downstream performance.
In the top panel of Figure 2, we visualize the representations of SimCLR pre-trained on ImageNet-1K [38]. Each row denotes the representation vector (h i ) of a random sample drawn from the the ImageNet-1K validation set. There are 2048 columns corresponding to the representation size of a ResNet-50 [27] encoder. In all our analysis, we perform L2 normalization over every representation vector h to ensure fair comparison of features.
First, we study some visual properties of these repre-sentations. We observe that each representation is nearly sparse, i.e., most feature values are close to zero [30]. However, there exists a select few features that are strongly deviated from the remaining features in any given representation. We verify this observation in the second panel of Figure 3 where, we plot the distribution of all the SimCLR features of the same samples as the top panel, as well that of other self-supervised models including, DINO [12], SwaV [11], MoCo [14], VICReg [3] and Barlow Twins [49]. In each distribution, a very large number of features have a magnitude of 0 or very close to 0. In the zoomed version of the same plot, we can see a relatively small number of features that show strong activations. For any given representation h i ∈ R r , we formally define the set of highly
activating features (L i ) as L i := {j : h ij > µ i + σ i },
where µ i and σ i denote the mean and standard deviation of h i respectively and is a hyperparameter that is empirically selected. We use = 4 in our experiments. For every feature j, the percentage of highly activating samples is denoted by, A j = 100 N N i=1 1 j∈Li where N is the size of the population. In Figure 1 top panel, we plot A j for all features j in the SimCLR representation space of the ImageNet-1K validation set. The x-axis is ordered in ascending order of A j . The top panel shows a heatmap of SimCLR representations of random ImageNet-1K samples. In the second panel, we plot the distribution of the features of the visualized samples for various models. We observe that representations are mostly sparse with a small number of strongly activated coordinates.
Discriminative Features
Based on Figure 1, we can define three broad categories of highly activating features: (i) Features that are highly activating across a very small fraction of the population, corresponding to the lower tail features in Figure 1. We take the example of features 621, 251 and 1021 and visualize their highly activating samples and gradient heatmaps (using GradCAM [39]). Since these features activate very Figure 1) and plot their top-1 accuracy for various self-supervised baselines. We compare these results to the baseline and the accuracy on randomly selected features of matching sizes (averaged over 4 random seeds). Classifiers trained using discriminative features consistently outperform those of randomly selected features. We can achieve up to 40% reduction in representations size using discriminative features without significantly affecting the top-1 accuracy.
few samples, they likely correspond to image-specific or uncommon attributes. Such features would also not be useful in classification tasks as these are not shared, class-relevant attributes. (ii) Features that highly activate a very large number of samples in the population i.e, the upper tail features in Figure 1. Like feature 2021 and 401, such features are likely to encode very broad and general characteristics (like texture, color etc.) common to most samples (spanning various classes) and therefore, are not class-discriminative. The third category includes, (iii) Features that are highly activating across a moderate number of samples in the population (i.e. the middle part in Figure 1). These features are most likely to encode unique physical attributes associated with particular classes. For example, feature 98 corresponds to the "stripe" pattern which is an important property of the zebra class. Similarly, feature 970 corresponds to the style of the daisy class, and feature 1266 corresponds to water fountains in different scenes. We refer to this subset of highly activating features as discriminative features. Note that we did not use any label information for this analysis. We can identify discriminative and non-discriminative features in a fully unsupervised manner by simply observing their percentage activations (A). The bar plots in Figure 1, show that these features activate more than 80% of particular classes which confirms that these features are strongly class-correlated.
We justify the described method of selection in Figure 3, where we plot the top-1 accuracy of a linear classifier trained on ImageNet-1K using subsets of discriminative features of varying sizes as chosen from Figure 1 (middle portion). We also plot the top-1 accuracy when random subsets of features are selected. We observe that discriminative features perform significantly better compared to randomly selected features. We also observe that we can reduce the representation size up to 40% using the discriminative features, with minimal reduction in performance. In practice, for a given model and dataset, we find that the discriminative features selected between the 55 th and the 97 th percentile of A (as shown in Figure 1), consistently gives us the best performance. Since our analysis assumes that highly activating features are axis-aligned, we also perform a PCA analysis on the representation space (see Appendix) to partially validate this assumption. We show that the gradient heatmaps of the highly activating features are strongly correlated with that of highly activating PCA features. Moreover, we show that discriminative features and PCA features perform comparably in downstream linear evaluation, up to 40% reduction in representation size.
Mis-classified Representations
We now study how discriminative features play a key role in detecting potential mis-classifications in a fully unsupervised manner, without requiring to train a linear classification head. In Figure 4, we take SimCLR ImageNet-1K representations and visualize the discriminative features.
On the left, we show the average representations of correctly classified samples in a subset of classes, while on the right, we show the same for the mis-classified samples in those classes. The subset of features we display is the same for correct and incorrect classifications.
As we can see, in Figure 4, in the first panel, there is a clear difference between representations of correctly and incorrectly classified examples. Both correct and misclassified representations are nearly sparse, however, the discriminative features are significantly more activated in correct classifications. This is especially interesting because we can visually distinguish between correct and incorrect classifications, just by observing the discriminative features, without using any label information.
The correlation of discriminative features to unique physical attributes as studied in Section 3.1, suggests that their presence may be useful in correctly classifying representations. In Figure 4, our claim is confirmed as we observe that mis-classified representations do not show high activations on these features. Therefore, for any given sample, we can consider discriminative features as strong signals indicating classification outcome. We would like to emphasize that our results only indicate an association between these structural properties and classification accuracy and we do not claim any causal relationship between the two. In the next Section 4, we show that enhancing these properties, through regularization, improves the overall quality of representations, followed by improved classification accuracy and interpretability.
Self-Supervised Representation Q-Score
Our study of learned representation patterns helps us discover discriminative features in an unsupervised manner. Discriminative features commonly activate class-specific attributes and help us visually distinguish between correct and incorrect classifications. We combine these observations to design a sample-wise quality score for selfsupervised representations. Let us define D, such that |D| < r, as the set of discriminative features for a given self-supervised model trained on a given dataset. For the i th sample, we have h i (representation), µ i (mean of h i ), σ i (standard deviation of h i ) and the set of highly activating features L i = {j : h ij > µ i + σ i }, |L i | < r. We define our Self-Supervised Quality Score for sample i as,
Q i := 1 |L i ∩ D| j∈Li∩D (h ij − µ i )(1)
where, L i ∩ D is the set of discriminative features specific to the i th sample. Intuitively, higher Q i implies that the representation contains highly activated discriminative features which are strongly deviated from the mean. Our objective with this metric is to compute a sample-specific score in an unsupervised manner indicating the quality of its representations. Ideally, we would like to argue that samples with higher Q-score have improved representations and thus are more likely to be classified correctly in the downstream task. This is a general score that can be applied to any self-supervised model trained on any dataset. See Appendix for a discussion on Q-Score in supervised models. Next, we measure how effective our score is in differentiating between correctly and incorrectly classified representations in an unsupervised manner. In Figure 5, we plot the Precision-Recall (PR) curve and the Receiver Operating Characteristic (ROC) curve of Q-Score when used as a predictor of classification outcome (correct or incorrect). We show this for SimCLR, SwaV, MoCo, BYOL, DINO and SimSiam for the validation set of ImageNet-100 containing 5000 samples (top panel) and ImageNet-1K containing 50000 samples (bottom panel). We also compute the AUROC (area under receiver operating characteristic curve) and AUPRC (area under precision-recall curve) of these curves. We observe AUPRC up to 91.45 on ImageNet-100 and 78.78 on ImageNet-1K on BYOL. On SimCLR, we observe AUROC up to 73.26 on ImageNet-100 and 65.44 on ImageNet-1K. Based on these results we can conclude that, Q-Score is a reliable metric in assessing the quality of representations, meaning that representations with lower Q-Score (quality), are more likely to be mis-classified.
We now check if promoting Q-Score on pre-trained representations is helpful. To do so, we take state-of-the-art pre-trained self-supervised models and further train them for a small number if iterations with Q-Score as a regularizer. For example, we can apply this regularizer to the SimCLR optimization as follows,
max θ 1 2N 2N i=1 log sim(z i ,z i ) 2N j=1 1 j =i sim(z i , z j ) + λ 1 1 Qi<α (Q i )(2)
where z is the latent vector computed by passing h Table 1. Boosting linear classification performance with Q-Score regularization: We tabulate the top-1 accuracy of linear evaluation on SimCLR [13], SwaV [11], MoCo [14], BYOL [26], DINO [12], SimSiam [15], VICReg [3] and Barlow Twins [49] with and without Q-Score regularized fine-tuning. We also compare with a simple lasso regularization [42]. We observe that Q-Score regularization consistently improves each self-supervised state-of-the-art baseline achieving up to 5.8% relative improvement on ImageNet-100 and 3.7% on ImageNet-1K. through a projector network and sim(.) denotes the exponentiated cosine similarity of the normalized latent vector. α is a threshold with which we select the low-score samples whose Q-Scores should be maximized and λ 1 is the regularization coefficient. In other words the goal of this regularization is to improve low-quality representations, similar to the ones shown in Figure 4, by maximizing their discriminative features for downstream classification.
In practice, directly applying this regularization could lead to a trivial solution where a small set of features gets activated for all samples. This is not a favorable situation because these representations become harder to classify accurately and more importantly, the discriminative features are no longer informative because they are activated for all samples (similar to the upper tail in Figure 1). Such features have significantly large L1 norms across samples compared to the remaining features. Therefore, in our revised optimization, we penalize features that have large L1 norms across samples. Let us denote the representation matrix of a given batch by H ∈ R 2N ×r and H * ,k 1 represents the L1 norm of the k th column (corresponding to the k th feature). Our regularized objective would then be,
max θ 1 2N 2N i=1 log sim(z i ,z i ) 2N j=1 1 j =i sim(z i , z j ) + λ 1 1 Qi<α (Q i ) − λ 2 r k=1 1 H * ,k 1 >β ( H * ,k 1 )(3)
where the threshold β helps us select the uninformative features whose L1 norms should be minimized. In practice, we choose α and β for each batch as the mean values of Q i and H * ,k 1 respectively.
Experimental Setup
Our setup consists of state-of-the-art self-supervised ResNet encoders (f (.)) -SimCLR [13], SwaV [11], MoCo [14], BYOL [26], DINO [12] (ResNet-based), SimSiam [15], VICReg [3] and Barlow Twins [49] that are pre-trained on datasets -ImageNet-1K [38], ImageNet-100 [38]. We use a ResNet-50 encoder for our ImageNet-1K experiments and ResNet-18 encoder for all other datasets. We discover discriminative features for each pre-trained model using the validation set of each dataset. For Q-Score regularization, maintaining the same encoder architecture as the respective papers, we use the LARS [48] optimizer with warmup-anneal scheduling. We fine-tune each pre-trained model with and without Q-Score regularization (controlled by λ 1 and λ 2 ) using a low learning rate of 10 −5 , until convergence (which takes up to 50 epochs). We find that λ 1 = λ 2 = 10 −4 generally works well for fine tuning. We use a maximum of 4 NVIDIA RTX A4000 GPUs (16GB memory) for all our experiments. Using the implementations from solo-learn [18], we have tried to match our baseline numbers as much as possible within the error bars reported in the papers using the available resources. We perform standard linear evaluation by passing frozen pretrained representations through a linear classifier that predicts class labels. For all our gradient heatmap visualizations, we utilize GradCAM [39].
Q-Score Regularization
We tabulate our linear evaluation results of various selfsupervised baselines before and after Q-Score regularization in Table 1. We also include results on lasso (L1) regularization [42] on pre-trained models. Lasso promotes sparsity by minimizing the L1 norm of representations. Q-Score regularization improves the linear probing top-1 accuracy on all of the self-supervised state-of-the-art models. We observe the most improvement on BYOL showing 5.8% increase in accuracy on ImageNet-100 and 3.7% on ImageNet-1K. Lasso regularization shows degraded performance across most models since naively sparsifying representations can lead to loss of information. In contrast, Q-Score regularization promotes highly activating discriminative coordinates which we have shown to be essential for downstream classification. We include more results on CIFAR-10 [34], STL-10 [17] and CIFAR-100 [35] in the Appendix. We also include the results on the transfer performance of discriminative features and Q-Score regularized ImageNet-1K models on unseen datasets in the Appendix. Q-Score is therefore a powerful regularizer that can boost the performance of state-of-the-art self-supervised baselines.
In addition to top-1 accuracy, Q-Score also shows significant improvement in representation quality. In Figure 4, we compare the discriminative features of representa-tions before and after Q-Score regularization. We observe that the magnitude of discriminative features increases with Q-Score regularization on both correct and mis-classified representations. We also observe improved classification confidence as representations become more disentangled (see Appendix for a discussion on this). Our regularization produces better quality representations with clear discriminative features making them more distinguishable across classes and therefore, easier to classify. Due to this, we can attribute the improvement in performance to improved representation quality. Although Q-Score improves accuracy, it does not entirely prevent mis-classifications as misclassifications may occur due to a variety of reasons such as, hardness of samples, encoder capacity, dataset imbalance etc.
Our motivation for using discriminative features as discussed in Section 3 is because -a) they are at clear contrast between correct and incorrect classifications, and b) they show strong correlation to ground truth. We observed in Figure 4 in the baseline, that the discriminative features in correctly classified samples are not strongly activated in mis-classified samples. We now study some mis-classified samples and observe how their features may improve with Q-Score regularization. In Figure 6, we visualize the gradient heatmaps of the discriminative features of some misclassified examples in SimCLR. In the baseline, we observe that discriminative features do highlight portions of the image relevant to the ground truth, however, they may also activate other portions that are not necessarily important (see rock crab and green mamba). These heatmaps reflect low quality representations where the discriminative features are not strongly deviated from the mean. After Q-Score regularization, the maximization of discriminative features also leads to better gradient heatmaps that are more localized and cover almost all important portions of the image relevant to the ground truth. Therefore, these samples get classified correctly with higher confidence after regularization.
Quantifying Representation Interpretability with Salient ImageNet
We have observed that discriminative features in representations correspond to meaningful physical attributes through gradient heatmaps and they play a key role in deciding the downstream classification outcome. In this section, we quantify the interpretability of these features between correct and incorrect classifications. We utilize Salient Im-ageNet [40] as the ground truth baseline to compare our gradient heatmaps with. The Salient ImageNet dataset contains annotated masks for both "core" and "spurious" features extracted from a supervised robust ResNet-50 model for 6858 images spanning 327 ImageNet classes. It also contains some natural language keywords, provided by workers to explain each feature. Core features are those that are highly Figure 6. Discriminative features in mis-classified samples: The discriminative features' heatmaps on the SimCLR (baseline) activate portions that may not be relevant to the image ground truth, leading to incorrect predictions. After Q-Score regularization on these representations, the heatmaps become more localized and less noisy, whilst improving predictions and confidence. correlated with the ground truth of the image, whereas, spurious features are those that activate portions irrelevant to the ground truth. In Figure 7, we study some correct and mis-classified samples in the SimCLR baseline. We plot the gradient heatmaps of the discriminative features (combining each individual feature heatmap) of SimCLR for each respective image. We also plot the core and spurious masks of the same images from the Salient ImageNet dataset. We observe that discriminative SimCLR features mostly capture relevant and defining characteristics of the images, therefore are highly correlated with the ground-truth. Moreover, for every correctly classified image, these heatmaps overlap more with core features than spurious features in Salient ImageNet. discriminative features in mis-classified images also overlap with core features in most cases. Since discriminative features are very closely related (in terms of overlap) to core features, we can potentially explain these features better with the help of worker annotations in Salient Ima-geNet. Therefore, these features can be considered as interpretable.
We quantitatively measure the interpretability of a given representation of a given model by computing the Intersection over Union (mIoU) between the heatmap of discriminative features and the core or spurious mask of that image in Salient ImageNet. We can extend this to measure the overall interpretability of a given model by computing the mean Intersection over Union (mIoU) over the population. For the i th image , we define Ar i as the area of the heatmap of its discriminative features. Let Ar core i and Ar sp i be the area of the core and spurious masks respectively. The mIoU scores are defined as follows, Figure 7. Comparing discriminative features with Salient ImageNet core and spurious features: We compare the gradient heatmaps of discriminative features correct and incorrect classifications of SimCLR on ImageNet-1K with the core and spurious masks of the same images in Salient ImageNet [40]. We observe that discriminative features generally overlap more with core features in Salient ImageNet.
mIoU core = 1 N i s(Ar i ∩ Ar core i ) s(Ar i ∪ Ar core i ) mIoU sp = 1 N i s(Ar i ∩ Ar sp i ) s(Ar i ∪ Ar sp i )
where s(.) calculates the sum of the pixel values of the discriminative features' heatmap in the given area. Higher mIoU core % indicates that, on an average higher percentage of the feature heatmap overlaps with the annotated core region, meaning that the model features are more interpretable.
In Figure 8, we show that for all self-supervised baselines, mIoU core > mIoU sp for both correct and incorrect classifications which confirms that discriminative features generally encode important and core attributes over the whole population. Among correct and mis-classified samples in the baselines, we observe that the mIoU core of correct classifications is higher than mis-classifications. This aligns with our observations in Figure 6, which shows that discriminative features in mis-classified samples may not be strongly deviated from the mean and therefore, may correspond to less important portions of the image. After Q-Score regularization, we observe an increase in mIoU core for both correct and mis-classified samples compared to the baseline. This shows that our regularization which enhances discriminative features produces better gradient heatmaps which are more overlapped with core portions of images and therefore, improves the overall model interpretability.
Conclusion
We studied the representation space of self-supervised models to identify discriminative features, a subset of features that correspond to unique physical attributes and show . mIoU scores with core and spurious Salient ImageNet features: We compute the mean mIoU core and mIoU sp scores of SSL baselines (using their discriminative features) before and after Q-Score regularization. We observe that discriminative features for all models generally show higher % IoU with core features than spurious features. Mis-classified representations show relatively lower % IoU with core features. After Q-score regularization, we observe that mIoU core generally improves for both correct and mis-classified representations.
strong class-correlation although they are selected in a fully unsupervised manner. Using discriminative features, we can compress the representation space by up to 40% without affecting the downstream linear evaluation performance to a large extent. Moreover, discriminative features are more strongly activated in correct classifications compared to mis-classified ones. Building on these observations, we define an unsupervised sample-wise score, Self-Supervised Representation Quality Score (Q-Score) that is effective in determining how likely samples are to be correctly or incorrectly classified. With the help of Q-Score regularization during pre-training, we remedied low-quality samples by improving their Q-Scores, thereby, improving the overall accuracy of state-of-the-art self-supervised models on ImageNet-1K by up to 3.7%. We also quantify representation interpretability with an IoU metric using Salient Ima-geNet masks as ground truth. With this metric, we confirm that highly activating features are more overlapped with core attributes. We also observed that regularization improves over model interpretability due to enhancing highly activating features. Our paper poses important questions for future studies such as: 1) what are the causes of misclassifications, apart from representation quality, 2) how can we better explain self-supervised features, given that there are no labels, 3) how can we utilize the better representations space for other tasks besides classification. In the first two plots, we compute the ROC and PR curves (similar to Figure 5) of Q-score on the representations of a supervised ResNet-18 model and a robust ResNet-18 trained on ImageNet-100. In the last two plots, we show the same for ResNet-50 trained on ImageNet-1K. We observe that robust ResNet performs better for Q-score when used as a predictor for correct or mis-classified representations. Figure 4, we observe that the regularization enhances discriminative features, thereby leading to an improvement in performance. Average |Li| (left) and classification confidence (right) before and after regularization: On the left we plot the average value of |Li| (number of highly activating features) and on the right we plot the average classification confidence over the population of ImageNet-1K. We observe that both the number of highly activating features and classification confidence consistently improve on every self-supervised baseline with Q-Score regularization. This improvement is due to the nature of Q-Score regularization which maximizes highly activating discriminative features over the course of pre-training leading to a higher number of such features and improved classification confidence.
A.7. More Gradient Heatmaps of SimCLR
In Figures A.8, A.9, A.10 and A.11, we plot more heatmaps of highly and lowly activating features of SimCLR for 4 different ImageNet-1K classes. We observe that the highly activating features correspond to unique physical properties that are correlated with the ground truth, whereas, lowly activating features, map to spurious portions that do not contribute to useful information. We plot the gradient heat maps of the top activating discriminative feature (by magnitude) for the given class and a lowly activating feature of the same class. We observe that discriminative features are more correlated with ground truth labels compared to lowly activating features in both correct and incorrect classification. The discriminative feature in correct classifications correspond to a unique physical attribute that may not exist (or be obfuscated) in mis-classified images. We plot the gradient heat maps of the top activating discriminative feature (by magnitude) for the given class and a lowly activating feature of the same class. We observe that discriminative features are more correlated with ground truth labels compared to lowly activating features in both correct and incorrect classification. The discriminative feature in correct classifications correspond to a unique physical attribute that may not exist (or be obfuscated) in mis-classified images. We plot the gradient heat maps of the top activating discriminative feature (by magnitude) for the given class and a lowly activating feature of the same class. We observe that discriminative features are more correlated with ground truth labels compared to lowly activating features in both correct and incorrect classification. The discriminative feature in correct classifications correspond to a unique physical attribute that may not exist (or be obfuscated) in mis-classified images. We plot the gradient heat maps of the top activating discriminative feature (by magnitude) for the given class and a lowly activating feature of the same class. We observe that discriminative features are more correlated with ground truth labels compared to lowly activating features in both correct and incorrect classification. The discriminative feature in correct classifications correspond to a unique physical attribute that may not exist (or be obfuscated) in mis-classified images.
Figure 2 .
2Visualizing the self-supervised representation space:
Figure 3 .
3Linear classification accuracy on discriminative features: We train linear classifiers after selecting subsets of discriminative features of various sizes (middle portion of
Figure 4 .
4Comparing correct and mis-classified representations: In these heatmaps, we visualize the discriminative features of average SimCLR representations of several ImageNet-1K classes -correct (left) and incorrect (right) classifications. In the baseline, we observe that discriminative features are strongly activated only in correctly classified representations. Q-Score regularization improves discriminative features' activations, even in mis-classified representations.
Figure 5 .
5Precision-Recall and ROC curves of Q-Score: We measure the effectiveness of Q-Score when used as a predictor in distinguishing between correct and mis-classified representations on ImageNet-100 and ImageNet-1K on each self-supervised model. Q-Score shows an AUPRC of up to 91.45 on ImageNet-100, 78.78 on ImageNet-1K and AUROC of 73.26 on ImageNet-100, 65.44 on ImageNet-1K.
Figure 8
8Figure 8. mIoU scores with core and spurious Salient ImageNet features: We compute the mean mIoU core and mIoU sp scores of SSL baselines (using their discriminative features) before and after Q-Score regularization. We observe that discriminative features for all models generally show higher % IoU with core features than spurious features. Mis-classified representations show relatively lower % IoU with core features. After Q-score regularization, we observe that mIoU core generally improves for both correct and mis-classified representations.
Figure A. 1 .Figure A. 4 .
14Hyper-parameter Search on λ1 and λ2: We set λ2 = 0 and search across various values of λ1 to find the best performing experiment. Next, we set λ1 to the best performing value and search over λ2. Precision-Recall and ROC curves of Q-Score on supervised setups:
Figure A.7. Average |Li| (left) and classification confidence (right) before and after regularization: On the left we plot the average value of |Li| (number of highly activating features) and on the right we plot the average classification confidence over the population of ImageNet-1K. We observe that both the number of highly activating features and classification confidence consistently improve on every self-supervised baseline with Q-Score regularization. This improvement is due to the nature of Q-Score regularization which maximizes highly activating discriminative features over the course of pre-training leading to a higher number of such features and improved classification confidence.
Figure A. 8 .
8Heatmaps of discriminative and lowly activating features of SimCLR (Class -Ski Mask):
Figure A. 9 .
9Heatmaps of discriminative and lowly activating features of SimCLR (Class -Park Bench):
Figure A. 10 .
10Heatmaps of discriminative and lowly activating features of SimCLR (Class -Head Cabbage):
Figure A. 11 .
11Heatmaps of discriminative and lowly activating features of SimCLR (Class -Dutch Oven):
Table A . 2 .
A2Transfer learning performance of various state-of-the-art self-supervised models trained on ImageNet-1K with and without Q-Score regularization: We observe that fine-tuning with Q-Score regularization improves the average transfer accuracy on all self-supervised models.Figure A.6. Comparing correct and mis-classified representations in Flowers dataset: In these heatmaps, we visualize the discriminative features of several Flowers [37] dataset samples. In the top panel, we display the correct (left) and incorrect (right) classifications of SimCLR (trained on ImageNet-1K) and in the bottom panel, we visualize the same when pre-trained using Q-Score regularization. Similar to the observations inTransfer
SimCLR
SwaV
MoCo
BYOL
DINO
SimSiam
VICReg
Barlow Twins
A.4. Q-Score on Supervised LearningWe note that we select discriminative features and compute Q-Score on self-supervised representations without using any label information. Thus, our study to show correlation between Q-score and classification outcome is non-trivial since self-supervised models learn without labels. Nevertheless, we have included an experiment inFigure A.4, where we analyze Q-score as a predictor of classification outcome (correct vs incorrect) on supervised ResNet-18 (ImageNet-100) and ResNet-50 (ImageNet-1K) representations as well as their robust versions (l2 threat model). Self-supervised representations generally perform better than supervised representations on Q-score indicating that the representational properties we have identified may be mainly prominent in self-supervised learning. We observe that non-robust supervised ResNet shows lower AUROC and AUPRC compared to robust ResNet on both ImageNet-100 and ImageNet-1K setups. This is in line with observations in[22]and[40]that show that robust models provide better axis-alignment of features.A.5. Transfer Performance of Q-Score RegularizationInTable A.2, we tabulate the transfer learning performance (linear evaluation) of various unseen datasets[34,35,17,36,37,7,33,16]on 6 self-supervised models trained on ImageNet-1K with and without Q-Score regularization. We use frozen ResNet-50 representations for each transfer dataset (using actual image size) and perform linear evaluation using a classifier. We observe that the average accuracy of unseen datasets improves on all setups, especially on SimCLR, SwaV and MoCo.InFigure A.5, we visualize the gradient heatmaps of some discriminative features discovered on SimCLR on ImageNet-1K on both ImageNet-1K and unseen datasets, Aircraft[36], Food[7]and Cars[33]. We observe that the physical meaning associated with each feature is consistent between both the training and unseen data. The heatmaps also correspond to informative features, strongly correlated with the ground truth. These gradients indicate that discriminative features areA. Appendix A.1. Results on Other DatasetsAs an extension to the results shown inTable 1, we include results on more datasets including CIFAR-10[34], CIFAR-100[35]and STL-10 [17] on 8 self-supervised baselines when fine-tuned (further trained) with and without Q-Score regularization. InTable A.2, we observe that Q-Score regularization helps boost the performance of all state-of-the-art models across datasets.A.2. Ablation on λ 1 and λ 2In this section, we discuss how we can perform a hyper-parameter search on λ 1 and λ 2 to find the best performing pair of values. We take the baseline of SimCLR trained on ImageNet-1K and further train this model under the setup outlined in Section 4.1. We train keeping both λ 1 = λ 2 = 0 and run experiments by gradually increasing λ 1 to find the best performing value. Next, we search over λ 2 keeping the best performing value of λ 1 . In these experiments we find that λ 1 = λ 2 = 10 −4 is the best performing pair. We find that this pair shows improved performance across most experiments. Due to lack of resources, we do not heavily tune these hyper-parameters in our experiments, however, we can expect improved performance if tuning is performed. We also plot the discriminative PCA features for the same images. We observe that both sets of features activate the same portions of the images meaning that discriminative features can we viewed as axis-aligned.A.3. Discriminative Features and Principal ComponentsIn our analysis, we select discriminative features independently and observe their heatmaps and activations across the population. Our analysis is based on the assumption that axis-aligned features can provide meaningful information regarding the quality of the feature representations for self-supervised models. To (partially) validate this assumption, we have conducted a PCA analysis where we select principal components of feature representations and perform linear evaluation on top of them. InFigure 3, we observe that, until 40% reduction of the representation size, PCA and (axis-aligned) discriminative features perform comparably in terms of the linear classification accuracy while discriminative features significantly outperforms random features across the board. We also plot the gradients of the highly activating PCA features and compare them to discriminative features in the full representation space inFigure A.3. We observe that both sets of features activate the same portions of the images between both correct and incorrect classifications. These results indicate that axis-aligned discriminative features capture a fair amount of information in the feature representations and thus (partially) validating our underlying assumption.
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Linear classification performance with Q-Score regularization (more datasets): Similar to Table 1, we tabulate our results on CIFAR-10. A Table, CIFAR-100 [35] and STL-10 [17] Model CIFAR-10 CIFAR-100 STL-101Table A.1. Linear classification performance with Q-Score regularization (more datasets): Similar to Table 1, we tabulate our results on CIFAR-10 [34], CIFAR-100 [35] and STL-10 [17] Model CIFAR-10 CIFAR-100 STL-10
Baseline Lasso Q-Score Baseline Lasso Q-Score Baseline Lasso Q-Score. Baseline Lasso Q-Score Baseline Lasso Q-Score Baseline Lasso Q-Score
. Imagenet-1k, 15Wing Class) Aircraft FeatureImageNet-1K (Wing Class) Aircraft Feature: 15
. Imagenet-1k, Chocolate SauceImageNet-1K (Chocolate Sauce)
Imagenet-1k, Pickup) Cars Feature. 31ImageNet-1K (Pickup) Cars Feature: 31
We observe that discriminative features correspond to the same physical attributes as the training data and are core and informative. transferable across unseen datasets, which support the improvement we observe in Table A.2. We also visualize the representations of correct and incorrect classifications of the Flowers [37] dataset in Figure A.6. We use SimCLR pre-trained on ImageNet-1K (top panel) and the same model pre-trained with Q-Score regularization (bottom panel). We observe that the same properties as Figure 4 on ImageNet-1K (train dataset) transfer at test time to Flowers, an unseen dataset. Before regularization, representations, especially the mis-classified ones. Figure A.5. Discriminative features on unseen datasets: We visualize the discriminative features discovered on ImageNet-1K classes on unseen datasets like Aircraft. 36do not contain highly activating discriminative features. These features get more enhanced after Q-Score regularization leading to improved top-1 accuracy as shown in Table A.2Figure A.5. Discriminative features on unseen datasets: We visualize the discriminative features discovered on ImageNet-1K classes on unseen datasets like Aircraft [36], Food [7] and Cars [33]. We observe that discriminative features correspond to the same physical attributes as the training data and are core and informative. transferable across unseen datasets, which support the improvement we observe in Table A.2. We also visualize the representations of correct and incorrect classifications of the Flowers [37] dataset in Figure A.6. We use SimCLR pre-trained on ImageNet-1K (top panel) and the same model pre-trained with Q-Score regularization (bottom panel). We observe that the same properties as Figure 4 on ImageNet-1K (train dataset) transfer at test time to Flowers, an unseen dataset. Before regularization, representations, especially the mis-classified ones, do not contain highly activating discriminative features. These features get more enhanced after Q-Score regularization leading to improved top-1 accuracy as shown in Table A.2.
we plot the mean of |L i | (left), i.e., number of highly activating features in the i th sample, and the mean linear classification confidence (right) over the population for each self-supervised model pre-trained with and without Q-Score regularization. We observe an increase in the average number of highly activating features (L i ) and as a result. A.6. Q-Score and Classification Confidence In Figure A.7an improvement in classification confidence, due to more enhanced featuresA.6. Q-Score and Classification Confidence In Figure A.7, we plot the mean of |L i | (left), i.e., number of highly activating features in the i th sample, and the mean linear classification confidence (right) over the population for each self-supervised model pre-trained with and without Q- Score regularization. We observe an increase in the average number of highly activating features (L i ) and as a result, an improvement in classification confidence, due to more enhanced features.
| {'fraction_non_alphanumeric': 0.04074622938105065, 'fraction_numerical': 0.01925127764774317, 'mean_word_length': 5.140840278959007, 'pattern_counts': {'":': 0, '<': 4, '<?xml version=': 0, '>': 4, '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': 'Self-supervised learning has shown impressive results in downstream classification tasks. However, there is limited work in understanding their failure modes and interpreting their learned representations. In this paper, we study the representation space of state-of-the-art selfsupervised models including SimCLR, SwaV, MoCo, BYOL, DINO, SimSiam, VICReg and Barlow Twins. Without the use of class label information, we discover discriminative features that correspond to unique physical attributes in images, present mostly in correctly-classified representations. Using these features, we can compress the representation space by up to 40% without significantly affecting linear classification performance. We then propose Self-Supervised Representation Quality Score (or Q-Score), a model-agnostic, unsupervised score that can reliably predict if a given sample is likely to be mis-classified during linear evaluation, achieving AUPRC of 91.45 on ImageNet-100 and 78.78 on ImageNet-1K. Q-Score can also be used as a regularization term on any pre-trained self-supervised model to remedy low-quality representations. Fine-tuning with Q-Score regularization can boost the linear classification performance of state-of-the-art self-supervised models by up to 5.8% on ImageNet-100 and 3.7% on ImageNet-1K compared to their baselines. Finally, using gradient heatmaps and Salient ImageNet masks, we define a metric to quantify the interpretability of each representation. We show that discriminative features are strongly correlated to core attributes and enhancing these features through Q-score regularization makes representations more interpretable across all self-supervised models.', 'arxivid': '2203.01881', 'author': ['Neha Kalibhat \nDepartment of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI\n', 'Kanika Narang \nDepartment of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI\n', 'Hamed Firooz \nDepartment of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI\n', 'Maziar Sanjabi \nDepartment of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI\n', 'Soheil Feizi \nDepartment of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI\n'], 'authoraffiliation': ['Department of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI', 'Department of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI', 'Department of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI', 'Department of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI', 'Department of Computer Science\nUniversity of Maryland\nCollege Park 2 Meta AI'], 'corpusid': 258547331, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 18763, 'n_tokens_neox': 16513, 'n_words': 10198, 'pdfsha': 'd168ef97244b74186a723199255b978003248519', 'pdfurls': ['https://export.arxiv.org/pdf/2203.01881v4.pdf'], 'title': ['Measuring Self-Supervised Representation Quality for Downstream Classification using Discriminative Features', 'Measuring Self-Supervised Representation Quality for Downstream Classification using Discriminative Features'], 'venue': []} |
arxiv |
A NOTE ON THE WORST CASE APPROACH FOR A MARKET WITH A STOCHASTIC INTEREST RATE
2018
Dariusz Zawisza
A NOTE ON THE WORST CASE APPROACH FOR A MARKET WITH A STOCHASTIC INTEREST RATE
Published in Appl. Math. (Warsaw)
45201810.4064/am2348-2-2018
We solve robust optimization problem and show the example of the market model for which the worst case measure is not a martingale measure. In our model the instantaneous interest rate is determined by the Hull-White model and the investor employs the HARA utility to measure his satisfaction.To protect against the model uncertainty he uses the worst case measure approach. The problem is formulated as a stochastic game between the investor and the market from the other side. PDE methods are used to find the saddle point and the precise verification argument is provided.
Introduction
We consider a portfolio problem embedded into a game theoretic problem. We assume that the investor does not trust his model much and believes it is only the best guess based on existing data. In such situation we say that the investor faces the model uncertainty (or the model ambiguity). In this work we would like to put more light into the portfolio optimization problem under the assumption that the short term interest rate exhibits some stochastic nature. We consider a financial market consisting of n assets and a bank account. The interest rate on the bank account follows the Hull-White model, which is extended version of the Vasicek model. The investor chooses between holding cash in a bank account and holding risky assets. The same model has been considered first by Korn and Kraft [4] but without the model uncertainty assumption. Instead of supposing that we have the exact model, we assume here the whole family of equivalent models, which will be described later. To determine robust investment controls the investor maximizes the total expected HARA utility of the final wealth after taking the infimum over all possible models. The robust optimization in the diffusion setting has been popularized especially by A. Schied and his coauthors (e.g. Schied [10] and references therein). The model ambiguity in the Vasicek model and its extensions has been considered already by Flor and Larsen [2], Sun et al. [11], Munk and Rubtsov [6], Wang and Li [12]. However, their objective function is different, because it includes the expression (along the lines of Maenhout [5]) which penalize the expected utility for divergence from the reference probability measure. Our model is in fact their limiting model, when their ambiguity coefficients are passing to +∞ (0 respectively). In the current paper the problem is formulated as a theoretic stochastic game between the market and the investor and the saddle point of this game is determined, despite of the fact we do not include the penalizing term into the objective function. Moreover, in addition to aferomentioned papers we provide correct and precise verification reasoning. First, we consider the full problem, without any constraints on the set of uncertainty measures. Further, we investigate what are the properties of the restricted model. To solve the game, we use the Hamilton-Jacobi-Bellman-Isaacs equation. After several substitutions we are able to solve the equation and use suitable version of the verification theorem to justify the method. Previously the same method has been used by Zawisza [13], [14], but in the model with a deterministic interest rate and with a different objective function. The major motivation for considering such model is to provide an example in which results of Oksendal and Sulem [7], [8] do not hold. In the papers they have considered the jump diffusion model but without assuming the stochastic nature of the interest rate, and have discovered that in that game the investor should always choose to invest only in the bank account and at the same time optimal market strategy is to choose a martingale measure. It is interesting because the martingale measure plays prominent role in derivative pricing. Our paper proves that in our framework the worst case measure is different from the martingale measure.
Model description
Let (Ω, F , P ) be a probability space with filtration (F t , 0 ≤ t ≤ T ) (possibly enlarged to satisfy usual assumptions) spanned by n-dimensional Brownian motion
(W t = (W 1 t , W 2 t , . . . W n t ) T , 0 ≤ t ≤ T ).
We have the initial measure P , but our investor concerns model uncertainty, so the measure should be treated only as a proxy for the real life measure. Further, we will consider a whole class of equivalent measures, which will describe the model uncertainty. Our agent has an access to the market with a bank account (B t , 0 ≤ t ≤ T ) and risky assets (S t = (S 1 t , S 2 t , . . . , S n t ), 0 ≤ t ≤ T ). Under the measure P the system is given by
(2.1) dB t = r t B t dt, dS t = diag(S t )[(r t e + Σ t λ T
the mixed stock-bond model (e.g. Korn and Kraft [4, Section 2.2]):
dS 1 t = (r t + λ 1 t σ 1,1 t + λ 2 t σ 1,2 t )S 1 t dt + σ 1,1 t S 1 t dW 1 t + σ 1,2 t S 1 t dW 2 t , dS 2 t = (r t + λ 2 t σ 2,2 t )S 2 t dt + σ 2,2 t S 2 t dW 2 t , dr t = (b t − κr t )dt + a t dW 2 t . Here S 2
t is the price of the bond in the Vasicek model with the maturity
T ′ > T , which means that σ 2,2 t = − a κ (1 − e −κ(T ′ −t)
). The portfolio process evolves according to
dX π t = r t X π t dt + π t Σ t λ T t X π t dt + X π t π t Σ t dW t .
The symbol A t denotes the class of progressively measurable processes π = (π 1 , π 2 , . . . , π n ) such that T t |π s | 2 ds < +∞ a.s. To describe the model uncertainty or model ambiguity issues we assume that the probability measure is not precisely known and the investor considers a whole class of possible measures. We follow the approach of Oksendal and Sulem [7] or Schied [10] in defining the set
(2.2) Q T := Q η T ∼ P | dQ η T dP = E η t dW t T , η ∈ M ,
where E(·) t denotes the Doleans-Dade exponential and M denotes the set of all, progressively measurable processes η = (η 1 , η 2 , . . . , η n ), such that
E dQ η T dP 2 < +∞.
In the latter part of the paper we assume that the process η takes his values in a fixed compact and convex set Γ. It is convenient to use the Q η T dynamics of the stochastic system (X t , r t ) i.e. preferences (Gilboa and Schmeidler [3] ), i.e. his aim is to maximize
(2.4) J π,η (x, r, t) = inf η∈M E η x,r,t U(X π T ).
The symbol E η x,r,t is used to denote the expected value under the measure Q η T when system starts at (x, r, t). Here we assume that U(x) = x γ γ with 0 < γ < 1. The solution for γ < 0 will be the same but due to the fact that U has negative values, it is needed to use few more restrictions and technicalities to complete the proof.
Here we are interested not only in the optimal portfolio π * , but also in the measure Q η * T for which the infimum is attained. Therefore, we are looking for a saddle point (π * , η * ) i.e. J π,η * (x, r, t) ≤ J π * ,η * (x, r, t) ≤ J π * ,η (x, r, t).
The solution
To solve the problem we will use the Hamilton-Jacobi-Bellman-Isaacs operator given by
L π,η V (x, r, t) :=V t + 1 2 a 2 t V rr + 1 2 πΣ t Σ T t π T x 2 V xx + πΣ t a t xV xr (3.1) + πΣ t (λ T t + η T )xV x + ηa T t V r + (b t − κ t r)V r + rxV x .
It should be considered together with the verification theorem. The reasoning behind its proof is of standard type (see for instance Zawisza [13,Theorem 3.1]). Here we present only short sketch, just to emphasis some minor differences.
V ∈ C 2,2,1 ((0, +∞) × R × [0, T )) ∩ C([0, +∞) × R × [0, T ])
and a Markov control
(π * (x, r, t), η * (x, r, t)) ∈ A t × M, such that L π * (x,r,t),η V (x, r, t) ≥ 0, (3.2) L π,η * (x,r,t) V (x, r, t) ≤ 0, (3.3) L π * (x,r,t),η * (x,r,t) V (x, r, t) = 0, (3.4) V (x, r, T ) = x γ γ (3.5)
for all η ∈ R, π ∈ R, (x, r, t) ∈ (0, +∞) × R × [0, T ), and
(3.6) E η x,r,t sup t≤s≤T V (X π * s , r s , s) < +∞ for all (x, r, t) ∈ [0, +∞) × R × [0, T ], π ∈ A t , η ∈ M.
Then J π,η * (x, r, t) ≤ V (x, r, t) ≤ J π * ,η (x, r, t) for all π ∈ A t , η ∈ M, and V (x, r, t) = J π * ,η * (x, r, t).
Proof. Let us fix first π ∈ A t . Consider Q η * T -dynamics of the system (X t , r t ) and apply the Itô formula using the function V . By using inequality (3.3) and taking the expectation from both sides, we obtain
V (x, r, t) ≥ E η * V (X (T −ε)∧τn , r (T −ε)∧τn , (T − ε) ∧ τ n ),
where (τ n , n ≥ 0) is a localizing sequence of stopping times. The function V is positive, thus the Fatou Lemma implies
V (x, r, t) ≥ E η * x,r,t V (X π T , r T , T ) = E η * x,r,t U(X π T ) = J π,η * (x, r, t).
To prove the reverse inequality we fix η ∈ M and consider Q η T -dynamics of the system (X t , r t ). After applying the Itô rule we get
V (x, r, t) ≤ E η
x,r,t V (X π * (T −ε)∧τn , r (T −ε)∧τn , (T − ε) ∧ τ n ) and the same is true with the equality (3.6) and the dominated convergence theorem finish the proof.
V (x, r, t) = E η * x,r,t V (X π * (T −ε)∧τn , r (T −ε)∧τn , (T − ε) ∧ τ n ). Property
Following Korn and Kraft [4] we predict that conditions (3.2) -(3.6) are satisfied by the function of the form
V (x, r, t) = x γ γ e f (t)r+g(t) , f (T ) = 0, g(T ) = 0.
Substituting it into (3.2)-(3.4) and dividing the expression by x γ γ e f (t)r+g(t) , we get
H (π,η * ) (r, t) ≤ H (π * ,η * ) (r, t) = 0 ≤ H (π * ,η) (r, t), π, η ∈ R n . where H (π,η) (r, t) := f ′ (t)r + g ′ (t) + 1 2 a 2 t f 2 (t) + 1 2 γ(γ − 1)πΣ t Σ T t π T + πΣ t a T t γf (t) + γπΣ t (λ T t + η T ) + ηa T t f (t) + (b t − κ t r)f (t) +
γr. Now, it is possible to determine the saddle point. Suppose first that we already have the saddle point (π * , η * ). Therefore, H (π,η * ) (r, t) ≤ H (π * ,η * ) (r, t), π, η ∈ R n and consequently
π * t = 1 (1 − γ) (λ t + η * + a t f (t))Σ −1 t .
On the other hand,
H (π * ,η * ) (r, t) ≤ H (π * ,η) (r, t), η ∈ R n .
We should notice first that H forms a linear function in η. In that case, the only chance to find η * is to delete the expression with η i.e.
γπ * Σ t + a t f (t) = 0.
This means that
π * = − f (t) γ a t Σ −1 t . So, we should have f (t) (1 − γ) a t Σ −1 t + λ t + η * t (1 − γ) Σ −1 t = − a t f (t) γ Σ −1 t ,
which yields
η * t = −λ t − f (t) γ a t .
Substituting π * and η * into the equation and using the fact that the expression with η is equal to 0, we get
f ′ (t)r + g ′ (t) + 1 2 |a t | 2 f 2 (t) + 1 2 |a t | 2 f 2 (t) (γ − 1) γ − |a t | 2 f (t) − λ t a T t f (t) + (b t − κ t r)f (t) + γr = 0. Thus, f ′ (t) − κ t f (t) + γ = 0, g ′ (t) + 1 2 |a t | 2 f 2 (t) + 1 2 |a t | 2 f 2 (t) (γ − 1) γ − |a t | 2 f (t) − λ t a T t f (t) + b t f (t) = 0.
More explicit forms are:
f (t) = γe − T t κsds T t e T k κsds dk, g(t) = T t 1 2 f 2 (s)|a s | 2 + 1 2 |a s | 2 f 2 (s) (γ − 1) γ − |a s | 2 f (s) − λ s a T s f (s) + b s f (s) ds.
We can now summarize our preparatory calculations.
Proposition 3.2. The pair (π * , η * ) given by
π * t = − f (t) γ a t Σ −1 t , η * t = −λ t − f (t) γ a t
is a saddle point for problem (2.4).
Proof. Note that π * t and Σ t are deterministic functions. To complete the proof we need only to verify that E η x,r,t sup t≤s≤T V (X π * s , r s , s) < +∞, η ∈ M.
We have E η
x,r,t sup t≤s≤T V (X π * s , r s , s) = E x,r,t dQ η dP sup t≤s≤T V (X π * s , r s , s) .
By the Cauchy -Schwarz inequality
E x,r,t dQ η dP sup t≤s≤T V (X π * s , r s , s) ≤ E dQ η dP 2 1 2 E x,r,t sup t≤s≤T V 2 (X π * s , r s , s) 1 2 .
The explicit formula for the function V leads to
V (X π * s , r s , s) = 1 γ X π *
s γ e f (s)rs+g(s) .
The portfolio process X t is a solution to the linear equation, so
X π * s = xe s t [r l +π * l Σ l λ T l − 1 2 (π * l Σ l Σ T l π T * l )]dl+ s t π * l Σ l dW l .
Note that the process ζ s = e s t κ l dl r s has the dynamics dζ s = e s t κ l dl b s ds + e s t κ l dl a s dW s . We have
r s = e − s t κ l dl r + s t b l dl + s t a l dW l .
By the stochastic Fubini theorem, the expression V 2 (X π * s , r s , s) can be rewritten in the form V 2 (X π * s , r s , s) = xZ s e β(s)rs+ξ(s) , where the process Z s is a square integrable martingale, β, ξ are bounded and deterministic functions.
After repeating the Cauchy -Schwarz inequality once more it is now sufficient to prove that for any bounded deterministic functionβ we have
Concluding remarks
To conclude the result we show that the measure Q η * T is not a martingale measure i.e. the process S t e − t 0 rsds is not a Q η * T -martingale. To see this, it is sufficient to write Q η * T dynamics of S t :
dS t = diag(S t ) r t e − f (t) γ Σ t a T t dt + Σ t dW t .
At the end, it is worth to compare the robust investment strategy
π * t = 1 (1 − γ) (λ t + η * t + f (t)a t )Σ −1 t , η * t = −λ t − f (t) γ a t
with the solution to the traditional utility maximization problem
π * t = 1 (1 − γ) (λ t + f (t)a t )Σ −1 t .
It is worth noticing as well that π * can be rewritten as and it does not depend on the risk aversion coefficient γ. The same property is true for η * .
π * t = − f (t) γ a t Σ −1 t = −e − T
Then, (π * , η * ) is a saddle point for the function H (π,η) (r, t). In particular, H (π,η * ) (r, t) ≤ H (π * ,η * ) (r, t), π ∈ R n .
The unique function π * which satisfy the above condition is given by
π * t = 1 (1 − γ) (λ t + η * t + f (t)a t )Σ −1 t .
Proposition 4.1. Suppose that η * is a minimizer of (4.2) and
π * t = 1 (1 − γ) (λ t + η * t + f (t)a t )Σ −1 t .
Then the pair (π * , η * ) is a saddle point for problem (2.4) with the restrictions imposed by the set Γ.
The proof is omitted because it is the repetition of the steps from the proof of Proposition 3.2.
Theorem 3. 1 (
1Verification Theorem). Suppose there exists a positive function
eβ (s)ζs < +∞.Note that the following inequality is true eβ (s)ζs ≤ eβ maxζs + eβ min ζs ,whereβ max = max t≤s≤Tβ (s),β min = min t≤s≤Tβ (s).Both processes eβ maxζs , eβ min ζs are solutions to linear equations with bounded coefficients and thus usual Lipschitz and linear growth conditions are satisfied. Property (3.7) follows from standard estimates for stochastic differential equations (see Pham [9, Theorem 1.3.16]).
κsds dk a t Σ −1 t .
t )dt + Σ t dW t ], dr t = (b t − κ t r t )dt + a t dW t .We assume that e = (1, 1, . . . , 1), coefficients κ t , b t , λ t = (λ 1 t , λ 2 t , . . . , λ n t ), a t = (a 1 t , a 2 t , . . . , a n t ), Σ t = [σ i,j t ] i,j=1...n are continuous deterministic functions, and in addition Σ t is invertible. For notational convenience we omit the term a t λ T t dt in the dynamics for r, and we assume it is already included in b t dt term. The representative example for the process (S t , t ∈ [0, T ]) is
(2.3) dX π t = r t X π t dt + π t Σ t (λ T t + η T t )X π t dt + π t Σ t X π t dW η t , dr t = [(b t − κ t r t ) + a t η T t ]dt + a t dW η t .Our investor takes into account the model ambiguity and has worst case
Model uncertainty with restrictionsFrom the practitioner's point of view, it might be interesting to solve the problem with restrictions imposed on the uncertainty set M. In this section we assume that the class M consists of all progressively measurable processes taking values in a compact and convex fixed set Γ ⊂ R n .We can use the same function HTo find the explicit saddle point for the function H, we start with solving the upper Isaacs equation We can determine a saddle point candidate (π * , η * ) by finding a Borel measurable function η * , such thatand a Borel measurable function π * , such that min η∈Γ max π∈R H (π,η) (r, t) = min η∈Γ H (π * ,η) (r, t).Because the variable η is separated from r, equation (4.1) can be split into two equations (the first one has already been solved):Therefore, to find η * , it is sufficient to determine any Borel measurable minimizer to the expression (4.2) − 1 2 γ 1 − γ |λ t +η +f (t)a t | 2 + γ 1 − γ (λ t +η +f (t)a t )(λ t +η) T +f (t)a t η T . Now, let π * be a Borel measurable maximizer of the function min η∈Γ H (π,η) (r, t).
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Robust optimal investment strategy for an AAM of DC pension plans with stochastic interest rate and stochastic volatility. P Wang, Z Li, Insur. Math. Econ. 80P. Wang, Z. Li Robust optimal investment strategy for an AAM of DC pension plans with stochastic interest rate and stochastic volatility, Insur. Math. Econ. 80 (2018), 67 -83.
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Target achieving portfolio under model misspecification: quadratic optimization framework. D Zawisza, Appl. Math. (Warsaw). 39D. Zawisza, Target achieving portfolio under model misspecification: quadratic optimization framework , Appl. Math. (Warsaw) 39, 425 - 443 (2012).
. Poland E Kraków, mail address: [email protected]ów, Poland E-mail address: [email protected]
| {'fraction_non_alphanumeric': 0.09231785387663856, 'fraction_numerical': 0.023066761507976832, 'mean_word_length': 3.1272803522751103, 'pattern_counts': {'":': 0, '<': 8, '<?xml version=': 0, '>': 1, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 2, '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 solve robust optimization problem and show the example of the market model for which the worst case measure is not a martingale measure. In our model the instantaneous interest rate is determined by the Hull-White model and the investor employs the HARA utility to measure his satisfaction.To protect against the model uncertainty he uses the worst case measure approach. The problem is formulated as a stochastic game between the investor and the market from the other side. PDE methods are used to find the saddle point and the precise verification argument is provided.', 'arxivid': '2001.01998', 'author': ['Dariusz Zawisza '], 'authoraffiliation': [], 'corpusid': 126057337, 'doi': '10.4064/am2348-2-2018', 'github_urls': [], 'n_tokens_mistral': 7250, 'n_tokens_neox': 6407, 'n_words': 3924, 'pdfsha': 'c778c9949529d9c4e5622aeaa8e33d1fa522f359', 'pdfurls': ['https://arxiv.org/pdf/2001.01998v1.pdf'], 'title': ['A NOTE ON THE WORST CASE APPROACH FOR A MARKET WITH A STOCHASTIC INTEREST RATE', 'A NOTE ON THE WORST CASE APPROACH FOR A MARKET WITH A STOCHASTIC INTEREST RATE'], 'venue': ['Published in Appl. Math. (Warsaw)']} |
arxiv |
Probing new physics in semileptonic Λ b decays
20 Dec 2018
Atasi Ray *[email protected]†[email protected]‡[email protected]
School of Physics
University of Hyderabad
Hyderabad-500046India
Suchismita Sahoo
Theoretical Physics Division
Physical Research Laboratory
Ahmedabad-380009India
Rukmani Mohanta
School of Physics
University of Hyderabad
Hyderabad-500046India
Probing new physics in semileptonic Λ b decays
20 Dec 20181
In recent times, several hints of lepton non-universality have been observed in semileptonic B meson decays, both in the charged-current (b → clν l ) and neutral-current (b → sll) transitions.Motivated by these intriguing results, we perform a model independent analysis of the semileptonic Λ b decays involving the quark level transitions b → (u, c)lν l , in order to scrutinize the nature of new physics. We constrain the new parameter space by using the measured branching ratios of B + c,u → τ + ν τ , B → πτ ν τ processes and the existing experimental results on R D ( * ) , R J/ψ and R l π parameters. Using the constrained parameters, we estimate the branching ratios, forward backward asymmetries, hadron and lepton polarization asymmetries of the Λ b → (Λ c , p)lν l processes.Moreover, we also examine whether there could be any lepton universality violation in these decay modes.
I. INTRODUCTION
Though the Standard Model (SM) is considered as the most fundamental theory describing almost all the phenomena of particle physics, still it is unable to shed light on some of the open issues, like matter-antimatter asymmetry, dark matter, dark energy, etc., which eventually necessitates to probe the physics beyond it. In this respect, the rare decays of B mesons involving the flavor changing neutral current (FCNC) transitions play an important role in the quest for new physics (NP). Even though the SM gauge interactions are lepton flavor universal, the violation of lepton universality has been observed in various semileptonic B decays. Recently, the LHCb Collaboration has reported a spectacular discrepancy of 1.9σ (3.3σ) [1][2][3][4][5][6] and 2σ [7] on the lepton non-universality (LNU) parameters R D ( * ) = Br(B →D ( * ) τν τ )/Br(B →D ( * ) lν l ) and R J/ψ = Br(B c → J/ψτν τ )/Br(B c → J/ψlν l ) respectively from their corresponding SM values. Analogous LNU parameters are also observed in b → sll processes i.e., R K ( * ) = Br(B →K ( * ) µ + µ − )/Br(B →K ( * ) e + e − ) with discrepancies of 2.6σ (2.2 − 2.4)σ [8,9]. The SM predictions as well as the corresponding experimental values of various LNU parameters along with their deviations are presented in Table I .
LNU parameters
Experimental value SM prediction Deviation R K | q 2 ∈[1,6] GeV 2 0.745 +0.090 −0.074 ± 0.036 [8] 1.003 ± 0.0001 [10] 2.6σ R K * | q 2 ∈[0.045,1.1] GeV 2 0.66 +0.11 −0.07 ± 0.03 [9] 0.92 ± 0.02 [11] 2.2σ R K * | q 2 ∈[1.1,6] GeV 2 0.69 +0.11 −0.07 ± 0.05 [9] 1.00 ± 0.01 [11] 2.4σ R D 0.391 ± 0.041 ± 0.028 [6] 0.300 ± 0.008 [12] 1.9σ R D * 0.316 ± 0.016 ± 0.010 [6] 0.252 ± 0.003 [13,14] 3.3σ R J/ψ 0.71 ± 0.17 ± 0.184 [7] 0.289 ± 0.01 [15, 16] 2σ In addition, another discrepancy in b → ulν l transition is also noticed in the measured ratio R l π =
τ B 0 τ B − Br(B − → τ −ν τ ) Br(B 0 → π + l −ν l ) , l = e, µ ,(1)
where τ B 0 (τ B − ) is the life time of B 0 (B − ) meson. Using the experimental measured values of the branching ratios of B − u → τ −ν τ and B 0 → π + l −ν l decay processes
Br(B − u → τ −ν τ )| Expt = (1.09 ± 0.24) × 10 −4 ,(2)
Br(B 0 → π + l −ν l )| Expt = (1.45 ± 0.05) × 10 −4 ,
with τ B − /τ B 0 = 1.076 ± 0.004 from [17], one can obtain R l π | Expt = 0.699 ± 0.156, (4) which has also nearly 1σ deviation from its SM value R l π | SM = 0.583 ± 0.055. It is generally argued that, compared to the first two generations, the processes involving the third generation leptons are more sensitive to NP due to their reasonably large mass. As the LNU parameters are the ratio of branching fractions, the uncertainties arising due to the CKM matrix elements and hadronic form factors are expected to be reduced, as they cancelled out in the ratio. Hence, these deviations of various LNU parameters hint towards the possible interplay of new physics in an ambiguous manner.
On the other hand, around 20% of the total number of hadrons produced at LHCb are Λ b baryon [18,19], and hence the study of Λ b becomes quite interesting in these days. The b → qlν l (q = u, c) quark level transitions can be probed in both B and Λ b decays. Thus, as in B decays one can also scrutinize the presence of lepton universality violation in the corresponding semileptonic baryon decays Λ b → (Λ c , p)lν l to corroborate the results from B sector and thus, to probe the structure of NP. The heavy-heavy and heavy-light semileptonic decays of baryons can serve as an additional source for the determination of the Cabbibo-Kobayashi-Maskawa (CKM) matrix elements V qb [17,[20][21][22]. In the literature [23][24][25][26][27][28][29][30][31][32][33][34][35], the baryonic decay modes mediated by b → (u, c)lν l quark level transitions are studied both in model dependent and independent approaches. The analysis of Λ b → Λ c τν τ decay in the context of SM and various NP couplings are performed in [25]. In Ref. [27], the SM hadron and lepton polarization asymmetries are computed in the covariant confined quark model.
The precise lattice QCD calculation of Λ b → (Λ c , p) form factors and the investigation of semileptonic baryonic b → (u, c)lν l processes are performed in [28]. Ref. [34] investigates the impact of five possible new physics interactions, adopting five different form factors of Λ b → Λ c τν τ decay mode. Considering various real NP couplings, the differential decay distributions, forward-backward asymmetries and the ratios of branching fractions of these baryonic decay modes are investigated in [29]. In this work, we intend to analyse the effect of complex new couplings on Λ b → (Λ c , p)lν l decay processes in a model independent way.
The main goal of this work is to check the possible existence of lepton universality violation in baryonic decays. The new coefficients are constrained by using the branching ratios of B u,c → τν τ , B → πτν τ processes and the experimental data on R D ( * ) , R J/ψ , R l π ratios. We then compute the branching ratios, forward-backward asymmetries, lepton and hadron polarization asymmetries of these baryonic decay modes. We also check the LNU parameters by using the constrained new couplings. The main difference between our approach and the previous analyses in [25,32] is that, we investigate the impact of individual complex new couplings on all the angular observables including the lepton and hadron polarization asymmetries. We use the updated experimental limits on R D ( * ) , R l π ratios including new R J/ψ parameter to constrain the allowed parameter space.
The outline of our paper is follows. In section II, we present the general effective Lagrangian of b → (u, c)lν l processes in presence of NP, and the necessary theoretical framework for analysing these processes. The constraints on new parameter space associated with b → (u, c)lν l transitions are computed from the experimental data on R D ( * ) , R J/ψ , R l π , Br(B c,u → τν τ ) and Br(B → πτν τ ) observables in section III. In section IV, we discuss the branching ratios and all the physical angular observables of Λ b → (Λ c , p)lν l processes. Our findings are summarized in section V.
II. THEORETICAL FRAMEWORK
The most general effective Lagrangian associated with B 1 → B 2 lν l decay processes, where B 1 = Λ b , B 2 = Λ c , p mediated by the quark level transition b → qlν l , (q = u, c) is given by [36,37]
L eff = − 4 G F √ 2 V qb (1 + V L )l L γ µ ν LqL γ µ b L + V RlL γ µ ν LqR γ µ b R +S LlR ν LqR b L + S RlR ν LqL b R + T LlR σ µν ν LqR σ µν b L + h.c. ,(5)
where G F denotes the Fermi constant, V qb are the CKM matrix elements and q(l) L,R = P L,R q(l) are the chiral quark(lepton) fields with P L,R = (1 ∓ γ 5 )/2 as the projection operators. Here V L,R , S L,R , T L represent the vector, scalar and tensor type NP couplings, which are zero in the SM.
In the presence of NP, the double differential decay distribution for B 1 → B 2 lν l processes and N =
G 2 F |V qb | 2 q 2 λ(M 2 B 1 , M 2 B 2 , q 2 ) 2 10 π 3 M 3 B 1 ,
λ(a, b, c) = a 2 + b 2 + c 2 − 2(ab + bc + ca). (9) Here M B 1 (2) and m l are the masses of B 1 (2) baryon and charged leptons respectively. The helicity amplitudes in terms of the various form factors and the NP couplings are given as [25,33] H V
1 2 0 = (1 + V L + V R ) √ Q − q 2 (M B 1 + M B 2 ) f 1 (q 2 ) − q 2 f 2 (q 2 ) , H A 1 2 0 = (1 + V L − V R ) √ Q + q 2 (M B 1 − M B 2 ) g 1 (q 2 ) + q 2 g 2 (q 2 ) , H V 1 2 + = (1 + V L + V R ) 2 Q − − f 1 (q 2 ) + (M B 1 + M B 2 ) f 2 (q 2 ) , H A 1 2 + = (1 + V L − V R ) 2 Q + − g 1 (q 2 ) − (M B 1 − M B 2 ) g 2 (q 2 ) , H V 1 2 t = (1 + V L + V R ) √ Q + q 2 (M B 1 − M B 2 ) f 1 (q 2 ) + q 2 f 3 (q 2 ) , H A 1 2 t = (1 + V L − V R ) √ Q − q 2 (M B 1 + M B 2 ) g 1 (q 2 ) − q 2 g 3 (q 2 ) , H S 1 2 0 = (S L + S R ) √ Q + m b − m q (M B 1 − M B 2 ) f 1 (q 2 ) + q 2 f 3 (q 2 ) , H P 1 2 0 = (S L − S R ) √ Q − m b + m q (M B 1 + M B 2 ) g 1 (q 2 ) − q 2 g 3 (q 2 ) , H T 1 2 ,+,0 = −T L 2 q 2 f T Q + (M B 1 − M B 2 ) + g T Q − (M B 1 + M B 2 ) , H T 1 2 ,+,− = −T L f T Q + + g T Q − , H T 1 2 ,+,t = T L − 2 q 2 f T Q − (M B 1 + M B 2 ) + g T Q + (M B 1 − M B 2 ) + 2q 2 f V T Q − − g V T Q + , H T 1 2 ,0,t = T L − f T Q − − g T Q + + f V T Q − (M B 1 + M B 2 ) − g V T Q + (M B 1 − M B 2 ) +f S T Q − Q + + g S T Q + Q − , H T − 1 2 ,+,− = T L f T Q + − g T Q − , H T − 1 2 ,0,− = T L 2 q 2 f T Q + (M B 1 − M B 2 ) − −g T Q − (M B 1 + M B 2 ) , H T − 1 2 ,−,t = T L − 2 q 2 f T Q − (M B 1 + M B 2 ) − g T Q + (M B 1 − M B 2 ) + 2q 2 f V T Q − + g V T Q + , H T − 1 2 ,0,t = T L − f T Q − + g T Q + + f V T Q − (M B 1 + M B 2 ) + +g V T Q + (M B 1 − M B 2 ) +f S T Q − Q + − g S T Q + Q − .(10)
where
Q ± = (M B 1 ± M B 2 ) 2 − q 2 and f (a) i , g (b)
i , (i = 1, 2, 3, T & a, b = V, S) are the various form factors. After integrating out cos θ l in Eqn. (6), one can obtain the q 2 dependent differential decay rate. Besides the branching ratios, other interesting observables in these decay modes are
• Forward-backward asymmetry parameter:
A F B (q 2 ) = 0 −1 d cos θ l d 2 Γ dq 2 d cos θ l − 1 0 d cos θ l d 2 Γ dq 2 d cos θ l dΓ dq 2 .(11)
• Convexity parameter:
C l F (q 2 ) = 1 dΓ/dq 2 d 2 d(cos θ l ) 2 d 2 Γ dq 2 d cos θ l .(12)
• Longitudinal hadron polarization asymmetry parameter:
P h L (q 2 ) = dΓ λ 2 =1/2 /dq 2 − dΓ λ 2 =−1/2 /dq 2 dΓ/dq 2 ,(13)
where dΓ λ 2 =±1/2 are the individual helicity-dependent differential decay rates, whose detailed expressions are given in Appendix A [33].
• Longitudinal lepton polarization asymmetry parameter:
P τ L (q 2 ) = dΓ λτ =1/2 /dq 2 − dΓ λτ =−1/2 /dq 2 dΓ/dq 2 ,(14)
where dΓ λ 2 =±1/2 are the individual helicity-dependent differential decay rates, whose detailed expressions are given in Appendix A [33].
• Lepton non-universality parameter:
R B 2 = Br(B 1 → B 2 τ −ν τ ) Br(B 1 → B 2 l −ν l )
, l = e, µ.
• The LHCb Collaboration has measured the ratio of the partially integrated decay rates of Λ 0 b → p µν l over the Λ 0 b → Λ + c µν l process as
R µ Λc p = q 2 max 15 GeV 2 dΓ(Λ b → p µν l ) dq 2 dq 2 q 2 max 7 GeV 2 dΓ(Λ b → Λ c µν l ) dq 2 dq 2
= (1.00 ± 0.04 ± 0.08) × 10 −2 (16) and put constraint on the ratio |V ub |/|V cb | = 0.083 ± 0.004 ± 0.004 [20]. Similarly, we define the following parameter, to investigate if there is any possible role of NP
R τ Λc p = q 2 max 15 GeV 2 dΓ(Λ b → p τν τ ) dq 2 dq 2 q 2 max 7 GeV 2 dΓ(Λ b → Λ c τν τ ) dq 2 dq 2 .(17)
III. CONSTRAINTS ON NEW COUPLINGS
After assembling the expressions for all the interesting observables in presence of NP, we now proceed to constrain the new coefficients by using the experimental bounds on Br(B u,c → τν τ ), Br(B → πτν τ ), R l π , R D ( * ) and R J/ψ parameters. In this analysis, the new Wilson coefficients are considered as complex. We further assume that only one new coefficient to present at a time and accordingly compute the allowed parameter space of these couplings.
The branching ratios of B q → lν l processes in the presence of NP couplings are given by [38] Br(B q → lν l ) =
G 2 F |V qb | 2 8 π τ Bq f 2 Bq m 2 l M Bq 1 − m 2 l M 2 Bq 2 × (1 + V L − V R ) − M 2 Bq m l (m b + m q ) (S L − S R ) 2 ,(18)
where M Bq is the mass of B q meson. By using the masses of all the particles, lifetime of B q meson, CKM matrix elements from [17] and decay constants f Bu = 190.5 ± 4.2 MeV, f Bc = 489 ± 4 ± 3 MeV from [39,40], the branching ratios of B + u,c → τ + ν τ processes in the SM are found to be Br(B + u → τ + ν τ )| SM = (8.48 ± 0.5) × 10 −5 ,
Br(B + c → τ + ν τ )| SM = (3.6 ± 0.14) × 10 −2 .
Using the current world average of the B c lifetime, the upper limit on the branching ratio of B + c → τ + ν τ process is [41] Br(B + c → τ + ν τ ) 30%. (21) The branching ratios of B q → P lν l (P = π, D ) are given as [42,43] dBr(B q → P lν l )
dq 2 = τ Bq G 2 F |V qb | 2 192π 3 M 3 Bq q 2 λ P (q 2 ) 1 − m 2 l q 2 2 × |1 + V L + V R | 2 1 + m 2 l 2q 2 H 2 0 + 3 2 m 2 l q 2 H 2 t + 3 2 |S L + S R | 2 H 2 S + 8|T L | 2 1 + 2m 2 l q 2 H 2 T +3Re[(1 + V L + V R )(S * L + S * R )] m l q 2 H S H t −12Re[(1 + V L + V R )T * L ] m l q 2 H T H 0 ,(22)
where the helicity amplitudes in terms of form factors (F 0,+ ) are expressed as
H 0 = λ(M 2 Bq , m 2 P , q 2 ) q 2 F + (q 2 ), H t = M 2 Bq − M 2 P q 2 F 0 (q 2 ) , H S = M 2 Bq − M 2 P m b − m q F 0 (q 2 ) H T = − λ P (q 2 ) M Bq + M P F T (q 2 ).(23)
Using the values of the B → π form factors from [44][45][46][47], the obtained branching ratios of B q → πlν l processes, in the SM are given as
Br(B 0 → π + µ −ν µ )| SM = (1.35 ± 0.10) × 10 −4 ,(24)
Br(B 0 → π + τ −ν τ )| SM = (9.40 ± 0.75) × 10 −5 .
It should be noted that, the branching ratio of the muonic channel agrees reasonably well with the experimental value as given in Eqn. (3), whereas the tau-channel is within its current experimental limit [17] Br(B 0 → π + τ −ν τ )| Expt < 2.5 × 10 −4 .
The branching ratios of B q → V lν l , where V = D * , J/ψ, are given as [42,43] dBr(B → V lν l )
dq 2 = τ Bq G 2 F |V qb | 2 192π 3 M 3 Bq q 2 λ V (q 2 ) 1 − m 2 l q 2 2 × (|1 + V L | 2 + |V R | 2 ) 1 + m 2 l 2q 2 H 2 V,+ + H 2 V,− + H 2 V,0 + 3 2 m 2 l q 2 H 2 V,t −2Re[(1 + V L )V * R ] 1 + m 2 l 2q 2 H 2 V,0 + 2H V,+ H V,− + 3 2 m 2 l q 2 H 2 V,t + 3 2 |S L − S R | 2 H 2 S + 8|T L | 2 1 + 2m 2 l q 2 H 2 T,+ + H 2 T,− + H 2 T,0 +3Re[(1 + V L − V R )(S * L − S * R )] m l q 2 H S H V,t −12Re[(1 + V L )T * L ]
where H V,± , H V,0 , H V,t and H S are the hadronic amplitudes [42,43].
In this analysis, we consider the new physics contribution to third generation lepton only and the couplings with light leptons are assumed to be SM like. By allowing only one coefficient at a time, we constrain its real and imaginary parts by comparing the theoretically predicted values of Br(B + u → τ + ν τ ) and R l π with their corresponding 3σ range of observed experimental results for b → uτν τ transitions. We have also used the upper limit of the branching ratio of B 0 → π + τ −ν τ process. In Fig. 1 , we show the constraints on real and imaginary parts of new coefficients V L (top-left panel), V R (top-right panel), S L (middle-left panel) and S R (middle-right panel) obtained from the Br(B + u → τ + ν τ ), Br(B 0 → π + τ −ν τ ) and R l π observables. Since the branching ratio of B + u → τ + ν τ process does not receive any contribution from tensor operator, the allowed region of real and imaginary parts of tensor coupling (T L ) obtained only from the upper limit on Br(B 0 → π + τ −ν τ ), and is presented in the bottom panel of this figure. Now imposing the extrema conditions, the allowed range of the new couplings associated with b → uτν τ transition are presented in Table II the constraints on T L coupling is obtained from the experimental data on R D ( * ) , which is shown in the bottom panel of Fig. 2. In Table II , The constraints on these parameters are obtained earlier from various B decays in Refs. [13,14,25,29,38,43,[48][49][50]. Our analysis is similar to Refs. [25,32]. In Ref. [25], the authors have considered the couplings to be complex and constrained the new coefficients associated with b → cτν τ from R D ( * ) data. However, they have not includeed the tensor couplings in their analysis, and found that the effects produced by the pseudoscalar coef-ficient are larger than those obtained from the scalar coefficient. In Ref. [29], the author assumed the couplings as real and computed the allowed parameter space by comparing the R D ( * ) , R l π parameters with their corresponding 3σ experimental data. In [50], the authors have considered the covariant confined quark model and studied the effect of new physics in theB 0 → D * τ −ν τ . They took the new coefficients as complex and constrained them using the experimental values of R D and R D * within their 2σ range. Recently, the decay process B c → (J/ψ)τ ν τ has been studied, in the covariant confined quark model [49], where the parameter space is constrained by using the experimental values of R D , R D * , R J/ψ within 2σ range. The new coefficients are considered to be complex and their best fit values are V L = −1.05 + i1.15, V R = 0.04 + i0.60, T L = 0.38 − i0.06. Though our analysis is similar to these approaches, but we get more severe bounds on the phases and strengths of the couplings due to additional constraints from Br(B c → τ ν τ ) and R J/ψ parameters for b → cτν τ case and from Br(B u → τ ν τ ) and Br(B → πτ ν τ ) observables for b → uτν τ process.
IV. NUMERICAL ANALYSIS AND DISCUSSION
In this section, we present the numerical results for semileptonic Λ b decay modes with third generation leptons in the final state. The masses of all the particles and the lifetime of Λ b are taken from [17]. The q 2 dependence of the helicity form factors (f +,⊥,0 , g +,⊥,0 , h +,⊥ , h +,⊥ ) in the lattice QCD calculation can be parametrized as [28,32] f i (q 2 ) = 1
1 − q 2 /(m f pole ) 2 a f 0 + a f 1 z(q 2 ) , (i = +, ⊥, 0)(28)
where m f pole is the pole mass and
z(q 2 ) = t + − q 2 − √ t + − t 0 t + − q 2 + √ t + − t 0 ,(29)with t ± = (M B 1 ± M B 2 ) 2 .
The values of the parameters m f pole , a f 0,1 associated with (axial)vector and (pseudo)scalar form factors (f +,⊥,0 , g +,⊥,0 ) are taken from [28]. In the lattice QCD approach, the m f pole , a f 0,1 parameters linked to tensor form factors (h +,⊥ , h +,⊥ ) of Λ b → Λ c lν l process are computed in [32]. However, currently no lattice results are available on the tensor form factors associated with Λ b → plν l process. Hence, we relate the tensor form factors of Λ b → plν l decay mode with its (axial)vector form factors by using the HQET relations as [33,51,52],
f T = g T = f 1 = (M B 1 + M B 2 ) 2 f + − q 2 f ⊥ (M B 1 + M B 2 ) 2 − q 2 , f V T = g V T = f S T = g S T = 0 .(30)
The detailed relation between the helicity form factors (f +,⊥,0 , g +,⊥,0 , h +,⊥ , h +,⊥ ) with other various hadronic form factors (f 1,2,3 , g 1,2,3 , f T , g T , f
V (S) T , g V (S) T
) are listed in Appendix B [51]. Using all these input parameters, the predicted branching ratios of Λ b → (Λ c , p)µν µ processes in the SM are given by
Br(Λ b → pµ −ν µ )| SM = (4.31 ± 0.345) × 10 −4 , Br(Λ b → Λ c µ −ν µ )| SM = (4.994 ± 0.4) × 10 −2 ,(31)
which are in reasonable agreement with the corresponding experimental data [17] Br
(Λ b → pµ −ν µ ) = (4.1 ± 1.0) × 10 −4 , Br(Λ b → Λ c l −ν l ) = 6.2 +1.4 −1.3 × 10 −2 .(32)
The values of the forward-backward asymmetries in these channels are found to be
A µ F B | SM Λ b →p = 0.316 ± 0.025, A µ F B | SM Λ b →Λc = 0.19 ± 0.0152.(33)
In Eqn. (31 , 33), the theoretical uncertainties are mainly due to the uncertainties associated with the CKM matrix elements and the form factor parameters. After having idea on all the required input parameters and the allowed parameter space of new couplings, we now proceed to discuss various new physics scenarios and their impact on Λ b → (Λ c , p)τν τ decay modes in a model independent way.
A. Scenario A: Only V L coefficient
In this scenario, we assume that the additional new physics contribution to the SM result is coming only from the coupling associated with the left-handed vector like quark currents i.e., V L = 0 and V R , S L,R , T L = 0. Since in this case, the NP operator has the same Lorentz structure as the SM operator, the SM decay rate gets modified by the factor and imaginary parts of V L coefficient from Table II , we present the differential branching ratios of Λ b → pτ −ν τ (left panel) and Λ b → Λ c τ −ν τ (right panel) processes with respect to q 2 in Fig. 3 . In these figures, the blue dashed lines represent the SM contribution, the orange bands are due to the presence of new V L coefficient and the grey bands stand for the theoretical uncertainties associated with the input parameters like form factors, CKM matrix elements etc. The branching ratios of Λ b → (Λ c , p)τ −ν τ deviate significantly from their corresponding SM values due to the NP contribution. In addition to the decay rate, other interesting observables, which can be used to probe new physics, are the zero crossing of the forward-backward asymmetry and the convexity parameters. From Eqn. (12), one can notice that the convexity parameter depends only on the V L,R and T L couplings. The values for forward-backward asymmetries of Λ b → (Λ c , p)τν τ processes in the SM are
|1 + V L | 2 . Imposing 3σ constraint on Br(B + u,c → τ + ν τ ), Br(B 0 → π + τ −ν τ ), R l π , R DA τ F B | SM Λ b →p = 0.115 ± 0.0092 , A τ F B | SM Λ b →Λc = −0.09 ± 0.007 ,(34)
and the corresponding values for the convexity parameters are
C τ F | SM Λ b →p = −0.157 ± 0.013 , C τ F | SM Λ b →Λc = −0.098 ± 0.008 .(35)
We found no deviation from SM results for the forward-backward asymmetry and convexity parameters due to the presence of V L coefficient. In q 2 variation of lepton universality violating parameters R p (R Λc ). We observe that the NP contribution coming from the V L coupling has significant impact on R p and R Λc parameters.
The variation of R τ Λcp parameter with q 2 for this case, is presented in the left panel of Fig. 7 . The numerical values of the branching ratios and the LNU parameters for both the SM and the V L -type NP scenario are given in Table III . Besides the branching ratios, forwardbackward asymmetry and LNU parameters of Λ b → (Λ c , p)τν τ processes, the NP effects can also be observed in the hadron and lepton polarization asymmetries. However, no deviation has been found in the presence of V L coupling from their corresponding SM results.
B. Scenario B: Only V R coefficient
Here, we assume that only the new V R coefficient is present in addition to the SM contribution, in the effective Lagrangian (5). To investigate the effect of NP coming from V R coefficient, we first constrain the new coefficient by imposing 3σ experimental bound on the b → (u, c)τν τ anomalies. Using the values from Table II , In these figures, the cyan bands are due to the additional contribution from V R coefficient.
We notice significant deviation in the branching ratios from their corresponding SM results.
The predicted values of the branching ratios for V R coefficient are presented in Table III . Apart from branching ratios, we are also interested to see the effect of this new coefficient on various q 2 dependent observables. The q 2 variation of the forward backward asymmetry and the convexity parameters for Λ b → pτ −ν τ (left) and Λ b → Λ c τ −ν τ (right) decay processes are depicted in the middle and bottom panels of Fig. 5 , respectively. The deviation of convexity parameters from their SM prediction are quite noticeable in these plots. In the presence of V R coefficient, the numerical values of the C τ F parameters are
C τ F | V R Λ b →p = −0.169 → −0.147 , C τ F | V R Λ b →Λc = −0.105 → −0.094 .(36)
The effect of V R coefficient is found to be rather significant on the forward-backward asymmetry observables of both Λ b → p(Λ c )τ −ν τ decay modes and the corresponding numerical values are
A τ F B | V R Λ b →p = −0.248 → 0.115 , A τ F B | V R Λ b →Λc = −0.23 → −0.09 .(37)
Left and right panels of Fig. 6, depict the variation of R p and R Λc parameters with respect to q 2 . Though there are no experimental limits on these parameters, significant deviation from their SM values are noticed in the scenario with only V R coupling. The right panel of Table III .
Though the presence of V L coefficient has no effect on the lepton and hadron polarization asymmetries of b → (u, c)τν τ decay modes, the V R coefficient has significant impact on these parameters. In the top panel of Fig. 8
P τ L | V R Λ b →p = −0.577 → −0.433 , P τ L | V R Λ b →Λc = −0.25 → −0.146 .(40)
C. Scenario C: Only S L coefficient
Here, we explore the impact of only S L coefficient on the angular observables of heavyheavy and heavy-light semileptonic decays of Λ b baryon. In section III, we discussed the constraints on the S L coupling. In the top panel Fig. 9 , we present the plots for the differential branching ratios of Λ b → pτν τ (left) and Λ b → Λ c τν τ (right) decay processes with respect to q 2 in the presence of S L coefficient. The corresponding plots for the forward- The lepton polarization asymmetry parameters provide profound deviation from the SM in comparison to their longitudinal hadron polarization parameters. The top-left panel of Fig. 18 shows the variation of R τ Λcp parameter with q 2 . In Table IV , we report the numerical values of all these parameters. In this subsection, we perform an analysis for semileptonic decay modes of Λ b baryon with the additional S R coupling. Using the allowed ranges of the real and imaginary part of S R coupling from Table II , We notice significant deviation of hadron and lepton polarization asymmetries from their corresponding SM values due to additional contribution from S R coupling. The plot for the R τ Λcp parameter with q 2 in the presence of only S R coefficient is presented in the right panel of Fig. 18 . The numerical values of all these parameters are presented in Table IV . Since the convexity parameters are independent of scalar type couplings, the S L,R coefficients play no role for this parameter.
The plots for the lepton nonuniversality parameter R p (left panel) and R Λc (right panel) are shown in Fig. 16 . The top panel of Fig. 17 represents the hadron polarization asymmetry parameters of Λ b → pτν τ (left panel) and Λ b → Λ c τν τ (right panel) process and the corresponding plots for lepton polarization asymmetries are given in the bottom panel of this figure. We observe that, the LNU parameter, longitudinal hadron and lepton polarization asymmetries of Λ b → pτν τ process have large deviation from their SM values due to the presence of tensor coupling, whereas negligible deviations (R Λc has some deviation from its SM result) are noticed for the observables of Λ b → Λ c τν τ decay mode. The q 2 variation of R τ Λcp parameter is depicted in the bottom panel of Fig. 18 . Table IV shows the integrated values of all these angular observables.
V. CONCLUSION
In this work, we have performed a model independent analysis of baryonic Λ b → (Λ c , p)lν l decay processes by considering the generalized effective Lagrangian in the presence of new physics. We considered the new couplings to be complex in our analysis. In order to constrain the new couplings, we have assumed that only one coefficient to be present at a time and constrained the new coefficients by comparing the theoretical predictions of Br(B + u,c → τ + ν τ ), Br(B → πτν τ ), R l π , R D ( * ) and R J/ψ observables with their measured experimental data. Using the allowed parameter space, we estimated the branching ratios, forward-backward asymmetries, convexity parameters of Λ b → (Λ c , p)lν τ decay processes. We also investigated the longitudinal polarization components of the daughter baryon (p, Λ c ) and the final state rameter, forward-backward asymmetries, lepton and hadron polarization asymmetries. We further, noticed profound deviation in the branching ratios and all other angular observables of semileptonic baryonic b → (u, c)τν l decay processes due to the additional contribution of V R coupling to the SM. The branching ratios, forward-backward asymmetries, longitudinal hadron and lepton polarization asymmetry parameter and the LNU observables deviate significantly from their corresponding standard model results in the presence of S L,R coefficients. These coefficients do not have significant effect on R Λcp parameter. We have also computed the branching ratio, forward-backward asymmetry, convexity parameter, hadron and lepton polarization asymmetries and LNU parameter of Λ b → p(Λ c )τν τ decay process by using the additional contribution from new tensor (T L ) coupling. All the angular observables of Λ b → pτν τ process receive significant deviations from their SM values, compared to the corresponding parameters of Λ b → Λ c τν τ decay mode. To conclude, we have explored the effect of individual complex V L,R , S L,R and T L couplings on the angular observables of baryonic decays of Λ b baryon. We found profound deviation from the standard model results due to the presence of these new couplings. We noticed that the V R and S L couplings
T,0 H V,0 + H T,+ H V,+ − H T,− H V,− ) +12Re[V R T * L ] m l q 2 (H T,0 H V,0 + H T,+ H V,− − H T,− H V,+ ) ,
FIG. 1 :
1. For the case of b → cτν τ decay processes, the constraints on the real and imaginary parts of individual V L (top-left panel), V R (top-right panel), S L (middle-left panel) and S R (middleright panel) coefficients obtained from R D ( * ) and R J/ψ parameters are shown in Fig. 2 . Till now, there is no precise determination of the form factors associated with tensorial operators for B c → J/ψlν l process both from the theoretical and experimental sides. In addition, the leptonic B c meson decays do not receive any contribution from tensor coupling. Therefore, Constraints on V L (top-left panel), V R (top-right panel), S L (middle-left panel), S R (middleright panel) and T L (bottom panel) coefficients associated with b → uτν τ transitions, obtained fromBr(B + u → τ + ν τ ), Br(B → πτν τ ), R l π observables.Here the constraint on T L coupling is obtained from Br(B → πτν τ ) experimental data.
FIG. 2 :
2we have presented the allowed values of (Re[V L(R) ] − Im[V L(R) ]) and (Re[S L(R) ] − Im[S L(R) ]) coefficients, which are compatible with the 3σ range of the experimental data. Constraints on V L (top-left panel), V R (top-right panel), S L (middle-left panel), S R (middleright panel) and T L (bottom panel) new coefficients associated with b → cτν τ transitions, obtained from Br(B + c → τ + ν τ ), R D ( * ) and R J/ψ observables. Here the constraint on T L coupling is obtained from R D ( * ) experimental data.
FIG. 3 :
3( * ) and R J/ψ observables, the allowed parameter space of V L couplings associated with b → (u, c)τ ν τ are shown in Figs. 1 and 2 respectively. Using the minimum and maximum values on real The q 2 variation of branching ratio of Λ b → pτ −ν τ (left panel) and Λ b → Λ + c τ −ν τ (right panel) processes in the presence of only V L new coefficient. Here the orange bands represent the new physics contribution. Blue dashed lines stand for the SM and the theoretical uncertainties arising due to the input parameters are presented in grey color.
Fig. 4 FIG. 4 :
44The variation of R p (left panel) and R Λc (right panel) LNU parameters with respect to q 2 in the presence of only V L new coefficient.
we show the plots for the branching ratios of Λ b → p (Λ c )τν τ process in the top-left panel (top-right panel) of Fig. 5 .
Fig. 7
7represents the q 2 variation of R τ Λcp parameter. The corresponding numerical values are listed in
FIG. 5 :
5, the distribution of the longitudinal polarization components of the daughter baryon p (left panel) and Λ c (right panel) are shown both Top panel represents the q 2 variation of branching ratio of Λ b → pτ −ν τ (left panel) and Λ b → Λ + c τ −ν τ (right panel) for only V R new coefficient. The corresponding plots of forward backward asymmetry and the convexity parameters are shown in the middle and bottom panels respectively. Here cyan bands are due to the additional new physics contribution coming from only V R coefficient. in the SM and in the presence of only V R coefficient, and the corresponding plots for the charged τ lepton are presented in the bottom panel. The integrated values of the hadron longitudinal polarization asymmetry parameters in the full physical phase space are P p L | SM Λ b →p = −0.897 , P Λc L | SM Λ b →Λc = −0.797 ,
P p L | V R Only Λ b →p = −0.897 → 0.276 , P Λc L | V R Only Λ b →Λc = −0.797 → −0.068 ,
FIG. 6 :FIG. 7 :
67The variation of R p (left panel) and R Λc (right panel) LNU parameters with respect to q 2 in the presence of only V R new coefficient. The variation of R τ Λcp parameter with respect to q 2 in the presence of only V L (left panel) and V R (right panel) new coefficients.and the corresponding numerical values for the charged lepton τ , areP τ L | SM Λ b →p = −0.514 , P τ L | SM Λ b →Λc = −0.207 ,
FIG. 8 :
8The plots in the left panel represent the longitudinal polarizations of daughter light baryon p (left-top panel) and the charged τ lepton (left-bottom) with respect to q 2 for only V R coefficient. The corresponding plots for Λ b → Λ c τ −ν τ mode are shown in the right panel. backward asymmetry are shown in the bottom panel. In these figures, the red bands stand for the NP contribution from S L coefficient. The additional contributions provide deviation in the branching ratios and forward-backward asymmetries from their SM values. The q 2 variation of the R p (left panel) and R Λc (right panel) LNU parameters in the presence of S L coupling are given in Fig. 10 . In the presence of only S L coupling, the longitudinal polarization components of the p (top-left panel) and Λ c (top-right panel) daughter baryons with respect to q 2 are presented in the top panel of Fig. 11 and the bottom panel depicts the longitudinal lepton polarization asymmetry parameters for Λ b → p(Λ c )τν τ processes.
FIG. 9 :
9Top panel represents the q 2 variation of branching ratios of Λ b → pτ −ν τ (left panel) and Λ b → Λ + c τ −ν τ (right panel) decay modes in the presence of only S L new coefficient. The corresponding plots for forward-backward asymmetries are shown in the bottom panel. Here red bands are due to the additional new physics contribution coming from only S L coefficient. D. Scenario D: Only S R coefficient
FIG. 10 :)
10the branching ratios of Λ b → pτν τ (left) and Λ b → Λ c τν τ The variation of R p (left panel) and R Λc (right panel) with respect to q 2 in the presence of only S L coefficient. FIG. 11: The plots in the left panel represent the longitudinal polarizations of daughter light baryon p (left-top panel) and the charged τ lepton (left-bottom) with respect to q 2 for only S L coefficient. The corresponding plots for Λ b → Λ c τ −ν τ mode are shown in the right panel.
(
right) decay processes with respect to q 2 are presented inFig. 12. The bottom panel of this figure represents the q 2 variation of the forward-backward asymmetry for Λ b → pτν τ (left) and Λ b → Λ c τν τ (right). In these figures, the green bands are due to the additional new contribution of S R coefficient to the SM. We observe profound deviation in the branching ratios and forward-backward asymmetries of these decay modes from their SM values. Left(right) panel of Fig. 13, show the effect of S R coupling on the q 2 variation of R p (R Λc ) parameter. The longitudinal polarization components of the p (top-left panel) and Λ c (topright panel) daughter baryons with respect to q 2 in the presence of contribution from only S R coefficient, are presented in the top panel of Fig. 14 and the bottom panel depict the longitudinal lepton polarization asymmetry parameters for Λ b → p(Λ c )τν τ processes.
)FIG. 13 :)
13FIG. 12: Top panel represents the q 2 variation of branching ratios of Λ b → pτ −ν τ (left panel) and Λ b → Λ + c τ −ν τ (right panel) decay processes in the presence of only S R coefficient. The corresponding plots for the forward-backward asymmetries are shown in the bottom panel. Here green bands stand for the additional new physics contribution coming from only S R coefficient. The variation of R p (left panel) and R Λc (right panel) with respect to q 2 in the presence of only S R coefficient. FIG. 14: The plots in the left panel represent the longitudinal polarizations of daughter light baryon p (left-top panel) and the charged τ lepton (left-bottom) with respect to q 2 for only S R coefficient. The corresponding plots for Λ b → Λ c τ −ν τ mode are shown in the right panel. E. Scenario E: Only T L coefficient The sensitivity of tensor coupling on various physical observables associated with semileptonic baryonic b → (c, u)τν τ decay processes will be investigated in this subsection. The allowed region of real and imaginary parts of the tensor coupling are presented in section III. Using all the input parameters and the constrained new tensor coefficient, we show the q 2 variation of branching ratio (left-top panel), forward-backward asymmetry (left-middle panel) and convexity parameter (left-bottom panel) of Λ b → pτν τ decay mode in the left panel of Fig. 15 . The right panel of this figure represents the corresponding plots for Λ b → Λ c τν τ process. Here the magenta bands represent the additional contribution coming from the new T L coefficient. For Λ b → pτν τ process, as the bound on T L is weak, the branching ratio, forward-backward asymmetry and the convexity parameter deviate significantly from their SM predications compared to the observables for Λ b → Λ c τν τ process. For Λ b → Λ c τν τ process, the deviations are quite minimal as the coefficient T L is severely constrained. In the presence of T L coefficient, the numerical values of the convexity parameters are C τ F | T L Λ b →p = −0.017 → −0.027 , C τ F | T L Λ b →Λc = −0.121 → −0.098 .
)
FIG. 15: Top panel represents the q 2 variation of branching ratio of Λ b → pτ −ν τ (left panel) and Λ b → Λ + c τ −ν τ (right panel) for only T L new coefficient. The corresponding plots of forward backward asymmetry and the convexity parameters are shown in the middle and bottom panels respectively. Here magenta bands are due to the additional new physics contribution coming from only T L coefficient.
FIG. 16 :))
16charged lepton, τ . The convexity parameter only depend on the (axial)vector and tensor type couplings and are independent of the S L,R , T L coefficients. Inspired by the observation of lepton non-universality parameters in various B meson decays, we have also scrutinized The variation of R p (left panel) and R Λc (right panel) with respect to q 2 in the presence of only T L coefficient. FIG. 17: The plots in the left panel represent the longitudinal polarizations of daughter light baryon p (left-top panel) and the charged τ lepton (left-bottom) with respect to q 2 for only T L coefficient. The corresponding plots for Λ b → Λ c τ −ν τ mode are shown in the right panel. the lepton universality violating parameters (R p , R Λc , R τ Λcp ) in the baryonic decay modes. We found significant deviation in the branching ratios and the R p , R Λc , R Λcp parameters from their corresponding standard model values, in the presence of additional new vector like coupling (V L coefficient). However, such coupling does not affect the convexity pa-FIG. 18: The variation of R τ Λcp parameter with respect to q 2 in the presence of only S L (top-left panel), S R (top-right panel) and T L (bottom panel) coefficients.
R
IV: The predicted values of branching ratios, forward-backward asymmetries, longitudinal hadron ad lepton polarization asymmetries and lepton non-universality parameters of Λ b → (Λ c , p)τν τ processes in the SM and in the presence of only S L,R and T L new coefficients.ObservablesValues for S L coupling Values for S R coupling Values for T L couplingBr(Λ b → pτ −ν τ )(2.98 − 5.25) × 10 −4 (2.98 − 3.48) × 10 −4 (0.298 − 6.68) Λcp (1.693 − 1.95) × 10 −2 (1.582 − 1.693) × 10 −2 0.0192 − 0.367 significantly affect all the observables and the tensor coupling plays a vital role in the case of Λ b → pτν τ decay mode. Though there is no experimental measurement on these baryonic b → (u, c)τν τ decay processes, the study of these modes are found to be very crucial in order to shed light on the nature of new physics.
TABLE I :
IList of measured lepton non-universality parameters.
TABLE II :
IIAllowed ranges of the new coefficients.Decay processes New coefficients Minimum value Maximum Valueb → uτν τ
(Re[V L ], Im[V L ])
(−2.489, −1.5)
(0.504, 1.48)
(Re[V R ], Im[V R ]) (−0.478, −1.185)
(0.645, 1.198)
(Re[S L ], Im[S L ]) (−0.136, −0.396)
(0.672, 0.398)
(Re[S R ], Im[S R ]) (−0.6743, −0.398) (0.1265, 0.398)
(Re[T L ], Im[T L ]) (−0.473, −0.773)
(1.07, 0.773)
b → cτν τ
(Re[V L ], Im[V L ]) (−2.224, −1.228)
(0.225, 1.225)
(Re[V R ], Im[V R ]) (−0.129, −0.906)
(0.173, 0.89)
(Re[S L ], Im[S L ]) (−0.116, −0.788)
(0.474, 0.8)
(Re[S R ], Im[S R ]) (−1.076, −0.809)
(0.06, 0.807)
(Re[T L ], Im[T L ]) (−0.0094, −0.028) (0.0467, 0.028)
TABLE III :
IIIThe predicted values of branching ratios and lepton non-universality parameters of Λ b → (Λ c , p)τν τ processes in the SM and in the presence of only V L,R coefficients.Observables SM prediction Values for V L coupling Values for V R coupling Br(Λ b → pτ −ν τ ) (2.98 ± 0.238) × 10 −4 (0.298 − 1.34) × 10 −3 (2.98 − 8.17) × 10 −4R p
0.692
0.692 − 3.09
0.692 − 1.895
Br(Λ b → Λ +
c τ −ν
τ ) (1.76 ± 0.14) × 10 −2
(1.76 − 5.29) × 10 −2
(1.76 − 3.4) × 10 −2
R Λc
0.353
0.353 − 1.06
0.353 − 0.68
R Λcp
(1.693 ± 0.19) × 10 −2 (1.693 − 2.533) × 10 −2 (1.693 − 2.4) × 10 −2
V R Only
TABLE
Appendix A: Helicity-dependent differential decay ratesThe expressions for the helicity-dependent differential decay rates required to analyze the longitudinal hadron and lepton polarization asymmetries are given by[33]Appendix B: Form factors relationsThe relation betwen various form factors are given as[51,52]f
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| {'fraction_non_alphanumeric': 0.0860639712207711, 'fraction_numerical': 0.0684875263939939, 'mean_word_length': 3.2978741282245188, 'pattern_counts': {'":': 0, '<': 1, '<?xml version=': 0, '>': 0, 'https://': 1, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 122, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'In recent times, several hints of lepton non-universality have been observed in semileptonic B meson decays, both in the charged-current (b → clν l ) and neutral-current (b → sll) transitions.Motivated by these intriguing results, we perform a model independent analysis of the semileptonic Λ b decays involving the quark level transitions b → (u, c)lν l , in order to scrutinize the nature of new physics. We constrain the new parameter space by using the measured branching ratios of B + c,u → τ + ν τ , B → πτ ν τ processes and the existing experimental results on R D ( * ) , R J/ψ and R l π parameters. Using the constrained parameters, we estimate the branching ratios, forward backward asymmetries, hadron and lepton polarization asymmetries of the Λ b → (Λ c , p)lν l processes.Moreover, we also examine whether there could be any lepton universality violation in these decay modes.', 'arxivid': '1812.08314', 'author': ['Atasi Ray *[email protected]†[email protected]‡[email protected] \nSchool of Physics\nUniversity of Hyderabad\nHyderabad-500046India\n', 'Suchismita Sahoo \nTheoretical Physics Division\nPhysical Research Laboratory\nAhmedabad-380009India\n', 'Rukmani Mohanta \nSchool of Physics\nUniversity of Hyderabad\nHyderabad-500046India\n'], 'authoraffiliation': ['School of Physics\nUniversity of Hyderabad\nHyderabad-500046India', 'Theoretical Physics Division\nPhysical Research Laboratory\nAhmedabad-380009India', 'School of Physics\nUniversity of Hyderabad\nHyderabad-500046India'], 'corpusid': 119273234, 'doi': '10.1103/physrevd.99.015015', 'github_urls': [], 'n_tokens_mistral': 19997, 'n_tokens_neox': 16553, 'n_words': 9641, 'pdfsha': '850c78c3fb326a58f5177f8cfc7caf2063c32d4a', 'pdfurls': ['https://arxiv.org/pdf/1812.08314v1.pdf'], 'title': ['Probing new physics in semileptonic Λ b decays', 'Probing new physics in semileptonic Λ b decays'], 'venue': []} |
arxiv |
A Comparison of natural (english) and artificial (esperanto) languages. A Multifractal method based analysis
(Dated: February 3, 2008)
J Gillet
M Ausloos
Institut de Physique Nucléaire, Atomique et de Spectroscopie
GRAPES
B5a Sart-Tilman
B-4000LiègeBelgium
Université de Liège
B-4000LiègeBelgium
A Comparison of natural (english) and artificial (esperanto) languages. A Multifractal method based analysis
(Dated: February 3, 2008)
We present a comparison of two english texts, written by Lewis Carroll, one (Alice in wonderland) and the other (Through a looking glass), the former translated into esperanto, in order to observe whether natural and artificial languages significantly differ from each other. We construct one dimensional time series like signals using either word lengths or word frequencies. We use the multifractal ideas for sorting out correlations in the writings. In order to check the robustness of the methods we also write (!) (consider? ) the corresponding shuffled texts. We compare characteristic functions and e.g. observe marked differences in the (far from parabolic) f (α) curves, differences which we attribute to Tsallis non extensive statistical features in the frequency time series and length time series. The esperanto text has more extreme vallues. A very rough approximation consists in modeling the texts as a random Cantor set if resulting from a binomial cascade of long and short words (or words and blanks). This leads to parameters characterizing the text style, and most likely in fine the author writings.
We present a comparison of two english texts, written by Lewis Carroll, one (Alice in wonderland) and the other (Through a looking glass), the former translated into esperanto, in order to observe whether natural and artificial languages significantly differ from each other. We construct one dimensional time series like signals using either word lengths or word frequencies. We use the multifractal ideas for sorting out correlations in the writings. In order to check the robustness of the methods we also write (!) (consider? ) the corresponding shuffled texts. We compare characteristic functions and e.g. observe marked differences in the (far from parabolic) f (α) curves, differences which we attribute to Tsallis non extensive statistical features in the frequency time series and length time series. The esperanto text has more extreme vallues. A very rough approximation consists in modeling the texts as a random Cantor set if resulting from a binomial cascade of long and short words (or words and blanks). This leads to parameters characterizing the text style, and most likely in fine the author writings.
PACS numbers:
As soon as modern fractals appeared in order to describe physical objects, it was evident that some generalization was in order: multifractals spurred up, e.g., since obvioulsy a fractal dimension D is not enough to describe an object [1,2]. The more so in non equilibrium systems, characterized by some unusual dynamics. Through a generator and from an initiator one can produce a fractal object with a given dimension. How to produce realistic and meaningful multifractal models is a challenge. Do they really exist [3]? Do multifractal model exist nowadays [4]? These questions come in parallel with the measurement of the fractal dimension, ... and its distribution. One question of interest is whether the apparently multifractal nature of an object is due to its finite size or to a complex dynamical feature or something else! Some attempt in this direction results from observation of multifractal features in meteorology and climate studies [5,6,7], but also in many other fields [8], like mathematical finance [4,9,10,11,12,13] Let us recall that one has basically to obtain a D(q) function or the f (α) spectrum, where q represents the degree of some moment distribution of some variable, and α is some sort of critical exponent at phase transitions, also called the Holder exponent; f (α) being its distribution.
There is a need for experimental work leading to reliable D(q) and f (α) data, before modeling. Interesting pioneering data should be here recalled : see work on DLA [14,15], DNA [16,17], SOI [18], NAO [19], .... It appears that most of the time some "signal is either directly a time series or is transformed into a time series; more generally, the signal is called a text, because it can be decomposed through level thresholds which can be thought to be a set of characters taken from an alphabet. Here below we take real texts in fact as the source of experimental observations and follow the multifractal ideas to make an analysis of such texts. The main question concerns whether multifractals are indeed found in real texts; a question raised in [20]; another is whether the technique of analysis can give some insight on a logical construction [21], from which stems the possible connection of such ideas with coding, transcription, machine translation, social distances, network properties of languages, ... [21] [22] [23] and more classically in physics about identifying coherent structures in spatially extended systems [24]. We examine two classical english texts but also one translation, into an unusual language, and the corresponding shuffled texts. We focus on how local/structural properties develop into global ones.
Since Shannon [25], writings and codings are of interest in statistical physics. Writings are systems practically composed of a large number of internal components (the words, signs, and blanks in printed texts). In terms of complexity investigations, writings which are a form of recorded languages, like living systems, belong to the top level class involving highly optimized tolerance design, error thresholds in optimal coding, and financial markets [21]. Relevant questions pertain to the life time, concentration, distribution, .. complexity of these. One should distinguish two main frameworks. On one hand, language developments seem to be understandable through competitions, like in Ising models, and in self-organized systems. Their diffusion seems similar to percolation and nucleation-growth problems taking into account the existence of different time scales, for inter-and intra-effects; this is the realm of anthropology. The second frame originates from more classical linguistics studies; it pertains to the content and meanings of words and texts. Concerning the internal structure of a text, supposedly char-arXiv:0801.2510v1 [cs.CL] 16 Jan 2008 acterized by the language in which it is written, it is well known that a text can be mapped into a signal, of course through the alphabet characters. However it can be also reduced to less abundant symbols through some threshold, like a time series, which can be a list of +1 and -1, or sometimes 0. In fact, laws of text content and structures have been searched for a long time ago by Zipf and others, see many refs. in [26], through the least effort (so called ranking) method. The technique is now currently applied in statistical physics as a first step to obtain, when they exist, the primary scaling law. Yet long range order correlations (LROC) between words in text are searched for. In [27] it was claimed that LROC express an author's ideas, and in fine consist in some author's signature.
Interestingly writings can be thought as social networks [23]. Social networks have fractal properties [28]; most usually they should be multifractals; one can thus imagine/consider that a text which is a form of partially self-organized social network (for words) due to grammatical and style constraints present multifractal features. The properties of such texts taken as signals have already been examined, e.g., a multifractal analysis of Moby Dick letter distribution can be found in [20].
Even though we recognize such a pioneering paper, we stress that sentences made of words, not letters, are translated. Thus we present below an original consideration in this respect, i.e. the analysis and results about a translation between one of the most commonly used language, i.e. english, and a relatively recent language, i.e. esperanto. Esperanto is an artificially constructed language [29], which was intended to be an easy-to-learn lingua franca. Statistical analyses seem to indicate that esperanto's statistical proportions are similar to those of other languages [30]. It was found that esperanto's statistical proportions resemble mostly those of German and Spanish, and somewhat surprisingly least those of French and Italian. English seems to be the intermediary case.
Comparison of different languages (writings) arising from apparently different origins or containing different signs, e.g. greek [31], turkish [32], chinese [33], ... even somewhat artificial languages, like those used for simulation codes on computers [34] has also been made. To our knowledge few comparisons have been presented about written texts translated from one to another language [35,36] and in particular from the point of view of LROC in words.
The text used here was chosen for its wide diffusion, freely available from the web [37] and as a representative one of a famous scientist, Lewis Caroll, i.e.
Alice in wonderland (AWL) [38]. Moreover knowing the special (mathematical) quality of this author mind, and some, as we thought a priori, some possibly special way of writing, another text has been chosen for comparison, i.e. Through a looking glass (TLG) [39]; -to our knowledge only available in english (on the web). This will allow us to discuss whether the differences, if any, between esperanto and english, are apparently due to the transla-tion or to the specificity of this author work. Previous work on the english AWL version (AWL eng ) should be mentioned [40], but pertains to a mere Zipf analysis.
In Sect. I, we present some elementary facts on these texts and briefly expose the methodology, i.e. we emphasize that we distinguish between "frequency time series (FTS) and "length time series (LTS). We recall the multifractal technique for this specific application. In order to check the robustness of the method we also invent (write !) (or consider ) the corresponding shuf f led texts, to which we apply the same technique of analysis. Therefore, in Sect. II, we present the somewhat unusual results, and discuss them in Sect. III. For the simplicity of the discussion we very roughly approximate the texts as resulting from a binomial cascade of short and long words. We obtain parameters in fine characterizing the text style and the author's writings. We observe that such multifractals have a deep connection to Tsallis non extensive statistics as pointed out in ref. [41] in another framework.
I. DATA AND METHODOLOGY
For our empirical considerations we have selected the two texts here above mentioned downloading them from a freely available site [37], resulting obviously into three files, called. Next, we have removed the chapter heads. All our analyses are carried over this reorganized file for each text. Thereafter, we have shuffled these texts, in the files, without taking into account the punctuation [42].
There are two ways to construct a time series from such documents Obviously there is a large number of ways to map a text to a time series, but in the present study we only consider the above two since some physical meaning can be thought to arise in the mapping. As indicated in [43] the length of the word is associated with speaker effort, meaning that the longer the word the higher the effort required to pronounce it. The frequency of the word is also associated with the hearer effort as frequently used words require less effort to be understood by the hearer. These time series are thereby analyzed along the multifractal ideas, for which we briefly recall the formulae of interest in order to set the notations.
A. Multifractal Analysis
Let the (LTS ou FTS) time series having N data points (words, here), i.e. y i (1 ≤ i ≤ N ).
Transform the series as follows: if the length of a word mot in LTS (or its frequency in FTS) is smaller than the next one, the former word gets a value = 2; if it is greater, it gets the value = 1; and 0 if both are equal.
The new series is called M i (1 ≤ i ≤ N − 1). Each M i is cut into N s subseries of size s , where N s is the smallest integer in N/s. The ordering starts from the beginning of the text (contrary to analyses in which some forecasting is expected, and for which the end "points are more relevant).
Next one calculates the probability
P (s, ν) = Σ s i=1 M (ν−1)s+i Σ Ns ν=1 Σ s i=1 M (ν−1)s+i(1)
for every ν and s. Thereafter one calculates
χ(s, q) = Σ Ns ν=1 P (s, ν) q(2)
for each s value. A power law behaviour is expected
χ(s, q) ∼ s τ (q) ;(3)
where τ (q) plays the role of the partition function [1]. The generalized fractal dimension D(q) [1,2] is defined from
D(q) = τ (q) q − 1(4)
and the generalized Hurst exponent, h(q), from
h(q) = 1 + τ (q) q .(5)
Let
α = dτ (q) dq ,(6)
from which one obtains the generalized critical exponent, f (α) curve [44] from
f (α) = qα − τ (q).(7)
In the present work, we have calculated χ(s, q) for s between 2 and 200. The τ (q) values were calculated by a linear best fit on a log-log plot of χ(s, q) and s, for q values ranging from -25 till 25. As one may expect the q and q − 1 values at the denominator in Eq.(4)-Eq.(5) were leading to numerical singularities. A smooth interpolation can be visually made without difficulty. For conciseness we don't show χ(s, q).
II. RESULTS
The results of the FTS and LTS multifractal analysis for the three main texts and their shuffled corresponding ones are shown in Figs. 1-4 (a-b).
A mere perusing of the graphs indicate that the multifractal approach is in good order, e.g. since D(q) is not a single point, and should allow one to observe interesting LRO correlations and local ones.
A. D(q) plots: Figs. 1-2 In FTS, the generalized fractal dimension has a similar set of values for both english texts, decaying from ca. 1.2 to 1.0 for q increasing but negative; D(q) decays slowly for q positive, barely reaching a value 0.95 for q = 80 (Fig. 1). The value of D(q) is much greater along the negative q axis, for AWL esp but is identical to the other two for q ≥ 0. In LTS, even though the form of D(q) is that to be expected, it has to be stressed that the AWL eng and AWL esp are very quantitatively similar, but markedly differ from TLG eng . This already indicates that one can observe the high creativity of the author through these two books. Moreover the translation effect on style is much better seen on FTS than LTS.
The shuffled texts (Fig.2) remarkably have the same D(q) values; their range and variations being similar to those of the real texts. Slight quantitative differences occur, more markedly for the shuffled AWL esp FTS, but along a Baeysian reasoning these can be attributed to the finite size of the sample.
By the way,
C 1 = dτ (q) dq q=1 (8)
is a measure of the intermittency lying in the signal y(n); it can be numerically estimated by measuring τ q around q = 1. In all cases the value of C 1 is close to unity. Some conjecture on the role/meaning of C 1 is found in Ref. [45]. From some financial and political data analysis it seems that H 1 is a measure of the information entropy of the system. The same can be thought of here.
B. f (α) plots: Figs. 3-4
The f (α) spectra are shown in Figs. 3-4. They are markedly non symmetric, as was found for DLA [14,15], with very high positive skewness, i.e. for q ≤ 0. Interestingly, the esperanto text curve behaves differently from the english texts, in FTS, though TLG eng is different in the LTS case; the shuffled texts f (α) spectra behaving in a very similar qualitative and quantitative way. However the shuffling does not fully symmetrize the spectra.
It is interesting to observe that the f (α) curve is very sharp: it originates from negative values for α less than 1.0; reaches a maximum (=1.0) at 1.0, at the maximum so called box dimension, and decays rapidly for α positive; f (α) = 0 at α= 1.2 and 1.3 respectively for AWL eng and TLG eng ; the maximum is also reached at (1.0, 1.0) and the spectrum spans the (narrow) interval 0.90: 1.25 for AWL esp on the f (α) =0 line. It is worth noticing that the values are reasonable in view of their correspondence to the fractal dimensions. On the other hand the sharpness indicates a high lack of uniformity of the texts LROC.
III. DISCUSSION WITH CONCLUSION
In summary, one can observe similarities between the original and shuffled texts and their translations; see Table 1 for summarizing the similarities seen through D(q) and f (α). The english texts look more similar with each other than with respect to the esperanto translation. On the other hand, one physical conclusion arises from the above : the existence of a multifractal spectrum found for the examined texts indicates a multiplicative process in the usual statistical sense for the distribution of words length and frequency in the text considered as a time series. Thus linguistic signals may be considered indeed as the manifestation of a complex system of high dimensionality, different from random signals or systems of low dimensionality such as the financial and geophysical (climate) signals. Finally, the f (α) curve represents the measurable aspects of the word networks, be they considered through LTS or FTS. Our work confirms that texts could be seen as networks indeed [23].
Before suggesting a physical model describing the construction of a writing, let us consider implications from the f (α) spectrum in some detail. The not fully parabolic, to say the least, f (α) curve indicates non uniformity and strong LROC between long words and small words, -evidently arising from strong short range correlations between these. In some sense this is expected for classical writings. It is usually known that the left (right) hand side of the f (α) curve corresponds to fluctuations of the q ≥ 0 (q ≤ 0)-correlation function. In other words, they correspond to fluctuations in small (large) word distributions. Therefore the distribution of small and long words should be examined in further work in order to observe these local correlations, e.g. through a detrended fluctuation analysis.
Moreover, in order to characterize the writings, texts and/or authors, we propose a very rough approximation/model, i.e. let us consider (assume !) that the writings are made of only two types of words : small and large [46,47], appearing through some recursive process. In so doing one can consider the behavior of the atypical f (α) curve as originating form a binomial cascade of short and long words, on a support [0,1], with an arbitrary contraction ratio r i and a weight w i for the word in each successive subinterval, as for a multifractal Cantor set construction [1]. For an arbitrary number n of subin- tervals the generalized fractal dimension (or rather τ (q)) is obtained from
Series D(q) D s (q) f (α) f s (α)n i=1 w q i r −τ i = 1.(9)
The formula is easily generalized for random contraction ratios and weights. However it simplifies for the case of a simple binomial cascade, i.e.
w q 1 r −τ 1 + w q 2 r −τ 2 = 1.(10)
Whence the extremal α values read α − = log(w 2 )/ log(r 2 ) and α + = log(w 1 )/ log(r 1 ), from which the weights and ratios can be estimated by inversion (table I), thereby suggesting the author's somewhat systemic way used in his/her writings.
The physics connection can be obtained if one relates the f (α) curve extremal points through their physical meanings [48], i.e.
1 1 − Q = 1 α + − 1 α −(11)
where α − and α + are the extremes of the range of support for the (positive) multifractal spectrum f (α) and Q (instead of the usual q) is used to represent the parameter arising in the non extensive description of statistical physics [49]; see values in table I. By extension it is a measure of the attractor dimension or the number of so called degrees of freedom. It is obvious from the table that Q varies between 4 and 7, with interesting differences between the LTS and FTS cases, LTS's Q being systematically smaller, in the original or shuffled texts. Notice that the value of Q is more extreme though with the same order of magnitude in the case of the esperanto text for both types of series.
Finally we re-emphasize the remarkable difference for the esperanto text (Fig. 3a) with the english texts in the FTS analysis. Linguistics input should be searched at this level and is left for further discussion. The origin of differences between TLG and AWL needs more work at the linguistic level. However we have indicated the interest of the multifractal scheme in providing a measure of these correlations, thus a new measure of an author's style. This suggests a (binomial, at first) cascade model containing parameters characterizing (or reflecting, at least) the text style, and most likely in fine the author writings. It remains to be seen whether the f (α) curve and the (to be generalized) binomial cascade model, with the weight and ratio parameters hold through in other cases, and can characterize authors and texts, -and in general time series. Moreover the multifractal method should additionally be able to distinguish a natural language signal from a computer code signal [43] and help in improving translations by suggesting perfection criteria and indicators of text qualitative values.
1 .
1Take a document of N words. Select all the different words. Count the frequency f of appearance of each word in the document. The time of "appearance is played by the rank position of the word in the file. We map the word frequency to a time series f (t). Such a time series is called the frequency time series (FTS).2. Take a document of N words. Consider the length l (number of letters) of each word. Record where each word of length l is located in the text; the time is played by the position of the word in the document, i.e. the first word is considered to be emitted at time t= 1, the second at time t = 2, etc. A time series l(t) is so constructed. We refer to such a time series as the length time series (LTS).
FIG
. 1: D(q) for (a) FTS (b) LTS of the original texts
FIGFIG
. 2: D(q) for (a) FTS (b) LTS of the shuffled indicated texts . 4: f (α) for (a) FTS (b) LTS of the shuffled indicated texts
AWLeng AWLeng AWLeng AWLeng FTS TLGeng T LGeng T LGeng T LGeng AWLeng AWLeng AWLeng AWLeng LTS AWLesp T LGeng AWLesp T LGeng
TABLE I :
IComparing original texts quasi identical behaviors through functions D(q) and f (α), and their counterpart for shuffled (s) cases, i.e. D s (q) and f s (α)Original texts Q α − α + w 1 w 2 r 1 r 2AWLeng FTS 5.71 0.95 1.19 0.96 0.04 0.97 0.03
AWLesp FTS 4.39 0.94 1.30 0.99 0.01 0.99 0.01
TLGeng FTS 5.71 0.95 1.19 0.99 0.01 0.99 0.01
AWLeng LTS 4.65 0.92 1.23 0.89 0.11 0.91 0.09
AWLesp LTS 4.83 0.92 1.21 0.87 0.13 0.89 0.11
TLGeng LTS 3.94 0.92 1.34 0.96 0.04 0.97 0.03
Shuffled texts Q α − α + w 1 w 2 r 1 r 2
AWLeng FTS 6.94 0.95 1.13 0.89 0.11 0.90 0.10
AWLesp FTS 6.57 0.96 1.16 0.97 0.03 0.97 0.03
TLGeng FTS 6.59 0.94 1.13 0.82 0.18 0.84 0.16
AWLeng LTS 4.35 0.91 1.25 0.88 0.12 0.90 0.10
AWLesp LTS 4.56 0.92 1.24 0.90 0.10 0.92 0.08
TLGeng LTS 4.35 0.91 1.25 0.88 0.12 0.9 0.10
TABLE II :
IITsallis Q-parameter, derived from α− and α+ values, read on Figs. 3-4, from Eq. (11), for the original texts and for shuffled or translated corresponding texts, according to the type of series; the weights and ratios of the binomial cascade approximation (see text) are given
Acknowledgements The authors would like to thank D. Stauffer for as usual fruitful discussions .... This work has been supported by European Commission Project CREEN FP6-2003-NEST-Path-012864.
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The first "data point is exchanged with a following one, its location chosen from a generated random number. The second data point is exchanged with a following one, chosen from another random number, etc. The random number generator was check to lead to a rather uniform distribution. The shuffle algorithm is found on wikipedia. The algorithm was applied ten times on the texts to get the final shuffled texts we usedThe shuffle algorithm is found on wikipedia. The first "data point is exchanged with a following one, its loca- tion chosen from a generated random number. The sec- ond data point is exchanged with a following one, chosen from another random number, etc. The random number generator was check to lead to a rather uniform distribu- tion, for a number between 0 and 1. The algorithm was applied ten times on the texts to get the final shuffled texts we used.
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We apologize to all authors for this simplification of their work. Yet, in some sense this is equivalent to considering that the (+/-1) Ising model describes all magnetic features. Notice that this assumption on writings has been successfully made in the study of blogs. 47] where words occurring with an equivalent frequency are considered to be identical for studying the blogs statistical propertiesWe apologize to all authors for this simplification of their work. Yet, in some sense this is equivalent to considering that the (+/-1) Ising model describes all magnetic fea- tures. Notice that this assumption on writings has been successfully made in the study of blogs, [47] where words occurring with an equivalent frequency are considered to be identical for studying the blogs statistical properties.
Word statistics in Blogs and RSS feeds: Towards empirical evidence. R Lambiotte, M Ausloos, M Thelwall, J. Informetrics. 1R. Lambiotte, M. Ausloos, M. Thelwall, Word statistics in Blogs and RSS feeds: Towards empirical evidence, J. Informetrics 1 (2007) 277 -286.
Analysis of turbulence by statistics based on generalized entropies. T Arimitsu, N Arimitsu, Physica A. 295177T. Arimitsu and N. Arimitsu, Analysis of turbulence by statistics based on generalized entropies, Physica A 295 (2001) 177.
Nonextensive statistics: theoretical, experimental and computational evidences and connections, Braz. C Tsallis, J. Phys. 291C. Tsallis, Nonextensive statistics: theoretical, exper- imental and computational evidences and connections, Braz. J. Phys. 29 (1999) 1.
| {'fraction_non_alphanumeric': 0.05562333491024077, 'fraction_numerical': 0.030475811069890196, 'mean_word_length': 4.331032651977701, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 1, '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': 'We present a comparison of two english texts, written by Lewis Carroll, one (Alice in wonderland) and the other (Through a looking glass), the former translated into esperanto, in order to observe whether natural and artificial languages significantly differ from each other. We construct one dimensional time series like signals using either word lengths or word frequencies. We use the multifractal ideas for sorting out correlations in the writings. In order to check the robustness of the methods we also write (!) (consider? ) the corresponding shuffled texts. We compare characteristic functions and e.g. observe marked differences in the (far from parabolic) f (α) curves, differences which we attribute to Tsallis non extensive statistical features in the frequency time series and length time series. The esperanto text has more extreme vallues. A very rough approximation consists in modeling the texts as a random Cantor set if resulting from a binomial cascade of long and short words (or words and blanks). This leads to parameters characterizing the text style, and most likely in fine the author writings.', 'arxivid': '0801.2510', 'author': ['J Gillet ', 'M Ausloos ', '\nInstitut de Physique Nucléaire, Atomique et de Spectroscopie\nGRAPES\nB5a Sart-Tilman\nB-4000LiègeBelgium\n', '\nUniversité de Liège\nB-4000LiègeBelgium\n'], 'authoraffiliation': ['Institut de Physique Nucléaire, Atomique et de Spectroscopie\nGRAPES\nB5a Sart-Tilman\nB-4000LiègeBelgium', 'Université de Liège\nB-4000LiègeBelgium'], 'corpusid': 15532815, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 12052, 'n_tokens_neox': 10518, 'n_words': 6418, 'pdfsha': '11ea866095a0ca430591c439724ce14a5fb9fa71', 'pdfurls': ['https://arxiv.org/pdf/0801.2510v1.pdf'], 'title': ['A Comparison of natural (english) and artificial (esperanto) languages. A Multifractal method based analysis', 'A Comparison of natural (english) and artificial (esperanto) languages. A Multifractal method based analysis'], 'venue': []} |
arxiv |
CONFORMALLY EINSTEIN-MAXWELL KÄHLER METRICS AND STRUCTURE OF THE AUTOMORPHISM GROUP
Akito Futaki
Hajime Ono
CONFORMALLY EINSTEIN-MAXWELL KÄHLER METRICS AND STRUCTURE OF THE AUTOMORPHISM GROUP
Let (M, g) be a compact Kähler manifold and f a positive smooth function such that its Hamiltonian vector field K = Jgrad g f for the Kähler form ωg is a holomorphic Killing vector field. We say that the pair (g, f ) is conformally Einstein-Maxwell Kähler metric if the conformal metricg = f −2 g has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kähler manifolds. More generally we consider extensions of Calabi functional and extremal Kähler metrics, and prove an extension of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kähler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture. into type (1, 0)-part and type (0, 1)-part. Naturally T * J M = T * J M . For another complex structure J ∈ Z K we also have) and the last isomorphism uses the Kähler metric. As an element of the last term, v is in the symmetric part C ∞ (Sym(T J M ⊗ T J M )), see the proof of Lemma 2.1 in[13]. So, we havethe last term being the underlying real vector space of C ∞ (Sym(T J M ⊗ T J M )). The L 2 -inner product with respect to the volume form f −2m+1 ω m gives a Kähler structure on Z K . If J = J + δJ andThus, up to the first order,If J(t) is any smooth curve in Z K for t ∈ (− , ) with J(0) = J, (4) shows that we may regard J −1J ∈ C ∞ (End(T * J M, T * J M )) R . HereJ denotes the derivative of J(t) at t = 0. But (4) also shows
Introduction.
Let (M, J) be a compact complex manifold admitting a Kähler structure. A Hermitian metricg on (M, J) is said to be a conformally Kähler, Einstein-Maxwell (cKEM for short) metric if there exists a positive smooth function f on M such that g = f 2g is Kähler, that the Hamiltonian vector field K = Jgrad g f for the Kähler form ω g is Killing for g (and also forg necessarily), and that the scalar curvature sg ofg is constant.
K is necessarily a holomorphic vector field since any Killing vector field on a compact Kähler manifold is holomorphic. If f is a constant function then g is a Kähler metric of constant scalar curvature. When f is not a constant function, typical known examples are conformally Kähler, Einstein metrics by Page [27] on the one-point-blow-up of CP 2 , by Chen-LeBrun-Weber [8] on the two-point-blow-up of CP 2 , by Apostolov-Calderbank-Gauduchon [1], [2] on 4-orbifolds and by Bérard-Bergery [4] on P 1 -bundles over Fano Kähler-Einstein manifolds.
In the more recent studies, non-Einstein cKEM examples are constructed by LeBrun [20], [21] showing that there are ambitoric examples on CP 1 × CP 1 and the one-point-blow-up of CP 2 , and by Koca-Tønnesen-Friedman [18] on ruled surfaces of higher genus. The authors [15] also extended Le-Brun's construction on CP 1 × CP 1 to CP 1 × M where M is a compact constant scalar curvature Kähler manifold of arbitrary dimension.
In this paper we are interested in finding a Kähler metric g in a fixed Kähler class and a positive Hamiltonian Killing potential f such thatg = f −2 g is a conformally Kähler, Einstein-Maxwell metric. This point of view is taken by Apostolov-Maschler [3]. Thus we may call such a Kähler metric g a conformally Einstein-Maxwell Kähler metric. Fixing a Kähler class, the primary question is which choice of a Killing vector field K is the right one. The answer is given by the volume minimization as shown by [15] in the same spirit as Kähler-Ricci solitons by Tian-Zhu [28] and Sasaki-Einstein metrics by Martelli-Sparks-Yau [26], see also [16]. The critical points of the volume functional considered in [15] are precisely those Killing vector fields such that the natural obstruction defined in [3] vanishes. This obstruction will be introduced in Remark 2.5 and Remark 2.6 in this paper, and is an extension of the one given for constant scalar curvature Kähler metrics in [11], [12]. The secondary question is whether the Lichnerowicz-Matsushima theorem ( [24], [23]) asserting the reductiveness of the Lie algebra of holomorphic vector fields can be extended for conformally Einstein-Maxwell Kähler manifolds. The main theorem of this paper, Theorem 2.1, gives an affirmative answer to this question 1 .
More generally we consider extensions of Calabi functional and extremal Kähler metrics and prove an extension of Calabi's theorem for extremal Kähler manifolds. We call the extended metric an f -extremal metric where f stands for the Hamiltonian function of a Killing vector field K. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture. The arguments in the finite dimensional setting is given by Wang [29]. Wang's arguments were effectively employed in [14] to show an extension of Calabi's structure theorem for extremal Kähler metrics to perturbed extremal Kähler metrics defined in [13]. The proof of the present paper follows closely the lines of arguments in [14], but we will not avoid overlaps with [14] so as to make the exposition self-contained.
After this introduction, in section 2, we consider the Donaldson-Fujiki type formulation of the problem, and compute the first variation of the scalar curvature of the conformal metricsg. This computation and the resulting consequenes have been obtained by Apostolov-Maschler [3]. A novelty, if any, is just that the derivation of the formulae is closer to the original style of Calabi's computation [6], and this enables the computations in section 3 possible. In section 3, we prove a Hessian formula of the Calabi functional. As mentioned above, we follow the derivation of the formula given by Wang [29] and the first author [14]. As an example we consider the case of the one-point-blow-up of CP 2 . In section4 we give an example of f -extremal Kähler metrics.
Calabi type formulation and Donaldson-Fujiki set-up.
Let Ω ∈ H 2 DR (M, R) be a fixed Kähler class, and choose a Kähler metric g with its Kähler form ω g in Ω. Denote by Aut(M, Ω) the group of all biholomorphisms of M preserving Ω. Its Lie algebra, denoted by h(M ), is the same as the Lie algebra of the full automorphism group Aut(M ) and consists of all holomorphic vector fields on M . Consider the Lie subalgebra g of h(M ) defined as the kernel of the differential of the homomorphism of Aut(M ) to the Albanese torus induced by the Albanese map. Differential geometric expression of g is given by
(1) g = {X ∈ h(M ) | grad u = X for some u ∈ C ∞ C (M )} where C ∞ C (M )
denotes the set of all complex valued smooth functions on M and (2) grad u = g ij ∂u ∂z j ∂ ∂z i , refer to, for example, [22], [17]. The Lie subalgebra g is independent of the choice of the Kähler metric as can be seen from the Albanese map description, and is referred to as the reduced Lie algebra of holomorphic vector fields. For the purpose of the present paper, it is convenient to take (1) to be the definition of g. Then the independence of the choice of the metric can also be seen from the standard fact that a holomorphic vector field X belongs to g if and only if X has a zero, see [25], [7], [22], [17]. If ω g = ig ij dz i ∧ dz j is the Kähler form then grad u = X is equivalent to i(X)ω g = i∂u.
We shall write X u := grad u. The Lie algebra structure of g is then given by
[X u , Y w ] = X {u,w} where {u, w} = ∇ i u∇ i w − ∇ i w∇ i u, ∇ i = g ij ∇ j .
Let G be the connected subgroup of Aut(M ) corresponding to g, i.e. Lie(G) = g. G is referred to as the reduced automorphism group. G is then a subgroup of Aut(M, Ω) since G is the connected group containing the identity. Let T be a maximal torus of G, and T c be the complexification of T . Then T c is a subgroup of the maximal reductive subgroup G r of G.
Let K ∈ Lie(T ) be arbitrarily chosen. The problem is to find a Kähler form ω g ∈ Ω such that (i) f is a positive smooth function with Jgrad g f = K, and that (ii)g = f −2 g is a conformally Kähler, Einstein-Maxwell metric. At this point we could assume that K is a critical point of the volume functional introduced in [15] so that the obstruction in [3] vanishes. But this assumption is needless for the arguments in what follows, and we do not assume it.
Denote by Isom(M, g) the group of all isometries of (M, g). The following is the main theorem of this paper. Theorem 2.1. Letg be a conformally Kähler, Einstein-Maxwell metric so that (i) and (ii) above are satisfied. Then the the centralizer G K = {g ∈ G | Ad(g)K = K} of K in the reduced automorphism group G is the complexification of Isom(M, g) ∩ G K . In particular G K is reductive. When f is a constant function and (M, g) has a constant scalar curvature, Theorem 2.1 is a part of Lichnerowicz-Matsushima theorem ( [24], [23]) which asserts the reductiveness of the full automorphism group.
As in the case of the problem finding constant scalar curvature Kähler metrics, the Calabi type set-up fixing an integrable complex structure and varying Kähler forms in a fixed Kähler class can be turned through Moser's theorem into Donaldson-Fujiki type set-up fixing a symplectic form and varying integrable almost complex structures.
Let (M, J 0 , ω) be a compact Kähler manifold, g be the reduced Lie algebra as above, and T a maximal torus in the reduced automorphism group G. We fix a Hamiltonian Killing vector field K ∈ Lie(T ). Let T K be the subtorus obtained as the closure in T of {exp(tK) | t ∈ R}. We set
g K := {X ∈ g | [K, X] = 0} = {X ∈ g | X = grad u for some u ∈ C ∞ C (M ), Ku = 0}
. Now we consider ω as a fixed symplectic form on M . Then we can choose a fixed positive Hamiltonian function f of K by adding a positive constant if necessary. We may define
C ∞ (M ) K := {u ∈ C ∞ (M ) | Ku = 0}, C ∞ C (M ) K := {u ∈ C ∞ C (M ) | Ku = 0}, C ∞ (M ) K 0 := {u ∈ C ∞ (M ) K | M u f −2m−1 ω m = 0}, C ∞ C (M ) K 0 := {u ∈ C ∞ C (M ) K | M u f −2m−1 ω m = 0}.
We denote by Ham(M ) K the group of Hamiltonian diffeomorphisms for C ∞ (M ) K or equivalently C ∞ 0 (M ) K considered as Hamiltonian functions. Let Z K be the space of all T K -invariant ω-compatible integrable almost complex structures J ∈ C ∞ (End(T M )). Here, J is said to be ω-compatible if the following two conditions are satisfied:
(a) ω(JX, JY ) = ω(X, Y ), (b) g J := ω(·, J·) is positive definite. By (b), g J is a T K -invariant Kähler metric for each J ∈ Z K .
For any fixed J ∈ Z K we have the decomposition
J −1 δJα ≡ −2δα.(5)
Thus, as a real tensor field, J −1J is in the form of
J −1J = 2 (v i k ∂ ∂z i ⊗ dz k ).(6)
The corresponding curve of Kähler metrics g(t) = ω(·, J(t)·) may be expressed in the matrix form with respect to local coordinates as
g(t) = ωJ(t).
We thus have
(7) g −1ġ = J −1 ω −1 ωJ = J −1J , and (8) (g −1 ) · = −g −1ġ g −1 = −J −1J g −1 . Lemma 2.3. Let Θ(t) = ∂ t (g(t) −1 ∂ t g(t)
) be the curvature form of g(t) expressed in terms of time dependent local holomorphic coordinates. Then
(9) d dt | t=0 Θ(t) = d ∇ ∇(J −1J ).
Proof. Consider d dt | t=0 Θ(t) with normal coordinates with respect to g(0) = g at p ∈ M so that the derivative dg = 0 at p. Let θ t = g −1 t ∂ t g t be the connection form. Then at p we have from (7)
θ = g −1 ∇ġ = ∇(g −1ġ ) = ∇(J −1J ).
On the other hand, from Θ = dθ + θ ∧ θ, we have
(dθ + θ ∧ θ) · = dθ +θ ∧ θ + θ ∧θ = d ∇θ . ThusΘ = d ∇θ = d ∇ ∇(J −1J ).
Let S(J, f ) be the scalar curvature of the conformal metric f −2 g J . We often write S J,f and g J,f instead of S(J, f ) and f −2 g J when these are more convenient. The following theorem is due to Apostolov-Maschler [3], but we express and prove it in the form which fits to the aim of the present paper.
Theorem 2.4 ([3]). For any smooth curve J(t), − < t < , in Z K with J −1J = 2 (v i k ∂ ∂z i ⊗ dz k ) ∈ T J Z K ,(10)we put v := v i k ∂ ∂z i ⊗ dz k . Then for any smooth function h ∈ C ∞ (M ) we have (11) d dt | t=0 M S(J(t), f )hf −2m−1 ω m = 2 M (∇ i ∇ j h) v ij f −2m+1 ω m where v ij = g jk v i k . Proof. The Ricci form Ric J of g J is given by Ric J = itrΘ J .
Thus the scalar curvature S J of g J satisfies
S J ω m = 2Ric J ∧ ω m−1 (m − 1)! = 2i trΘ J ∧ ω m−1 (m − 1)! .
In Lemma 2.3, the 1-formθ with values in End(T M ) is written as
(∇ j v i k (dz j ⊗ ∂ ∂z i ) ⊗ dz k ).
Thus we haveṠ
J = 2 ∇ j ∇ i v i j = 2 ∇ j ∇ i v ij .
The scalar curvature S J,f of g J,f and the scalar curvature S J of g J are related by
S J,f = 2 2m − 1 m − 1 f m+1 ∆ J (f −m+1 ) + S J f 2 where ∆ J = d * g J d is
the Hodge Laplacian, see e.g. [5]. Then
M S J,f h f 2m+1 ω m = 2m(2m − 1) M h f 2m+1 g −1 J (df, df ) ω m −2(2m − 1) M f −2m g −1 J (dh, df )ω m (12) + M S J h f −2m+1 ω m . Noting ∇ i ∇ j f = ∇ i ∇ j f = 0
since K is a holomorphic vector field, and using (8) and
v ij = v ji we can compute the derivative d dt | t=0 M S J,f h f 2m+1 ω m = −2m(2m − 1) M h f 2m+1 g −1 ( t (J −1J )df, df ) ω m +2(2m − 1) M f −2m g −1 ( t (J −1J )dh, df )ω m + MṠ J h f −2m+1 ω m . = 2 [−2m(2m − 1) M h f 2m+1 g kj v i j ∇ i f ∇ k f ω m +2(2m − 1) M f −2m g kj v i j ∇ i h∇ j f ω m + M ∇ j ∇ i v i j h f −2m+1 ω m ] = 2 [ M ∇ i ∇ j h v ij f −2m+1 ω m ].
This completes the proof of Theorem 2.4.
Remark 2.5. Recall we fixed K ∈ Lie(T ), the symplectic form ω and the positive Hamiltonian function f of K with respect to ω. Let g R be the subalgebra of g consisting of all X = gradu with real smooth function u.
The linear map F ut f : g R → R defined by
(13) F ut f (X) = M S(J, f ) u X ω m f 2m+1 is independent of the choice of J ∈ Z where grad u X = X with (14) M u X f −2m−1 ω m = 0.
This follows by taking h = u X in Theorem 2.4.
Remark 2.6. By taking h = 1 in Theorem 2.4 we see (15) c
J,f = M S(J, f ) f −2m−1 ω m / M f −2m−1 ω m is independent of J ∈ Z.
Thus, we may remove the normalization (14) and define F ut f by
(16) F ut f (X) = M (S(J, f ) − c J,f ) u X ω m g f 2m+1 .
In the Calabi type setup, when K is fixed, f varies as ω g varies in Ω in the unique manner with a normalization
(17) M f ω m g = a.
When this normalization is satisfied we shall write f as f K,g,a . Then F ut K,a : g R → R defined by
(18) F ut K,a (X) = M (S(ω g , f K,g,a ) − c Ω,K,a ) u X ω m g f 2m+1 K,g,a with (19) c Ω,K,a := M S(ω g , f K,g,a ) 1 f K,g,a 2m+1 ω m m! M 1 f K,g,a 2m+1 ω m m! ,
is independent of the choice of ω g ∈ Ω where S(ω g , f ) is the scalar curvature ofg = f −2 g. If there exists a Kähler metric g with its Kähler form ω g in Ω and with S(ω g , f ) constant then F ut K,a = 0. Namely, F ut K,a is an obstruction to the existence of conformally Einstein-Maxwell Kähler metric in the Kähler class Ω.
Since the Hermitian inner product of the tangent space T J Z K of Z K at J is given by the L 2 -inner product (20) (
λ, ν) L 2 (f 2m+1 ) := M λ ij ν ij f −2m+1 ω m ,
the symplectic structure Ω J,f on Z K is given at J by
Ω J,f (λ, ν) = (λ, √ −1ν) = M λ ij √ −1ν ij f −2m+1 ω m .(21)C ∞ 0 (M ) K where (ϕ, ψ) L 2 (f −2m−1 ) = M ϕ ψ f −2m−1 ω m .
More precisely, the equality (23) below holds and the left hand side is equivariant under the action on Z K of the group of Hamiltonian diffeomorphisms generated by C ∞ 0 (M ) K
Proof. If X = X + X is a smooth complex vector field where X and X are type (1, 0) and (0, 1) part respectively, then by Lemma 2.3 in [13] (22)
L X J = 2 √ −1∇ J X − 2 √ −1∇ J X .
In particular, if X u is the Hamiltonian vector field of u ∈ C ∞ 0 (M ) K , we have
L Xu J = 2 √ −1∇ k u i ∂ ∂z i ⊗ dz k − 2 √ −1∇ k u i ∂ ∂z i ⊗ dz k .
Hence
χ u := 4 ( √ −1∇ k u i ∂ ∂z i ⊗ dz k )
defines a tangent vector in T J Z K . For α = α i dz i , we have
χ u (α) = 2 √ −1(∇ j u i )α i dz j .
If J −1J = v then by Theorem 2.4 we have
d dt | t=0 M S(J(t), f ) u f −2m−1 ω m = 2 M u ij v ij f −2m+1 ω m = M 2 √ −1 u i j √ −1v i j f −2m+1 ω m = Ω(χ u , v).(23)
Further, since w ∈ C ∞ 0 (M ) K is T K -invariant, the scalar curvature S(J, f ) defines a moment map on Z K equivariant under the action of Hamiltonian diffeomorphisms generated by X w for any w ∈ C ∞ 0 (M ) K .
Note in passing that (22) shows
(24) L JXu J = −2∇ J X u − 2∇ J X u .
and the corresponding tangent vector J −1J ∈ T J Z k is expressed as
(25) (J −1J )α = −2(∇ j u i )α i dz j for α = α i dz i .
Hessian formula for the Calabi functional
Consider the Calabi functional Φ :
Z K → R defined by Φ(J) = M S 2 J,f ω m .
If J is a critical point of Φ, the Kähler metric g = ωJ is called an f -extremal Kähler metric. From Theorem 2.4 we see
(26) d dt M S 2 J(t),f f −2m−1 ω m = 8 M ∇ i ∇ j S J,f √ −1∇ i ∇ j u f −2m+1 ω m when J −1J = χ u (27) = 2 √ −1∇ k u i ∂ ∂z i ⊗ dz k − 2 √ −1∇ k u i ∂ ∂z i ⊗ dz k .
Thus we obtain the following. We define the fourth order elliptic differential operator L :
C ∞ C (M ) → C ∞ C (M ) by (w, Lu) L 2 (f −2m−1 ) = (∇ ∇ w, ∇ ∇ u) L 2 (f −2m+1 ) .(28)
We further define the fourth order elliptic differential operator L :
C ∞ C (M ) → C ∞ C (M ) by Lu = Lu. (29) Lemma 3.2. If J −1J = 2 ∇ i ∇ j u ∂ ∂z i ⊗ dz j for a real valued smooth func- tion u ∈ C ∞ (M ), we have d dt | t=0 S J(t),f = (Lu + Lu).
Proof. For any real smooth function w we see from Theorem 2.4
d dt | t=0 M w S J(t),f f −2m−1 ω m = M ((∇ i ∇ j w, ∇ i ∇ j u) + ∇ i ∇ j w, ∇ i ∇ j u)f −2m+1 ω m = (w, Lu) + (w, Lu) = (w, Lu) + (w, Lu) = (w, Lu + Lu).
This completes the proof. Proof. This lemma is simply a restatement of Proposition 2.7. Let σ be in the Hamiltonian diffeomorphisms generated by the Hamiltonian vector field of w ∈ C ∞ (M ) K . Since S J,f gives a Ham(M ) K -equivariant moment map we have
(30) M u S(σJ, f ) f −2m−1 ω m = M u • σ −1 S J,f f −2m−1 ω m .
Taking the time differential of σ we obtain the lemma by (23).
(L − L)u = i 2 {u, S(J, f )} = 1 2 (u α S(J, f ) α − S(J, f ) α u α )
where u α = g αβ ∂u/∂z β for local holomorphic coordinates z 1 , · · · , z m .
Proof. It is sufficient to prove when u is real valued. For any real valued smooth function w ∈ C ∞ (M ) K it follows from Lemma 3.3 that
(w, Lu − Lu) L 2 (f −2m−1 ) = (∇ ∇ w, ∇ ∇ u) L 2 (f −2m+1 ) − (∇ ∇ w, ∇ ∇ u) L 2 (f −2m+1 ) = i 2 Ω(χ v , χ u ) = − i 2 ({u, w}, S J,f ) L 2 (f −2m−1 ) = i 2 (w, {u, S J,f }) L 2 (f −2m−1 ) = i 2 (w, X u S J,f ) L 2 (f −2m−1 ) = i 2 (w, ω(X u , JgradS J,f )) L 2 (f −2m−1 ) = i 2 (w, du(JgradS J,f )) L 2 (f −2m−1 ) = (w, S(J, f ) α u α − S(J, f ) α u α ) L 2 (f −2m−1 ) .
This completes the proof of Lemma 3.4.
Lemma 3.5. If u ∈ C ∞ (M ) K and J −1J = 2 ∇ i ∇ j u ∂ ∂z i ⊗ dz j ∈ T J Z K , then d dt | t=0 M S(J(t), f ) 2 f −2m−1 ω m = 4(u, LS(J, f )) L 2 (f − 2m−1) (31) = 4(u, LS(J, f )) L 2 (f − 2m−1) .
Proof. Since S(J, f ) ∈ C ∞ (M ) K we can apply Theorem 2.4 to show that the left hand side of (31) is equal to
4 (∇ ∇ S(J, f ), ∇ ∇ u) L 2 (f −2m+1 ) = 2((∇ ∇ S(J, f ), ∇ ∇ u) L 2 (f −2m+1 ) + (∇ ∇ u, ∇ ∇ S(J, f )) L 2 (f −2m+1 ) ) = 2(u, LS(J, f )) L 2 (f −2m−1 ) + 2(u, LS(J, f )) L 2 (f −2m−1 ) .
But Lemma 3.4 implies
LS(J, f ) = LS(J, f ).
Hence the left hand side of (31) is equal to
4(u, LS(J, f )) L 2 (f −2m−1 ) = 4(u, LS(J, f )) L 2 (f −2m−1 ) . Lemma 3.6. Suppose that (ω, J, f ) is an f -extremal Kähler metric so that JgradS(J, f ) is a holomorphic vector field. If J −1J = 2 ∇ i ∇ j u ∂ ∂z i ⊗ dz j for some real smooth function u ∈ C ∞ (M ) K then we have ( d dt | t=0 L)S(J, f ) = − 1 2 L(S(J, f ) α u α − u α S(J, f ) α ) = L(L − L)u
Proof. First note that if i(X u )ω = du then (32) L 1 2 JXu ω = i∂∂u. Let {f s } be the flow generated by − 1 2 JX u . Let S be a smooth function on M such that grad S is a holomorphic vector field. We shall compute Thus taking the derivative of (34), we obtain
(35) ( d ds | s=0 L)S + L(− 1 2 (JX u )S + S α u α ) = 0.
By an elementary computation we see
(JX u )S = g(JX u , grad S) = ω(X u , grad S) = du(grad S) = (∂u + ∂u)(∇ S + ∇ S) = u α S α + u α S α .
Thus, from (35) and the above computation, we obtain
( d ds | s=0 L)S = L( 1 2 (u α S α + u α S α ) − S α u α ) = 0 = 1 2 L(u α S α − u α S α ) = L(L − L)u.
This completes the proof of Lemma 3.6.
To express the Hessian formula, for a real smooth function Proof. Suppose J −1J = 2 ∇ i ∇ j w ∂ ∂z i ⊗ dz j , or ∇ ∇ w by our identification. Then by Lemma 3.5, Lemma 3.6 and Lemma 3.2 we obtain
u ∈ C ∞ (M ) K , we identify J −1J = 2 ∇ i ∇ j u ∂ ∂z i ⊗ dz j with ∇ ∇ u.Hess(Φ) J (∇ ∇ u, ∇ ∇ w) = d dt | t=0 4(u, LS(J, f )) L 2 (f −2m−1 ) = 4(u, ( d dt | t=0 L)S(J, f ) + L d dt | t=0 S(J(t), f )) L 2 (f −2m−1 ) = 4(u, L(L − L)w + L(L + L)w) L 2 (f −2m−1 ) = 8(u, LLw) L 2 (f −2m−1 ) .
Similarly, we obtain
Hess(Φ) J (∇ ∇ u, ∇ ∇ w) = d dt | t=0 4(u, LS(J, f )) L 2 (f −2m−1 ) = 4(u, ( d dt | t=0 L)S(J, f ) + L d dt | t=0 S(J(t), f )) L 2 (f −2m−1 ) = 4(u, L(L − L)w + L(L + L)w) L 2 (f −2m−1 ) = 8(u, LLw) L 2 (f −2m−1 ) .
This completes the proof of Theorem 3.7.
The following theorem extends a theorem of Calabi [6] for extremal Kähler metrics.
Theorem 3.8. If g = ωJ is an f -extremal Kähler metric with K = Jgradf then the centralizer g K of K in the reduced Lie algebra g of holomorphic vector fields has the following structure:
(a) g K 0 := (i(M ) ∩ g K ) ⊗ C is the maximal reductive subalgebra of g K where i(M ) denotes the real Lie algebra of all Killing vector fields.
(b) grad S J,f = g ij ∂S J,f ∂z j is in the center of g K 0 . (c) g K = g K 0 + λ =0 g K λ where g K λ is the λ-eigenspace of ad(grad S J,f ). Moreover, we have [g K λ , g K µ ] ⊂ g K λ+µ .
Proof. By Theorem 3.7, LL = LL on C ∞ C (M ) K . Therefore L maps ker L to ker L, and we have the direct sum decomposition ker L = λ E λ into the eigenspaces of 2L. Further by Lemma 3.4
λu = 2Lu = 2(L − L)u = S(J, f ) α u α − u α S(J, f ) α .
This shows [grad S(J, f ), grad u] = λgrad u.
Thus we have E λ = g K λ . For λ = 0 we have g K 0 = E 0 = ker L ∩ ker L. Since Lu = 0 is equivalent to Lu = 0, if u ∈ g K 0 then u satisfies both Lu = 0 and Lu = 0. This implies L u = 0 and L u = 0. In general if grad u is a holomorphic vector field for a real smooth function u then Jgrad u is a Killing vector field. Hence we obtain
g K 0 = (i(M ) ∩ g K 0 ) ⊗ C.
Now we are in a position to prove Theorem 2.1
Proof of Theorem 2.1. If g = ωJ is a conformally Einstein-Maxwell Kähler metric, then g is an f -extremal Kähler metric with S(J, f ) is a constant function. Therefore, in this case
g K = g K 0 = (i(M ) ∩ g K ) ⊗ C.
Since Isom(M, g) is compact, g K is reductive. This completes the proof of Theorem 2.1. Example 3.9. We consider the case of the one-point-blow-up CP 2 of CP 2 . Let ∆ p be the convex hull of (0, 0), (p, 0), (p, 1 − p), (0, 1), (0 < p < 1) in (µ 1 , µ 2 )-plane. Then for each p, ∆ p determines a Kähler class of CP 2 . The Hamiltonian function f of a holomorphic Killing vector field is an affine linear function f = aµ 1 + bµ 2 + c, which is determined uniquely up to the choice of c. Then f is positive on ∆ p if and only if
c, b + c, (1 − p)b + pa + c, pa + c > 0.
In [15] we showed that the obstruction F ut f in Remark 2.5 vanishes if and only if K = Jgradf gives a critical point of the volume functional defined in Theorem 1.1 in [15]. The Hamiltonian functions f which correspond to critical points are determined in section 4.3 in [15]. The following is the list of critical points.
(1) a = p+2
√ 1−p−2 2p 2 , b = 0, 0 < p < 1. (2) a = − √ 9p 2 −8p+p 4p 2 , b = 0. 8 9 < p < 1. (3) a = √ 9p 2 −8p−p 4p 2 , b = 0. 8 9 < p < 1. (4) a = − √ p 4 −4p 3 +16p 2 −16p+4−p 2 +4p−2 2p 3 −4p 2 +12p−8 , b = − √ p 4 −4p 3 +16p 2 −16p+4 p 3 −2p 2 +6p−4 . 0 < p < α. (5) a = √ p 4 −4p 3 +16p 2 −16p+4+p 2 −4p+2 2p 3 −4p 2 +12p−8 , b = √ p 4 −4p 3 +16p 2 −16p+4 p 3 −2p 2 +6p−4 . 0 < p < α. (6) a = 2 √ −9b 2 p 3 +(21b 2 +1)p 2 +(1−16b 2 )p+4b 2 −1+3bp 2 +(1−2b)p 6p 2 −4p . (7) a = − 2 √ −9b 2 p 3 +(21b 2 +1)p 2 +(1−16b 2 )p+4b 2 −1−3bp 2 +(2b−1)p 6p 2 −4p .
Here, α is the smallest positive root of p 4 − 4p 3 + 16p 2 − 16p + 4 = 0. In the cases (1), (2) and (3)
Construction of f -extremal Kähler metrics
In this section we give a construction of f -extremal Kähler metrics on CP 1 × M when M is an (m − 1)-dimensional compact complex manifold with a Kähler metric g 2 of constant scalar curvature s g 2 = c. This is an extension of a construction of conformally Kähler Einstein-Maxwell metrics given in section 3 of [15].
Let g 1 be an S 1 -invariant metric on CP 1 . Using the action-angle coordinates (t, θ) ∈ (a, b) × (0, 2π], the S 1 -invariant metric g 1 can be written as
g 1 = dt 2 Ψ(t) + Ψ(t)dθ 2
for some smooth function Ψ(t) where the Hamiltonian function of the generator of the S 1 -action is t. Therefore f = t in this case. We put g = g 1 + g 2 . We wish to construct Ψ such that the gradient vector field with respect to g of the scalar curvature s(g) of the Hermitian metricg = g/t 2 on CP 1 × M is a holomorphic vector field. This happens to be the case if (36) s(g) = dt + e for some constants d and e.
Since the scalar curvature of g 1 is given by
s 1 = ∆ g 1 log Ψ = −Ψ (t),
the scalar curvature of g is given by
s = s 1 + s 2 = c − Ψ (t).
It follows that the equation (36) is equivalent to
(37) dt + e = 2 2m − 1 m − 1 t m+1 ∆ g (t 1−m ) + (c − Ψ (t))t 2 . Using ∆ g 1 1 t m−1 = (m − 1) Ψ t m , the equation (37) reduces to the ODE (38) t 2 Ψ − 2(2m − 1)tΨ + 2m(2m − 1)Ψ = ct 2 − dt − e.
The general solution of the equation (38) is (i) A is given by
(39) Ψ = At 2m +Bt 2m−1 + c 2(m − 1)(2m − 3) t 2 − d 2(m − 1)(2m − 1) t− e 2m(2mA = − A 1 B −2ab 2m−1 m + 2 b 2m m + 2a 2m−1 bm − 2a 2m m − 2b 2m + 2a 2m with A 1 = 2b 2m−1 m − 2ab 2m−2 m + 2a 2m−2 bm − 2a 2m−1 m − 3b 2m−1 +ab 2m−2 − a 2m−2 b + 3a 2m−1 .
(ii) c is given by
c = (m − 1)(2m − 3)P Q B − 4(m − 1)(2m − 3) b − a with P = (2a m b m+1 m − 2a m+1 b m m − a m b m+1 − ab 2m + a m+1 b m + a 2m b) ·(2a m b m+1 m − 2a m+1 b m m − a m b m+1 + ab 2m + a m+1 b m − a 2m b), Q = ab(ab 2m+2 m − 2a 2 b 2m+1 m + a 3 b 2m m + a 2m b 3 m − 2a 2m+1 b 2 m +a 2m+2 bm − ab 2m+2 + a 2 b 2m+1 + a 2m+1 b 2 − a 2m+2 b).
(iii) d is given by
d = (2(m − 1)(2m − 1)R S B − 4(a + b)(m − 1)(2m − 1) b − a with R = 2a 2m b 2m+3 m 2 − 2a 2m+1 b 2m+2 m 2 − 2a 2m+2 b 2m+1 m 2 + 2a 2m+3 b 2m m 2 −3a 2m b 2m+3 m + 3a 2m+1 b 2m+2 m + 3a 2m+2 b 2m+1 m −3a 2m+3 b 2m m + a 2m b 2m+3 − a 3 b 4m + a 2m+3 b 2m − a 4m b 3 ), S = ab(ab 2m+2 m − 2a 2 b 2m+1 m + a 3 b 2m m + a 2m b 3 m − 2a 2m+1 b 2 m + a 2m+2 bm −ab 2m+2 + a 2 b 2m+1 + a 2m+1 b 2 − a 2m+2 b).
(iv) e is given by We used Maxima to obtain the above result. For example, if we put B = 0 then we obtain A = 0,
e = − m(2m − 1)T U B + 4abm(2m − 1) b − a with T = 4a 2m+1 b 2m+3 m 2 − 8a 2m+2 b 2m+2 m 2 + 4a 2m+3 b 2m+1 m 2 − 8a 2m+1 b 2m+3 m +16a 2m+2 b 2m+2 m − 8a 2m+3 b 2m+1 m + 4a 2m+1 b 2m+3 − 6a 2m+2 b 2m+2 +4a 2m+3 b 2m+1 − a 4 bc = − 4(m − 1)(2m − 3) b − a , d = − 4(a + b)(m − 1)(2m − 1) b − a , e = 4abm(2m − 1) b − a(40)
and
Ψ(t) = − 2(t − a)(t − b) b − a .
This Ψ is positive on (a, b) and satsifies the boundary conditions. As another example, we set a = 1, b = 2 for simplicity, and put A = 2 2 2m+1 m − 5 · 2 2m + 8m + 4 (−2 m+1 m + 2 2m + 2 m − 2) (2 m+1 m + 2 2m − 2 m − 2) , B = −8 2 2m m − 2 2m+1 + 2m + 2 (−2 m+1 m + 2 2m + 2 m − 2) (2 m+1 m + 2 2m − 2 m − 2) , c = 0, d = − 4 (m − 1) (2m − 1) −3m2 2m+1 + 2 4m + 3 · 2 2m − 4 (−2 m+1 m + 2 2m + 2 m − 2) (2 m+1 m + 2 2m − 2 m − 2) , e = 4m (2m − 1) −2 2m+3 m + 3 · 2 2m+1 + 2 4m − 8 (−2 m+1 m + 2 2m + 2 m − 2) (2 m+1 m + 2 2m − 2 m − 2) .
Then Ψ satisfies the boundary conditions. Further Ψ > 0 on the interval (1, 2) because of Ad < 0 and the following Lemma. Proof. Suppose that there is a c ∈ (a, b) such that f (c) ≤ 0. Then by the boundary conditions f has at least three critical points on (a, b). On the other hand, we have f (t) t = 2mαt 2m−2 + (2m − 1)βt 2m−3 + 2γ + δ t ,
and thus
(42) f (t) t = t 2m−4 {2m(2m − 2)αt + (2m − 1)(2m − 3)β − δ t 2m−2 }.
From αδ > 0, the right hand side of (42) changes sign only once on the region t > 0. This implies f /t has at most two zeros on the interval (a, b). This is a contradiction, and completes the proof of Lemma 4.1.
Remark 2. 2 .
2It is shown in [3] that, under the conditions (i) and (ii), Isom(M, g) ∩ Aut(M, Ω) is a subgroup of Isom(M, g) ∩ Aut(M, Ω), and the Killing vector field K is in the center of the Lie algebra of Isom(M, g) ∩ Aut(M, Ω). The proof is as follows. Since g = f 2 g, Isom(M, g) ∩ Aut(M, Ω) acts as conformal transformations of (M, g). But, since g is Kähler, Isom(M, g)∩ Aut(M, Ω) acts on (M, g) as homotheties. Since Aut(M, Ω) preserves Ω, Isom(M, g) ∩ Aut(M, Ω) acts as isometries of (M, g). This proves the first statement. Further, again since g = f 2 g, Isom(M, g) ∩ Aut(M, Ω) preserves f . This implies K is in the center of Isom(M, g) ∩ Aut(M, Ω).
The (·, ·) L 2 (f −2m−1 ) -dual of the scalar curvature S(J, f ) gives an equivariant moment map for the action of the group Ham(M ) K of Hamiltonian diffeomorphisms generated by
Lemma 3. 1 .
1The Kähler metric g = ωJ is an f -extremal Kähler metric if and only if grad J S J,f is a holomorphic vector field.
Lemma 3. 3 .
3For real valued smooth functions u and w in C ∞ (M ) K we have Ω(χ u , χ w ) = − M {w, u} S J,f f −2m−1 ω m .
Lemma 3. 4 .
4For any smooth complex valued smooth function u ∈ C ∞ (M ) K we have
d ds | s=0 L(f s J, ω)S, and apply to S = S(J, f ), and obtain the conclusion of Lemma 3.6. Let {S s } be a family of smooth functions such that S 0 = S, that grad s S s = grad S, where grad s denotes the gradient with respect to f * −s ω, f s J, ω)f * s S s = f * s (L(J, f * −s ω)S s ) = 0. On the other hand, in general, if f * −s ω = ω + i∂∂ϕ then S s = S + S α ϕ α . Therefore, since L 1 2 JXu ω = i∂∂u by (32) we have (34) S s = S + sS α u α + O(s 2 ).
Theorem 3 . 7 .
37Let J be a critical point of Φ so that (ω, J, f ) is an f -extremal Kähler metric. Let u and w be real smooth functions in C ∞ (M ) K . Then the Hessian Hess(Φ) J at J is given by Hess(Φ) J (∇ ∇ u, ∇ ∇ w) = 8(u, LLw) = 8(u, LLw).In particular, LL = LL on C ∞ C (M ) K at any critical point of Φ.
we have b = 0 which shows the solution has to have U (2)-symmetry. In fact LeBrun[21] constructed a solution in each cases. K and the centralizer G K are then of the form P GL(3, C) where α = β, α = 0 and β = 0. In the cases (4), (5), (6) and (7), the existence is not known at this moment of writing, but if a solution exists it must have U (1) × U (1)-symmetry. K and the centralizer G K are then of the form P GL(3, C) where α, β and γ are non-zero and mutually distinct.
a, b). The boundary conditions are Ψ(a) = Ψ(b) = 0, Ψ (a) = −Ψ (b) = 2, which reduce to a simultaneous linear equation for A, B, c, d and e. The space of solutions are 1-dimensional. If we express it in terms B we have the following expression.
4m − a 4m b 4 , U = ab(ab 2m+2 m − 2a 2 b 2m+1 m + a 3 b 2m m + a 2m b 3 m − 2a 2m+1 b 2 m + a 2m+2 bm −ab 2m+2 + a 2 b 2m+1 + a 2m+1 b 2 − a 2m+2 b).
Lemma 4. 1 .
1Let m be an integer greater than 1, and suppose 0 < a < b. If the real valued function f (t) = αt 2m + βt 2m−1 + γt 2 + δt + ε with αδ > 0 satisfies the boundary conditionsf (a) = f (b) = 0, f (a) > 0, −f (b) > 0then f is positive on the interval (a, b).
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The Einstein-Maxwell equations, Kähler metrics, and Hermitian geometry. C Lebrun, J. Geom. Phys. 91C. LeBrun, The Einstein-Maxwell equations, Kähler metrics, and Hermitian geome- try, J. Geom. Phys., 91, 163-171, (2015).
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A compact rotating gravitational instanton. D Page, Phys. Lett. B. 79D. Page : A compact rotating gravitational instanton, Phys. Lett. B 79 (1978), 235- 238.
A new holomorphic invariant and uniqueness of Kähler-Ricci solitons. G Tian, X.-H Zhu, Comment. Math. Helv. 77G. Tian and X.-H. Zhu : A new holomorphic invariant and uniqueness of Kähler-Ricci solitons, Comment. Math. Helv. 77(2002), 297-325.
Hessians of the Calabi functional and the norm function. L.-J Wang, Ann. Global Anal. Geom. 292L.-J. Wang : Hessians of the Calabi functional and the norm function, Ann. Global Anal. Geom., 29(2006), No.2, 187-196.
. Graduate School of Mathematical Sciences. 3The University of TokyoGraduate School of Mathematical Sciences, The University of Tokyo, 3-
Japan E-mail address: [email protected]. Komaba Meguro-Ku Tokyo, 380-8570jp Department of Mathematics. Saitama University, 255 Shimo-Okubo, Sakura-Ku, Saitama; Japan E-mail addressKomaba Meguro-ku Tokyo 153-8914, Japan E-mail address: [email protected] Department of Mathematics, Saitama University, 255 Shimo-Okubo, Sakura- Ku, Saitama 380-8570, Japan E-mail address: [email protected]
| {'fraction_non_alphanumeric': 0.10482683814984117, 'fraction_numerical': 0.04581493246558715, 'mean_word_length': 3.0423843825033927, 'pattern_counts': {'":': 0, '<': 17, '<?xml version=': 0, '>': 7, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 3, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 36, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': "Let (M, g) be a compact Kähler manifold and f a positive smooth function such that its Hamiltonian vector field K = Jgrad g f for the Kähler form ωg is a holomorphic Killing vector field. We say that the pair (g, f ) is conformally Einstein-Maxwell Kähler metric if the conformal metricg = f −2 g has constant scalar curvature. In this paper we prove a reductiveness result of the reduced Lie algebra of holomorphic vector fields for conformally Einstein-Maxwell Kähler manifolds, extending the Lichnerowicz-Matsushima Theorem for constant scalar curvature Kähler manifolds. More generally we consider extensions of Calabi functional and extremal Kähler metrics, and prove an extension of Calabi's theorem on the structure of the Lie algebra of holomorphic vector fields for extremal Kähler manifolds. The proof uses a Hessian formula for the Calabi functional under the set up of Donaldson-Fujiki picture. into type (1, 0)-part and type (0, 1)-part. Naturally T * J M = T * J M . For another complex structure J ∈ Z K we also have) and the last isomorphism uses the Kähler metric. As an element of the last term, v is in the symmetric part C ∞ (Sym(T J M ⊗ T J M )), see the proof of Lemma 2.1 in[13]. So, we havethe last term being the underlying real vector space of C ∞ (Sym(T J M ⊗ T J M )). The L 2 -inner product with respect to the volume form f −2m+1 ω m gives a Kähler structure on Z K . If J = J + δJ andThus, up to the first order,If J(t) is any smooth curve in Z K for t ∈ (− , ) with J(0) = J, (4) shows that we may regard J −1J ∈ C ∞ (End(T * J M, T * J M )) R . HereJ denotes the derivative of J(t) at t = 0. But (4) also shows", 'arxivid': '1708.01958', 'author': ['Akito Futaki ', 'Hajime Ono '], 'authoraffiliation': [], 'corpusid': 119319999, 'doi': '10.1007/s00209-018-2112-3', 'github_urls': [], 'n_tokens_mistral': 16270, 'n_tokens_neox': 14125, 'n_words': 7905, 'pdfsha': '73e7d33d0ade93b8b693b828f22b716246365579', 'pdfurls': ['https://arxiv.org/pdf/1708.01958v3.pdf'], 'title': ['CONFORMALLY EINSTEIN-MAXWELL KÄHLER METRICS AND STRUCTURE OF THE AUTOMORPHISM GROUP', 'CONFORMALLY EINSTEIN-MAXWELL KÄHLER METRICS AND STRUCTURE OF THE AUTOMORPHISM GROUP'], 'venue': []} |
arxiv |
EPISODIC SELF-SIMILARITY IN CRITICAL GRAVITATIONAL COLLAPSE
12 Dec 2000
CHJ Thornburg
Institut für Theoretische Physik
Department of Physics and Astronomy
Universität Wien
Boltzmangasse 5A-1090WienAustria
M Lechner [email protected]
Institut für Theoretische Physik
Department of Physics and Astronomy
Universität Wien
Boltzmangasse 5A-1090WienAustria
P C Pürrer
Institut für Theoretische Physik
Department of Physics and Astronomy
Universität Wien
Boltzmangasse 5A-1090WienAustria
Aichelburg
Institut für Theoretische Physik
Department of Physics and Astronomy
Universität Wien
Boltzmangasse 5A-1090WienAustria
S Husa [email protected]
University of Pittsburgh
3941 O'Hara Street15260PittsburghPAUSA
EPISODIC SELF-SIMILARITY IN CRITICAL GRAVITATIONAL COLLAPSE
12 Dec 2000
We report on a new behavior found in numerical simulations of spherically symmetric gravitational collapse in self-gravitating SU(2) σ models at intermediate gravitational coupling constants: The critical solution (between black hole formation and dispersion) closely approximates the continuously self-similar (CSS) solution for a finite time interval, then departs from this, and then returns to CSS again. This cycle repeats several times, each with a different CSS accumulation point. The critical solution is also approximately discretely self-similar (DSS) throughout this whole process.
Introduction
As summarized in companion papers in these proceedings (Lechner et al. 1 , Thornburg et al. 2 ), and described in detail elsewhere (Husa et al. 3 ), we are studying critical phenomena in the SU(2) nonlinear σ model in spherical symmetry. This model is parameterized by a dimensionless coupling constant η. We denote the matter field by φ = φ(u, r), where u is an outgoing null coordinate (normalized to proper time at the origin) and r is the areal radius.
This model is known to have a CSS solution for all η < 0.5. This solution can be explicitly constructed 4 , and takes the form φ = φ CSS (z; u * ), where z = r/(u * −u) and the (only) free parameter u * gives the location of the accumulation point.
We consider a 1-parameter family of initial data, and fine-tune this parameter so the initial data's evolution is very close to the threshold of black hole formation. At large (small) η the evolution of such "critical" initial data is DSS (CSS) for a time, before finally either dispersing or collapsing.
However, at intermediate η (≈ 0.16) a new behavior appears, which we call "episodic self-similarity": The field configuration closely approximates a CSS solution, φ ≈ φ CSS (z; u (1) * ) in the inner part of the slice for some finite time interval, then departs from CSS, and then returns to closely approximate a CSS solution, φ ≈ φ CSS (z; u (2) * ) in the inner part of the slice for another finite time interval, then departs, this cycle repeating several times. The u (k) * values increase from one CSS episode to the next.
In addition, a large region of the evolution (spanning several CSS episodes) is approximately DSS, but to a much lower degree of approximation than the approx-imate CSS. Figure 1 shows an example of this behavior. The field configuration never exactly matches a CSS solution, but on the u = 15.893 and u = 16.414 slices (where the fit is good and hence u * = u (k) * is well-defined), |φ − φ CSS | 10 −2 everywhere inside the self-similarity horizon (the backwards light cone of the accumulation point u (k) * ). This region of the evolution is DSS to within ∼ 0.05 in φ.
We do not yet have a full understanding of episodic self-similarity in terms of the standard phase-space model of self-similarity 5 , but we think this behavior is caused by a competition between nearby CSS and DSS attractors.
Figure 1 .
1This figure shows selected u = constant slices in a numerical evolution of η = 0.16 critical initial data (•), with the best-fitting CSS solutions (computed independently for each slice) superimposed (-). For each slice where the fit is good and hence u * = u (k) * is well-defined, the vertical dashed line shows the CSS solution's self-similarity horizon. thornburg2: submitted to World Scientific on March 24, 2022
Type II Critical Collapse of a Self-Gravitating Nonlinear σ Model. Ch, M Lechner, J Pürrer, P C Thornburg, S Aichelburg, Husa, these proceedingsCh. Lechner, M. Pürrer, J. Thornburg, P. C. Aichelburg, and S. Husa, "Type II Critical Collapse of a Self-Gravitating Nonlinear σ Model", these proceedings.
Numerical Methods for Spherically Symmetric Critical Collapse. J Thornburg, Ch Lechner, M Pürrer, P C Aichelburg, S Husa, these proceedingsJ. Thornburg, Ch. Lechner, M. Pürrer, P. C. Aichelburg, and S. Husa, "Numer- ical Methods for Spherically Symmetric Critical Collapse", these proceedings.
. S Husa, Ch Lechner, M Pürrer, J Thornburg, P C Aichelburg, Phys. Rev. D. 62104007S. Husa, Ch. Lechner, M. Pürrer, J. Thornburg, and P. C. Aichelburg, Phys. Rev. D 62, 104007 (2000).
. P Bizoń, A Wasserman, Phys. Rev. D. 6284031P. Bizoń and A. Wasserman Phys. Rev. D 62, 084031 (2000).
. C Gundlach, Adv. Theor. Math. Phys. 2C. Gundlach, Adv. Theor. Math. Phys. 2, 1-49 (1998).
| {'fraction_non_alphanumeric': 0.05674995157853961, 'fraction_numerical': 0.02576021692814255, 'mean_word_length': 4.356846473029045, 'pattern_counts': {'":': 1, '<': 1, '<?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': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'We report on a new behavior found in numerical simulations of spherically symmetric gravitational collapse in self-gravitating SU(2) σ models at intermediate gravitational coupling constants: The critical solution (between black hole formation and dispersion) closely approximates the continuously self-similar (CSS) solution for a finite time interval, then departs from this, and then returns to CSS again. This cycle repeats several times, each with a different CSS accumulation point. The critical solution is also approximately discretely self-similar (DSS) throughout this whole process.', 'arxivid': 'gr-qc/0012043', 'author': ['CHJ Thornburg \nInstitut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria\n', 'M Lechner [email protected] \nInstitut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria\n', 'P C Pürrer \nInstitut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria\n', 'Aichelburg \nInstitut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria\n', "S Husa [email protected] \nUniversity of Pittsburgh\n3941 O'Hara Street15260PittsburghPAUSA\n"], 'authoraffiliation': ['Institut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria', 'Institut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria', 'Institut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria', 'Institut für Theoretische Physik\nDepartment of Physics and Astronomy\nUniversität Wien\nBoltzmangasse 5A-1090WienAustria', "University of Pittsburgh\n3941 O'Hara Street15260PittsburghPAUSA"], 'corpusid': 44485224, 'doi': '10.1142/9789812777386_0349', 'github_urls': [], 'n_tokens_mistral': 1627, 'n_tokens_neox': 1437, 'n_words': 790, 'pdfsha': '869438f0439f1493c32f639e5ba038f36d0c76cd', 'pdfurls': ['https://export.arxiv.org/pdf/gr-qc/0012043v1.pdf'], 'title': ['EPISODIC SELF-SIMILARITY IN CRITICAL GRAVITATIONAL COLLAPSE', 'EPISODIC SELF-SIMILARITY IN CRITICAL GRAVITATIONAL COLLAPSE'], 'venue': []} |
arxiv |
Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model
27 Feb 2012
N V Antonov
Department of Theoretical Physics
St Petersburg University
Uljanovskaja 1198904St Petersburg, PetrodvorezRussia
N M Gulitsky
Department of Theoretical Physics
St Petersburg University
Uljanovskaja 1198904St Petersburg, PetrodvorezRussia
D. I. Mendeleev Institute of Metrology
Moskovsky pr. 19190005St PetersburgRussia
Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model
27 Feb 2012
The field theoretic renormalization group and operator product expansion are applied to the Kazantsev-Kraichnan kinematic model for the magnetohydrodynamic turbulence. The anomalous scaling emerges as a consequence of the existence of certain composite fields ("operators") with negative dimensions. The anomalous exponents for the correlation functions of arbitrary order are calculated in the two-loop approximation (second order of the renormalization-group expansion), including the anisotropic sectors. The anomalous scaling and the hierarchy of anisotropic contributions become stronger due to those second-order contributions.Much attention has been attracted to the problem of intermittency and anomalous scaling in developed magnetohydrodynamic (MHD) turbulence; see e.g.[1]-[9] and references therein. It has long been realized that in the so-called Alfvénic regime, the MHD turbulence demonstrates the behavior, analogous to that of the ordinary fully developed fluid turbulence: cascades of energy from the energy-containing range towards smaller scales, where the dissipation effects dominate the dynamics, and selfsimilar (scaling) behavior of the energy spectra in the intermediate (inertial) range. However, the intermittent character of the fluctuations in the MHD turbulence is much stronger pronounced than that in ordinary fluids.The solar wind provides an ideal wind tunnel in which various models and approaches to MHD turbulence can be tested [3]-[8]. In solar flares, the highly energetic and anisotropic large-scale events coexist with small-scale coherent (singular) structures, finally responsible for the dissipation. Thus modeling the way in which the energy is transferred along the spectra and then dissipated is a difficult task. The intermittency modifies the scaling behavior of the higher-order spectra, leading to anomalous scaling, characterized by infinite families of independent exponents.A simplified description of that situation was proposed in [2]: the large-scale field B 0 i = n i B 0 dominates the dynamics in the distinguished direction n, while the activity in the perpendicular plane can be approximated as nearly two-dimensional.
This picture allows for reliable numerical simulations which show that the turbulent fluctuations tend to organize in rare coherent structures separated by narrow current sheets. On the other hand, the observations and simulations show that the scaling behavior in the solar wind is more similar to the anomalous scaling of fully developed hydrodynamic turbulence, rather than to simple Iroshnikov-Kraichnan scaling suggested by two-dimensional picture with the inverse energy cascade; see e.g. the discussion in [3]. Thus further analysis of more realistic three-dimensional models is desirable.
In this paper, we study the inertial-range scaling behavior within the framework of a simplified three-dimensional model, known as the Kazantsev-Kraichnan kinematic model [10], in which the magnetic field is passive (no feedback on the velocity), while the velocity field is modeled by a Gaussian ensemble with prescribed statistics; see also [11]- [16].
In spite of their relative simplicity, the models of passive scalar fields advected by such "synthetic" velocity ensembles proved to be very interesting because of the insight they offer into the origin of intermittency and anomalous scaling in the real fluid turbulence on the whole: they reproduce many of the anomalous features of genuine turbulent mass or heat transport observed in experiments; see the review paper [17] and the literature cited therein.
Owing to the presence of a new stretching term in the dynamic equation, the behavior of the passive vector field appears much richer than that of the scalar field: "...there is considerably more life in the large-scale transport of vector quantities" (p. 232 of Ref. [18]). Indeed, passive vector fields reveal anomalous scaling already on the level of the pair correlation function [11,12]. They also develop interesting large-scale instabilities that can be interpreted as manifestation of the dynamo effect [10,11,16].
In the presence of a constant background field B 0 i , the dynamic equation for the fluctuating part θ i = θ i (t, x) of the full magnetic field B i = B 0 (n i + θ i ) has the form
∂ t θ i + ∂ k (v k θ i − θ k v i ) = κ∂ 2 θ i + n k ∂ k v i .(1)
Here
v i = v i (t,
x) is the velocity field, κ = c 2 /4πσ is magnetic diffusion coefficient, c is the speed of light, σ is the conductivity, ∂ 2 is the Laplace operator; summation over repeated tensor indices is understood. Equation (1) follows from the simplest form of Ohm's law for a moving medium j = σ(E + v × B/c) and the Maxwell equations neglecting the displacement current; see e.g. [19]. The last term in the right hand side of (1) maintains the steady state of the system and is a source of the anisotropy; in principle, it can be replaced by an artificial Gaussian noise with appropriate statistics.
In the real problem, the field v i satisfies the Navier-Stokes equation with the additional Lorentz force term ∼ (B × curl B). In the Kazantsev-Kraichnan model it obeys a Gaussian distribution with zero mean and correlation function (we consider below the incompressible fluid) (2) where P ij (k) = δ ij − k i k j /k 2 is the transverse projector (needed to ensure the incompressibility condition ∂ i v i = 0), k ≡ |k|, D 0 > 0 is an amplitude factor, d is the dimensionality of the x space. The parameter ξ can be viewed as a kind of Hölder exponent, which measures "roughness" of the velocity field [17]. The IR regularization is provided by the cutoff in the integral (2) from below at k = m, where m ≡ 1/L is the reciprocal of the integral scale L. The anomalous exponents are independent on the precise form of the IR regularization; the sharp cutoff is the most convenient choice from the calculational viewpoints. The natural interval for the exponent ξ is 0 < ξ < 2, when the so-called effective eddy diffusivity has a finite limit for m → 0. However, for the magnetic field a steady state exists only if ξ < ξ c ≤ 2. In the following, we consider the physical case d = 3, when ξ c = 1 [11].
v i (x)v j (x ′ ) = D 0 δ(t − t ′ ) k>m dk (2π) d 1 k d+ξ P ij (k) exp[ik · (x − x ′ )],
The model (1), (2) corresponds to the so-called kinetic regime, in which the effects of the magnetic field on the velocity statistics are neglected. In this connection, it is worth noting that in the full-scale model of the MHD turbulence such a regime is indeed realized in the so-called kinetic fixed point of the renormalization group (RG) equations [20]. Various generalizations of this model (finite correlation time, compressibility, more general form of the nonlinear terms) were also studied [21]- [24].
The RG approach to the Kazantsev-Kraichnan model is described in [14] in detail; here we only recall the main points. The original stochastic problem (1), (2) is reformulated as a certain field theoretic model. The ultraviolet divergences in the corresponding Feynman diagrams manifest themselves as poles in ξ. They are removed by multiplicative renormalization; as a byproduct of that procedure, differential RG equations are derived for various correlation functions. They have an IR attractive fixed point. This means that in the IR range Λr ≫ 1, where Λ is the reciprocal of the inner (dissipation) length, the correlation functions acquire scaling forms with certain critical dimensions ∆ F of all the fields and parameters F of the model.
The most important part is played by the critical dimensions ∆ n,l associated with the irreducible tensor composite fields ("local composite operators" in the field theoretic terminology) built solely of the fields θ at a single space-time point x = (t, x). They have the forms
F n,l ≡ θ i 1 (x) · · · θ i l (x) (θ i (x)θ i (x)) p + . . . ,(3)
where l ≤ n is the number of the free vector indices and n = l + 2p is the total number of the fields θ entering into the operator; the tensor indices and the argument x of the symbol F n,l are omitted. The ellipsis stands for the appropriate subtractions involving the Kronecker delta symbols, which ensure that the resulting expressions are traceless with respect to contraction of any given pair of indices, for example, θ i θ j − δ ij (θ k θ k /d) and so on. The numbers n and l are even or odd simultaneously. The quantities of interest are, in particular, the equal-time pair correlation functions of the operators (3). For these, solving the corresponding RG equations gives the following asymptotic expression
F n,l (t, x)F k,j (t, x ′ ) ≃ κΛ 2 −(n+k)/2 (Λr) −∆ n,l −∆ k,j ζ n,l;k,j (mr)(4)
with certain scaling functions ζ(mr). To simplify the notation, here and below in similar expressions we omit the tensor indices and the labels of the functions ζ(mr). The last expression in (4) is valid for Λr ≫ 1 and arbitrary values of mr. The inertial-convective range corresponds to the additional condition that mr ≪ 1. The forms of the functions ζ(mr) are not determined by the RG equations themselves; their behavior for mr → 0 is studied using Wilson's operator product expansion (OPE).
According to the OPE, the equal-time product F 1 (x)F 2 (x ′ ) of two renormalized composite operators at x = (x + x ′ )/2 = const and r = x − x ′ → 0 can be represented in the form
F 1 (x)F 2 (x ′ ) ≃ F C F (r)F (t, x),(5)
where the functions C F are the Wilson coefficients, regular in m 2 , and F are, in general, all possible renormalized local composite operators allowed by symmetry. More precisely, the operators entering the OPE are those which appear in the corresponding Taylor expansions, and also all possible operators that admix to them in renormalization. If these operators have additional vector indices, they are contracted with the corresponding indices of the coefficients C F . It can always be assumed that the expansion in Eq. (5) is made in operators with definite critical dimensions ∆ F . The correlation functions (4) are obtained by averaging equation of the type (5) with the weight exp S, where S is the De Dominicis-Janssen action functional for our stochastic problem; the quantities F appear on the right hand sides. Their asymptotic behavior for m → 0 is found from the corresponding RG equations and has the form F ∝ m ∆ F . From the expansion (5) we therefore find the following asymptotic expression for the scaling function ζ(mr) in the representation (4) for mr ≪ 1:
ζ(mr) ≃ F A F (mr) ∆ F ,(6)
where the coefficients A F = A F (mr) are regular in (mr) 2 . The feature specific of the models of turbulence is the existence of composite operators with negative critical dimensions. Such operators are termed "dangerous," because their contributions to the OPE diverge at mr → 0 [25].
The dimension of the primary field θ(x) is known exactly, ∆ θ = ∆ 1,0 = −1 + ξ/2; the dimensions of the other operators (3) are calculated as infinite series in ξ:
∆ n,l = ∆ (1) n,l ξ + ∆ (2) n,l ξ 2 + O(ξ 3 ).(7)
The dimensions ∆ 2,0 and ∆ 2,2 can be found in a closed form [14] by comparison with the exact results for the pair correlation function, obtained within the zero-mode approach in [11]- [13]. For general n and l, the first-order term in (7) was derived in [14] and has the form (for d = 3 and up to the notation):
∆ (1) n,l = −n(n + 3)/10 + l(l + 1)/5.
We have calculated the second term in (7), which corresponds to the two-loop approximation of the RG. The calculation is rather cumbersome and will be discussed elsewhere, and here we only present the final result:
∆ (2) n,l = − 2n(n − 2) 125 − n(n + 3) 30 + 22l(l + 1) 375 − − 3(n − 2) 175 − √ 3π + 82 15 2n(n − 4) + 3l(l + 1) + + 9(n − 2) 700 − √ 3π + 24 5 n(n + 3) − 2l(l + 1) .(9)
In particular, for the two special cases mentioned above this gives
∆ 2,0 = −ξ − ξ 2 /3 + O(ξ 3 ), ∆ 2,2 = ξ/5 + 7ξ 2 /375 + O(ξ 3 ),(10)
in agreement with the exact results of [11]- [13]. This confirms validity and mutual consistency of the zero-mode and the RG+OPE approaches. From the leading-order expression (8) it follows that the dimensions (7) satisfy the hierarchy relations: ∆ n,l > ∆ n,l ′ if l > l ′ , which are conveniently expressed as inequalities ∂∆ n,l /∂l > 0. This fact, first established in [13], has a deep physical meaning: in the presence of large-scale anisotropy, the leading contribution in the inertial-range behavior mr ≪ 1 of the correlation functions like (4) is given by the isotropic "shell" (l = 0). The corresponding anomalous exponent is the same as for the purely isotropic case. In their turn, the anisotropic contributions give only corrections which vanish for mr → 0 the faster the higher the degree of anisotropy l is. This effect gives some quantitative support for Kolmogorov's hypothesis of the local isotropy restoration and appears rather robust, being observed for the real fluid turbulence [26] and the scalar Kraichnan model [27].
From (9) it follows that the O(ξ 2 ) term in the dimension (7) satisfies the same inequality, ∂∆ (2) n,l /∂l ≃ (2l + 1)(0.015n + 0.028) > 0, and we conclude that the hierarchy of anisotropic contributions becomes stronger when the second-order correction is taken into account.
For the most important case of the scalar operator (3) with l = 0 from (9) one obtains ∆ (2) n,0 ≃ −0.009n 3 − 0.052n 2 − 0.025n, (11) which is negative for all n. Thus the anomalous scaling also becomes stronger pronounced when the O(ξ 2 ) term is included. In this respect, the magnetic model differs drastically from its scalar counterpart, where the higher-order corrections to the O(ξ) term are positive, which eventually leads to the disappearance of the anomalous scaling for ξ → 2; see the discussion in [28].
To conclude with, we calculated the anomalous exponents in the Kazantsev-Kraichnan kinematic dynamo model in the two-loop order (including the anisotropic "shells" in the presence of large-scale anisotropy). We found that both the anomalous scaling and the hierarchy of anisotropic contributions become stronger due to the secondorder corrections to the leading terms. It is interesting to see how these results are
affected by compressibility, anisotropy and the influence of the magnetic field on the velocity field dynamics. This work remains for the future and is partly in progress.The authors are indebted to L. Ts. Adzhemyan, Michal Hnatich, Juha Honkonen, Andrea Mazzino and Paolo Muratore Ginanneschi for discussions.
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| {'fraction_non_alphanumeric': 0.07349367716340194, 'fraction_numerical': 0.039920654599553684, 'mean_word_length': 4.023916292974589, 'pattern_counts': {'":': 0, '<': 3, '<?xml version=': 0, '>': 6, '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': 'The field theoretic renormalization group and operator product expansion are applied to the Kazantsev-Kraichnan kinematic model for the magnetohydrodynamic turbulence. The anomalous scaling emerges as a consequence of the existence of certain composite fields ("operators") with negative dimensions. The anomalous exponents for the correlation functions of arbitrary order are calculated in the two-loop approximation (second order of the renormalization-group expansion), including the anisotropic sectors. The anomalous scaling and the hierarchy of anisotropic contributions become stronger due to those second-order contributions.Much attention has been attracted to the problem of intermittency and anomalous scaling in developed magnetohydrodynamic (MHD) turbulence; see e.g.[1]-[9] and references therein. It has long been realized that in the so-called Alfvénic regime, the MHD turbulence demonstrates the behavior, analogous to that of the ordinary fully developed fluid turbulence: cascades of energy from the energy-containing range towards smaller scales, where the dissipation effects dominate the dynamics, and selfsimilar (scaling) behavior of the energy spectra in the intermediate (inertial) range. However, the intermittent character of the fluctuations in the MHD turbulence is much stronger pronounced than that in ordinary fluids.The solar wind provides an ideal wind tunnel in which various models and approaches to MHD turbulence can be tested [3]-[8]. In solar flares, the highly energetic and anisotropic large-scale events coexist with small-scale coherent (singular) structures, finally responsible for the dissipation. Thus modeling the way in which the energy is transferred along the spectra and then dissipated is a difficult task. The intermittency modifies the scaling behavior of the higher-order spectra, leading to anomalous scaling, characterized by infinite families of independent exponents.A simplified description of that situation was proposed in [2]: the large-scale field B 0 i = n i B 0 dominates the dynamics in the distinguished direction n, while the activity in the perpendicular plane can be approximated as nearly two-dimensional.', 'arxivid': '1202.5992', 'author': ['N V Antonov \nDepartment of Theoretical Physics\nSt Petersburg University\nUljanovskaja 1198904St Petersburg, PetrodvorezRussia\n', 'N M Gulitsky \nDepartment of Theoretical Physics\nSt Petersburg University\nUljanovskaja 1198904St Petersburg, PetrodvorezRussia\n\nD. I. Mendeleev Institute of Metrology\nMoskovsky pr. 19190005St PetersburgRussia\n'], 'authoraffiliation': ['Department of Theoretical Physics\nSt Petersburg University\nUljanovskaja 1198904St Petersburg, PetrodvorezRussia', 'Department of Theoretical Physics\nSt Petersburg University\nUljanovskaja 1198904St Petersburg, PetrodvorezRussia', 'D. I. Mendeleev Institute of Metrology\nMoskovsky pr. 19190005St PetersburgRussia'], 'corpusid': 33214104, 'doi': '10.1103/physreve.85.065301', 'github_urls': [], 'n_tokens_mistral': 6771, 'n_tokens_neox': 5750, 'n_words': 3314, 'pdfsha': '4b5ca974e3e94fff6f4a639b88aaf81ab854812b', 'pdfurls': ['https://arxiv.org/pdf/1202.5992v3.pdf'], 'title': ['Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model', 'Anomalous scaling and large-scale anisotropy in magnetohydrodynamic turbulence: Two-loop renormalization-group analysis of the Kazantsev-Kraichnan kinematic model'], 'venue': []} |
arxiv |
Super-Resolution Imaging via Angular Magnification
Yi Zhou
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Dingpeng Liao
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Kun Zhang
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Zijie Ma
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Shikai Wu
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Jun Ma
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Xuemei Dai
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Zhengguo Shang
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Zhongquan Wen
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Gang Chen
Ministry of Education
School of Optoelectronic Engineering
Key Laboratory of Optoelectronic Technology and Systems
Chongqing University
Chongqing University
174 Shazheng Street400044Shapingba, ChongqingChina
Super-Resolution Imaging via Angular Magnification
1super-resolutiondielectric metalensangular magnification
The far-field resolution of optical imaging systems is restricted by the Abbe diffraction limit, a direct result of the wave nature of light. One successful technological approach to circumventing this limit is to reduce the effective size of a point-spread-function. In the past decades, great endeavors have been made to engineer an effective point-spread-function by exploiting different mechanisms, including optical nonlinearities and structured light illumination. However, these methods are hard to be applied to objects in a far distance. Here, we propose a new way to achieve super-resolution in a far field by utilizing angular magnification.We present the first proof-of-concept demonstration of such an idea and demonstrate a new class of lenses with angular magnification for far-field super-resolution imaging. Both theoretical and experimental results demonstrate a more than two-fold enhancement beyond the angularresolution limit in the far-field imaging. The proposed approach can be applied to super-2 resolution imaging of objects in far distance. It has promising potential applications in superresolution telescopes and remote sensing.
INTRODUCTION
Due to the spreading nature of light wave propagation, the optical image of an ideal infinitely small point obtained through optical devices or systems becomes a central, bright disk of infinite size surrounded by concentric rings [1]. This sets a fundamental restriction on the resolution of optical imaging, which was first proposed by Ernst Abbe [2]. According to Rayleigh criterion, the diffraction limit restricts the ability of optical systems to resolve objectives smaller than 0.61λ/NA, where λ and NA are wavelength and numerical aperture. To break through this limitation, great efforts have been made in the past half century, among which near-field probing [3] was firstly proposed. By utilizing non-propagating high-frequency components in the optical field near the objective surface, small details beyond the diffraction limit have been observed through a scanning probe moving above the surface at a working distance much less than a wavelength. Hyperlenses and metalenses can convert near-field waves into propagation waves in specially designed high-effective-index metamaterials for far-field imaging, however it working distance is in sub-wavelength scale and cannot be applied to imaging for objects in far distance. [4,5] Far-field super-resolution optical microscopy has also experienced a rapid development, which can be divided into three major categories, i.e., single molecule localization, point-spreadfunction engineering and frequency shifting. Single molecule localization microscopy can achieve a resolution of 10 nm [6] by using photoactivation or photoswitching of single fluorophores and position determination; stimulated emission depletion microscopy can compress the effective point-spread-function far beyond the diffraction limit and achieve a resolution of 2.4 nm [7] by utilizing non-linear effect in the competition between stimulated emission and autofluorescence; structured light illumination microscopy can realize a resolution of λ/5 [8] by shifting the higher frequency information into propagation wave which is used to reconstruct the high resolution image of an objective. However, the first two approaches largely rely on fluorescence labeling, and the third one depends on reconstruction algorithms. Physically, these approaches essentially work on compressing the effective point-spread-function of the whole system, which involves not only the optical system itself but also the objective under test.
Microspheres-assisted super-resolution microscopy can realize far-field imaging by utilizing increased effective numerical aperture, however its working distance is limited to several micrometers in optical domain and cannot be applied to imaging for objects in far distance too. [9,10] The recent proposal of the concept of optical superoscillation [11][12][13] provides an alternative way to achieve far-field super-resolution by engineering the point-spread-function of optical devices or systems. According to the theory, an arbitrary small point-spread-function can be realized by carefully designing the complex optical transmittance function. Various types of superoscillation optical devices [14][15][16][17][18][19] have been demonstrated, showing promising potentials in realizing super-resolution telescopes [20][21][22] and far-field label-free super-resolution microscopy. [23][24][25][26] However, the inevitable strong sidebands significantly limit the field of view in the realistic applications. Especially in telescopes, the strong sidebands result in a comparatively weak super-resolution image surrounded by strong sidebands.
In traditional optics, the angular magnification of optical lens is equal to 1, when they work in a single medium, for example in air. Due to this fact, as having been done in the past decades, the natural and direct way to overcome the diffraction limit is to compress the effective size of the point-spread-function. In contrast, another possible solution is to enhance the angular magnification without increasing the effective size of the point-spread-function. Here, we have proposed such an approach to realizing far-field super-resolution imaging by angular magnification. The validity of the concept has been theoretically and experimentally proved and demonstrated with properly designed lenses with angular magnification greater than 1. We also prove that the strong sideband can be eliminated in the super-resolution image by using the proposed approach, showing its obvious advantages over super-resolution point-spread-function engineering. The proposed approach can be applied to super-resolution imaging of objects in far distance. It has promising potential applications in super-resolution telescopes and remote sensing. Figure 1a presents the schematic diagram of optical imaging of two ideal points A and B through a conventional optical lens. In the object space, the spacing between the two points A and B is do; the angle between the chief rays of the two points is θ; the object distance is l0; the refractive index is n. In the image space, the spacing between the two image A' and B' is di; the angle between the chief rays of the two image points is θ'; the image distance is li; the size of the point-spread-function is d; the refractive index is n'. The lens radius is R. Generally, when the refractive indices are equal, i.e. n=n', the angles in both sides of the lens are also equal, i.e. θ=θ'.
THEORY ON SUPER-RESOLUTION IMAGING VIA ANGULAR
MAGNIFICATION
According to the Rayleigh diffraction limit, the achievable minimum size of the point-spreadfunction on the image plane is given by d=0.61λ/NA, where NA=sin(atan(R/li)), and the two points A and B cannot be distinguished when the spacing between the two image spots is less than d, as shown in Figure 1a. Therefore, the corresponding angular resolution in the object space is restricted to δθDL =2atan(0.5d/li), which can be rewritten as equation (1). If li is much greater than R, it can be further simplified as equation (2).
δθDL(0) =1.22λ/D(1)
where D is the diameter of the lens, i.e. D=2R. In a more general case, if the two angles are not equal and θ' is greater than θ, or θ'=Kθ (K > 1), then the angular resolution δθK in the object space should be modified by adding an additional factor 1/K to equation (1), as expressed in equation (3). Similarly, it can be simplified as equation (4) when li is much greater than R.
δθK(NA) =2atan(0.5×0.61λ/( K•NA•li))(3)δθK(0)=1.22λ/(K•D)(4)
Therefore, the corresponding resolution can be enhanced by a factor of K. Such angular magnification can be realized by making the refractive index in the object space higher than that in the image space, [27] as shown in Figure 1b. This technique has been implemented in medium-immersion-microscopy, where the object space is immersed in a transparent medium with a high refractive index greater than the medium of imaging space. However, this enhancement of the angular resolution demands a higher refractive index in the object space, which thus poses severe limitations to the use of medium-immersion imaging in many practical applications when the objects are in far distance. curve; the angular-resolution-limit δθK(NA) of a lens with angular magnification of K=2 is given by the blue curve; the limitation of the traditional angular-resolution-limit δθDL(0) is presented as the numerical aperture approaching zero, given by the black dash-dotted line. It is found that the angular-resolutions given by the two curves increase as the numerical aperture decrease, and both achieve their minimum angle limitations of δθDL(0) and δθK(0) respectively at NA=0.
Is it possible to achieve angular magnification without using medium-immersion approaches?
As shown in Figure 1c, does such a type of lenses which has the ability of angular magnification K > 1 exist even when the refractive indices are equal in both object and image spaces? Here we report the first experimental and numerical demonstration of the possibility to realize such lenses for super-resolution imaging by angular magnification. Experimentally, we can resolve the object details at 1/K of the diffraction limit imposed by the conventional optical systems.
SUPER-RESOLUTION LENSES OF ANGULAR MAGNIFICATION
SLAM Singlet with Large Numerical Aperture. To prove our concept, a SLAM singlet with K = 1.9487 has been designed by using angular spectrum method [28] and optimization algorithm [29]. As shown in Figure 3a, to realize the lens phase profile, an a-Si Pancharatnam-Berry phase meta-atom [30] with a unit size of P×P has been adopted for the wavelength of and focuses the emergence wave on the focal plane. In the design, the angular spectrum method [28] is used to calculate the diffraction pattern of the lens, and the particle-swarm optimization algorithm [29] is used to find the desired phase profile of the lens. The phase profile is optimized in a way that guarantees the focal spot is (FWHM, full-width-at-half-maximum) no greater than The critical constraint is also applied to achieve the linear-relationship between the incident angle θi and the emergence angle θe, i.e. θe=Kθi. The optimized phase profile φ(r) is presented in Table 1. Figure 3a also illustrates a letter of "E" used as the object for imaging. The linewidth of the letter is W, the center-to-center distance between two neighboring lines is dc, and the width of the entire letter is L. Numerical simulations have been conducted to obtain the intensity distribution on the focal plane for 21 different incident angles with an equal interval of 0.1° between 0 and 2°. It is found that there is only one single spot on the designed focal plane at each simulated angle and the displacement of the focal spot monotonically increases with the increase of the incident angle.
Figure 4b
shows the major focusing parameters, i.e. peak intensity Ipeak, FWHM, r (the halfwidth at intensity of 0.01Ipeak) and SR, on the designed focal plane at different incident angles. It is seen that the both FWHM and r are smaller than 1.8λ, which is slightly greater than the Rayleigh diffraction limit 0.61λ/NA =1.386λ. Figure 4c plots the relation between the emergence angle θe and the incident angle θi. The emergence angle clearly shows an excellent linear relationship with the incident angle and the angular magnification K is approximately 1.88345. Figure 4e presents the incoherence imaging result of letter "E" at illumination wavelengths of 632.8 nm. The object distance is lo = 38.1 cm and the image distance is li =506.9
μm. The width of the entire letter is L = 1884 μm, the linewidth of the letter is W = 80 μm and the center-to-center distance between two neighboring lines is dc = 471 μm, which corresponds to an angle of δθ = 0.0708º at the given object distance of lo = 38.1 cm. This angle is only 0.707 times of angular resolution limit of 0.09734º at NA=0.44 (0.61λ/(NAf)), and it is smaller than the angular resolution limit of 0.08737º at NA=0 (1.22λ/D). Figure 4d plots the intensity distribution along the white-dashed line in Figure 3d, and it shows that the image of the letter of "E" can be clearly resolved with a contrast of 11.85%, which is greater than the contrast value required by the Rayleigh resolution criterion. Therefore, the SLAM doublet proves the superresolution ability of the lens with angular magnification at small numerical aperture. Experimental Setup. As shown in Figure 6, a LED is used as the incoherent illumination source for imaging. A pair of linear polarizer (LP) and quarter-wave plate (QWP) is used to generate desired circular polarization. By utilizing a positive lens, the light is then collected and focused on to a glass plate with an array of letter "E", which is used as the object for imaging.
The fabricated meta-lens is mounted on a three-dimensional linear translation stage (LNR25, Thorlabs Inc.) right after the object. The distance between the object and the SLAM singlet is kept greater than the designed focal length. The image formed on the image plane of the SLAM singlet is acquired with a high-numerical and high-magnification microscope, which is composed of an infinite objective lens (CF Plan 150×/0.95, Nikon), a one-dimension nanopositioner (P-721.CDQ, Physik Instrumente), a tube lens (ITL200, Thorlabs, Inc.) and a high-resolution digital camera (acA3800-14μm, Basler, Inc.). Between the objective lens and the tube lens, another pair of QWP and LP is used to filter out the unwanted polarization. The objective lens is mounted on the nano-positioner, which is used to conduct the scan in the z-axis to obtain the two-dimensional optical intensity distribution at different z coordinates. The image of the diffraction pattern is recorded through the digital camera. The "E" letter on the object has the entire width of L = 120 μm, linewidth of W = 24 μm and the center-to-center distance between two neighboring lines dc = 80 μm. According to our theory, as given by equation 3, the minimum angle that can be resolved by the SLAM singlet is 0.308º, which is roughly half of the angular resolution limit of 0.613º allowed by a conventional optical lense with the same focal length of 60λ and radius of 240λ. According to the results presented in Figure 7a, the character "E" can be clearly identified in all four cases. Figures 7c and 7d to the theoretical design value of 1.9487. It is also found that the size of the image decreases as the objective distance increases in both the experimental and numerical results, which is similar to that observed in conventional imaging lens. Interestingly, although the angular resolution achieves more than two-fold enhancement, unlike the super-resolution telescope based on the point-spread-function engineering and the concept of superoscillation, where a weak image is inevitable surrounded by a bright huge sideband, [20][21][22][23] there is no obvious sideband appearing in the image plane in present case. Therefore, both the experimental and numerical results prove the validity and advantages of the proposed super-resolution imaging by angular magnification.
As indicated in
CONCLUSION
In conclusion, we have proposed a new way to realize super-resolution optical imaging by angular magnification. We have also proposed a new class of diffractive lenses with angular magnification greater than 1. Our concept has been theoretically and experimentally proved by specially designed diffractive lenses with angular magnification. The results demonstrate a more than two-fold enhancement in angular resolution, compared with the conventional angular resolution limit. Although, the demonstrated lenses have angular magnification of approximate 2, higher angular magnification is also possible. It provides a promising way to overcome the challenge in breaking through the conventional optical diffraction limit and solve the problem
Figure 1 .
1(a) Imaging using the conventional optical lens with angular magnification K=1; (b) imaging using the conventional medium-immersion optical lens with angular magnification K>1; (c) imaging using the proposed super-resolution lens with angular magnification K>1. δθDL(NA)=2atan(0.5×0.61λ/(NA•li))
Figure 2 .
2Comparison of the different angular-resolution-limits. The traditional angular-resolution-limit at different numerical aperture is given by the red curve; the angular-resolution-limit of a lens with angular magnification of K=2 is given by the blue curve; the limitation of the traditional angularresolution-limit δθ DL (0) is presented as the numerical aperture approaching zero, given by the black dash-dotted line.
Figure 2
2gives the comparison of the different angular-resolution-limits. In the figure, the traditional angular-resolution-limit δθDL(NA) at different numerical apertures is given by the red
632.8 nm. It is an a-Si cubic block with a height of H, length of Ls and width of Lf on the top of a SiO2 substrate. For a circularly polarized incident wave, it introduces a phase shift of φ=2α in the cross-polarized emergent wave with a rotation angle of α. Based on the a-Si meta-atom, a phaseonly transmission function has been optimized for a lens with given focal length of f = 60λ and radius of R = 240λ at wavelength of λ=632.8 nm. The lens numerical aperture is NA=0.97. The corresponding angular resolution limits are δθDL(NA) = 0.6005º at NA = 0.97 and δθDL(0) = 0.1456º at NA=0, respectively.
Figure 3 .
3Simulation results of the SLAM singlet. (a) the diagram of the a-Si meta-atom, the diagram of the side view and top view of the SLAM singlet, and the letter "E" used as the object for imaging; (b) the peak intensity (red stars), FWHW (royal spheres), first-zero r0 (Magenta hexagons) and sidelobe ratio (Olive diamonds) of the focal spot with different incident angles, where solid and open markers denote the results in x-direction and y-direction respectively; (c) the incoherent image of letter "E" at the object distance of lo=18 mm and image distance of li=37.97 μm under the incoherent illumination at wavelength of 632.8 nm; (d) the intensity distribution along the white-dashed line in (d).As shown inFigure 3a, the proposed SLAM singlet consists of a series of ring belts with width of P. The central radius of the i-th ring belt is denoted by Ri, which is equal to i×P. The meta-atoms within the same ring belt have the same phase value. For right-circular (RC) polarization incident wave, the lens converts the polarization into left-circular (LC) polarization
FWHMmax=0
.5λ and sidelobe ratio (SR, the ratio of the maximum sidelobe intensity to the central lobe intensity) is no greater than SRmax = 0.2 on the focal plane under the illumination of the plane waves at incident angles within the given field-of-view (FOV) of 4° in the object space which correspond to a FOV of 7.7948° in the image space for angular magnification K = 1.9487.
Figure 3b
3bdepicts focal spot intensity, FWHM, first-zero of the intensity pattern r0, SR and θe with respect to the incident angle θi. It is seen that both FWHM and r0 are smaller than the Rayleigh's diffraction limit of 0.61λ/NA = 0.629λ and the SR is smaller than 0.181, for all incident angles. It is noted that the focal spot intensity only fluctuates slightly in the range of 8% of the maximum value within the designed field-of-view. According to the result, the emergence angle θe varies linearly with the incident angle θi and the angular magnification is K=1.9487.
Figure 3c
3cpresents the incoherence imaging result of letter "E" at illumination wavelength of 632.8 nm, where the object distance is lo = 18 mm and the image distance li =37.97 μm. The width of the entire letter is L = 120 μm, the linewidth of the letter is W = 24 μm and the centerto-center distance between two neighboring lines is dc = 80 μm which corresponds to an angle of δθ = 0.255º at the object distance of lo = 18 mm.Figure 3dplots the intensity distribution along the white-dashed line inFigure 3c. Because the Rayleigh resolution criterion requires a minimum contrast of 10.5%[28], the image of the letter of "E" can be clearly resolved due to the image contrast of 15.3%. Therefore, the SLAM singlet can achieve an angular resolution of 0.255º, which is only 0.425 time of the angular resolution limit δθDL(NA) = 0.6005º at NA = 0.97.The angular resolution shows more than two-fold enhancement, resulting from both subdiffraction spot size and angular magnification. The spot size is approximately 0.787 times of the Rayleigh's diffraction limit, while the angular magnification K=1.9487. Therefore, ideally, the angular resolution can achieve approximately 0.404 (0.787/K), which is slightly smaller than the simulation result. This difference is due to the fact that the SR of the focal spot of the SLAM singlet is greater than that of Airy spot. It is clearly seen that the angular magnification plays a major role in the enhancement of the angular resolution, which proves the super-resolution imaging ability of the proposed SLAM singlet.SLAM Doublet with Small Numerical Aperture. Although the SLAM singlet shows an enhancement of angular resolution by a factor of 2.353 compared with that of a perfect conventional imaging lens with the same numerical aperture of NA=0.97, its resolution remains above the theoretical limitation with the same radius and NA close to zero, as illustrated in
Figure 2 .
2To further verify our concept at smaller NA, a SLAM doublet has been designed with an angular magnification of K=1.88345 and a numerical aperture of NA=0.44 at wavelength of λ=632.8 nm using the same method. As presented inFigure 4a, the SLAM doublet consists of two Pancharatnam-Berry metasurfaces with the same radius of R on both sides of the glass substrate with a thickness of t. The refractive index of the glass substrate is 1.457 at wavelength of 632.8 nm. For the focal length of f=816λ and radius of R=400λ, the optimized substrate thickness of t is 675 μm. The corresponding angular resolution limit is δθDL(NA) = 0.0973º at NA = 0.44 and δθDL(0) = 0.0874º at NA=0, respectively. Table 2 displays the optimized phase distributions of the two functional metasurfaces in 32-based digitals.
Figure 4 .
4Simulation results of the SLAM doublet. (a) the diagram of the proposed lens; (b) the peak intensity I peak (red stars), FWHW (royal spheres), r (Magenta hexagons, the half-width at intensity of 0.01Ipeak) and sidelobe ratio (Olive diamonds) of the focal spot at different incident angles, where solid and open markers denote the results in x-direction and y-direction respectively; (c) the relation between the emergence angle θ e and the incident angle θ i ; (d) the image of letter "E" at the object distance of l o =38.1 cm and image distance of l i =506.9 μm under the incoherent illumination at wavelength of 632.8 nm; (e) the intensity distribution along the white-dashed line in (d).
Figure 5 .
5Simulation results of broadband incoherent imaging with the SLAM doublet. The image of letter "E" at different wavelengths of (a) 554 nm, (b) 590 nm, (c) 625 nm, (d) 635 nm, (e) 666 nm and (f) 680 nm, where the left is the two-dimensional intensity distribution on the corresponding image plane and the right is the intensity distribution along the corresponding white-dashed line. The object distance and image distance are denoted by lo and li respectively.For the SLAM doublet, simulations have also been conducted to investigate its broadband super-resolution imaging performance at different wavelengths of 554 nm, 590 nm, 625 nm, 635 nm, 666 nm and 680 nm. The simulation results are presented in Figure 5. The corresponding object distance and image distance (lo, li) are (42 cm, 1170.8λ=648.62 μm), (39.568cm, 987.8λ=582.81 μm), (37.4cm, 830.2λ=518.90 μm), (36.95 cm, 794.2λ=504.34 μm), (35.7 cm, 705.0λ=469.54 μm), and (34.4 cm, 647.69λ=440.43 μm) respectively. The angular resolutions can achieve 0.0643º, 0.0682º, 0.0722º, 0.073º, 0.0756º and 0.0784º respectively, which are smaller than the corresponding angular resolution limit δθDL(NA) of 0.0821, 0.0888, 0.0960, 0.0981, 0.1045 and 0.1083. They are even smaller than the angular resolution limit δθDL(0) of 0.07649, 0.08146, 0.08629, 0.09227, 0.09195, 0.09389º respectively at NA=0 (1.22λ/D). Therefore, the SLAM doublets can achieve super-resolution imaging in a broadband wavelength range of 126 nm. EXPERIMENTAL RESULTS AND DISCUSSIONS For easy fabrication, a SLAM singlet based on the above design has been fabricated with conventional nanofabrication techniques. The a-Si meta-atom has been optimized at wavelength of 632.8 nm. The pitch of the meta-atom is P = 300 nm, the length, width and height of the metaatom are L = 200 nm, W = 116 nm and H = 330 nm, respectively. The phase φ(rij) at point of rij can be realized through rotating the a-Si block by an angle α=φ(rij)/2. To experimentally prove the validity of our idea, super-resolution imaging experiments have been conducted for incoherent illumination.
Figure 6 .
6The setup of the imaging experiments, where LPs and QWPs are linear polarizers and quarter-wave plates, respectively.Results and Discussions.In the experiments, for the object distances of lo = 12 mm, 13 mm, 14 mm and 15 mm, the images were obtained through SLAM singlet at wavelength of 625 nm.These images are depicted inFigure 7a, in which the corresponding angles between the two neighboring horizontal lines in character "E" are 0.382º, 0.353º, 0.327º and 0.306º, respectively.The corresponding image distances are li = 40.0 µm, 37.5 µm, 35.7 µm and 39.1 µm respectively.
Figure 7b
7bplots the intensity distributions along the dashed lines in Figure 7a, based on which the contrast has been calculated for each case. The corresponding contrast values are 0.22, 0.20, 0.168 and 0.158, respectively. All the contrast values are greater than the minima contrast value of 0.105 for the resolvable condition required by Rayleigh criterion. Therefore, the experimentsgive the best resolution of 0.306º, which is slightly smaller than the predicted 0.308º by equation3. This is because a conventional Airy spot has a spot size of 0.61λ/NA and sidelobe ratio less than 1.57%, while the focal spot of the proposed metalens has a smaller spot size (FWHM) of approximately 0.47λ and a larger sidleobe ratio of approximately 20% in the experiments. It is found that the peak-to-peak distances between two outer lines of the image of letter 'E' are 1.501λ, 1.437λ, 1.334λ and 1.366λ for the above four different cases, which correspond to anglesof 1.344º, 1.372º, 1.291º and 1.254º in the image space. The corresponding angular magnifications are 1.759, 1.943, 1.974 and 2.109 respectively, which are close to the theoretical design value of 1.9487. Figures 7c and 7d give the corresponding numerical simulation results of a single character "E" with the same geometrical size and under the same object distances. The corresponding image distance is 39.1 μm for the largest contrast value. The simulation results give better contrast values of 0.22, 0.361, 0.407 and 0.408 for each case. Compared with the numerical results, the lower contrast values obtained in the experiments are due to two reasons. Firstly, it is caused by the experimental and fabrication errors; Secondly, there are influences caused by neighboring characters in the array of letter "E" in the experiments, while there is only single letter "E" in the numerical simulations.
Figure 7 .
7The imaging results of the proposed SLAM singlet at different objective distances of l o . (a) the experimental imaging results; (b) the intensity curve along the dashed line in (a); (c) the simulation imaging results; (d) the intensity curve along the dashed line in (c).
Figure 8 .
8The imaging results of the proposed SLAM singlet at different image distances of li. (a) the experimental imaging results; (b) the intensity curve along the dashed line in (a); (c) the simulation imaging results; (d) the intensity curve along the dashed line in (c). To investigate the tolerance of the image position, the on-axis scanning of the objective lens was conducted. The images have been acquired at different image distances of li = 37.5 μm, 38.3μm, 39.1 μm and 40.0 μm for a fixed object distance of lo= 1500μm, corresponding to the actual minimum resolvable angle of 0.306º of the SLAM in the experiment. The experimentally obtained images are presented in Figure 8a. In Figure 8b, the intensity distributions along the dashed lines in Figure 8a are plotted for those images. It is found that, in the range of the object distance between 37.5 μm and 40 μm, the images obtained with the lens can be clearly resolved with contrast values of 0.123, 0.14, 0.133 and 0.129 respectively, which are greater than the Rayleigh criterion contrast value of 0.105. It indicates a depth-of-focus of 2.5 μm (4λ), which is unexpectedly large for a lens with NA of 0.97. Figure 8c and 8d are the corresponding simulation results for comparison. Consequently, for the two reasons discussed above, the images obtained by the numerical simulations show better contrast values.
Figure 9 .
9The imaging results of the proposed SLAM singlet at different illumination wavelengths of 625 nm and 591 nm. The experimental results at wavelengths of (a) 625 nm and (b) 591 nm; the numerical results at wavelengths of (c) 625 nm and (d) 591 nm. To investigate the bandwidth of the fabricated SLAM singlet, the imaging experiments have also been conducted at two different wavelengths of 625 nm and 591 nm. Figure 9a and 9b present the experimentally obtained images at the two different wavelengths with the minimum resolvable angle of 0.306º and 0.327º for different objective distances of lo = 15 mm and lo = 14 mm respectively. The best image plane located at li =39 μm and at li = 45.8 μm. The image contrasts are 0.158 and 0.122 respectively. Therefore, they are greater than the value required by the Rayleigh criterion and the images are resolvable at both wavelengths under test. The corresponding angular magnifications are calculated to be 2.12 and 1.65 respectively. Figures 9c and 9d present the corresponding numerical results, which give better contrast values of 0.4 and 0.367. The corresponding angular magnifications are 2.02 and 1.614. Again, the experimental and numerical results demonstrate a good agreement with each other. It is worth emphasizing that, although the lens is optimized at wavelength of λ=632.8 nm, it both theoretically and experimentally demonstrates a super-resolution imaging performance at wavelengths of 625 nm and 591 nm, indicating a certain bandwidth of such a SLAM lens.
Table 1 .
1The phase distribution of the SLAM singlet along the radial direction, where the phase number is given by Ni in the base-32 numerical system for the meta-atom at the location (i-1)P away from the lens center; the corresponding phase value of φi is equal to 2πNi /32.No. of Ring Belt
Ni
#1~#169
1802G2801511T100TMS1D00G2650M0023006E00100000943A00O00N200004
1000A000304000706U00905C0100022809B000200A9F6S064E60020001000B1
0D0500M00040000K20G30017020E2100020200616020I
#169~#338
0F00B00D0K50110700205NF3030I10C0040B0F8R70F0080B1B703000N00F40
9071140705075060B6E5440227001O4000CAF23C215B6B8C42266020AU2C0
DREA1506260C154003K0C00020705000090I0000700870
#339~#506
6250C000700000020000001000000000070GO420N0270KC0017N00E01U0707
10010000025R000F009D00000003002000000B0D070Q20JC0F8K004F00E0000
N0000H00900R4V73E0DVH1N20702033M0F80000B00N
Table 2 .
2The phase distribution of the SLAM doublets along the radial direction, where the phase number is given by Ni in the base-32 numerical system for the meta-atom at the location (i-1)P away from the lens center; the corresponding phase value of φi is equal to 2πNi /32.No. of Ring Belt
Ni
(front lens)
Ni
(rear lens)
#1~#281
0008V02220T000J0000000010VJ00000U00
J000000000S000000000000000N0QS00000
B0OJ00Q000J0040F7FF0LB0H0FDB000H
022002100GG30000G400I40000J000D0HN
020V0KT00JT07GA0053U000A0000000N
H0G91T809D0080B00I0T500PLA022802K
1T0420SA0000380SQ20F1MDFR96I4003S
A00N0080J00M08V7LVQ7P00CM40K200
2CGP0009H7S03I42
0L0H0000000000000000000000I5000000
0000000010005040000000000010000000
0T0000000000000000M00000000000000
000000000000C000000001100000000000
00L00Q00S000000002000000000000000
0000000000000000000000000000000000
000000000B002000000000000600000200
0000000000000000000010000000000000
00000000000
, the peak-to-peak distances between two outer lines of the image of letter 'E' are 1.484λ, 1.364λ, 1.364λ and 1.264λ for the above four different cases, which correspond to angles of 1.36º, 1.25º, 1.25º and 1.142º in the image space respectively. The corresponding angular magnifications are 1.78, 1.77, 1.91 and 1.87 respectively, which are close
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| {'fraction_non_alphanumeric': 0.05033914838592591, 'fraction_numerical': 0.057689253873550964, 'mean_word_length': 4.651538461538461, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 4, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 23, 'word_list_counts': {'cursed_substrings.json': 2, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'The far-field resolution of optical imaging systems is restricted by the Abbe diffraction limit, a direct result of the wave nature of light. One successful technological approach to circumventing this limit is to reduce the effective size of a point-spread-function. In the past decades, great endeavors have been made to engineer an effective point-spread-function by exploiting different mechanisms, including optical nonlinearities and structured light illumination. However, these methods are hard to be applied to objects in a far distance. Here, we propose a new way to achieve super-resolution in a far field by utilizing angular magnification.We present the first proof-of-concept demonstration of such an idea and demonstrate a new class of lenses with angular magnification for far-field super-resolution imaging. Both theoretical and experimental results demonstrate a more than two-fold enhancement beyond the angularresolution limit in the far-field imaging. The proposed approach can be applied to super-2 resolution imaging of objects in far distance. It has promising potential applications in superresolution telescopes and remote sensing.', 'arxivid': '2305.10011', 'author': ['Yi Zhou \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Dingpeng Liao \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Kun Zhang \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Zijie Ma \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Shikai Wu \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Jun Ma \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Xuemei Dai \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Zhengguo Shang \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Zhongquan Wen \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n', 'Gang Chen \nMinistry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina\n'], 'authoraffiliation': ['Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina', 'Ministry of Education\nSchool of Optoelectronic Engineering\nKey Laboratory of Optoelectronic Technology and Systems\nChongqing University\nChongqing University\n174 Shazheng Street400044Shapingba, ChongqingChina'], 'corpusid': 258741101, 'doi': None, 'github_urls': [], 'n_tokens_mistral': 14195, 'n_tokens_neox': 11393, 'n_words': 6533, 'pdfsha': '321399ca6c7d7a3dc66ca8ccae249387260df23d', 'pdfurls': ['https://export.arxiv.org/pdf/2305.10011v1.pdf'], 'title': ['Super-Resolution Imaging via Angular Magnification', 'Super-Resolution Imaging via Angular Magnification'], 'venue': []} |
arxiv |
Grouping Method for mmWave Massive MIMO System: Exploitation of Angular Multiplexing Gain
25 May 2023
Peng Jiang
Pengcheng Zhu
Jiamin Li
Dongming Wang
Grouping Method for mmWave Massive MIMO System: Exploitation of Angular Multiplexing Gain
25 May 2023JOURNAL OF L A T E X CLASS FILES, VOL. 14, NO. 8, AUGUST 2021 1Index Terms-Massive MIMOuser schedulingADMA group- ingmillimeter-wave communicationdigital beamforming
A future millimeter-wave (mmWave) massive multiple-input and multiple-output (MIMO) system may serve hundreds or thousands of users at the same time; thus, research on multiple access technology is particularly important. Moreover, due to the short-wavelength nature of a mmWave, largescale arrays are easier to implement than microwaves, while their directivity and sparseness make the physical beamforming effect of precoding more prominent. In consideration of the mmWave angle division multiple access (ADMA) system based on precoding, this paper investigates the influence of the angle distribution on system performance, which is denoted as the angular multiplexing gain. Furthermore, inspired by the above research, we transform the ADMA user grouping problem to maximize the system sum-rate into the inter-user angular spacing equalization problem. Then, the form of the optimal solution for the approximate problem is derived, and the corresponding grouping algorithm is proposed. The simulation results demonstrate that the proposed algorithm performs better than the comparison methods. Finally, a complexity analysis also shows that the proposed algorithm has extremely low complexity.Index Terms-Massive MIMO, user scheduling, ADMA grouping, millimeter-wave communication, digital beamforming.
I. INTRODUCTION
M ILLIMETER-WAVE (mmWave) massive multipleinput and multiple-output (MIMO) communication is expected to be the key technology for future mobile communication systems [1]- [4]. There are plenty of available spectra in the mmWave band from 30 GHz to 300 GHz, which could be used for mobile communication. Since a mmWave has a shorter wavelength than a microwave, it gains a higher path loss and attenuation, making deploying mmWaves difficult [5], [6]. Fortunately, due to the short wavelength of mmWave, many mmWave antennas can be accommodated in the same space. Hence, large-scale mmWave antenna arrays are easier to deploy, which goes well with massive MIMO technology [5], [7], [8]. Moreover, the array gain brought by large-scale arrays can compensate for the high loss of mmWaves at a certain extent.
The precoding or beamforming design is the key to obtaining the multiantenna gain for massive MIMO systems. Through the reasonable design of precoding, we can improve the signal strength of users and reduce the interference between users to obtain better system performance. Moreover, massive MIMO systems with precoding can transmit independent data streams to different users at the same time and using the same frequency resources, which makes precoding the core of space division multiple access (SDMA) in centimeter-wave This paper was produced by the IEEE Publication Technology Group.
systems [9]- [11] or angle division multiple access (ADMA) in mmWave systems with highly angular dependent channels [12]- [14].
A large and growing body of literature has investigated the hybrid precoding system of mmWave systems [5]- [10], [12], [15]- [17]. Due to the limitation of the current technology level, a hybrid precoding system requires a compromise between performance and implementation difficulty [5], [15]. Although the implementation complexity is high, full-digital precoding with better performance is still an important direction of mmWave system research, and many researchers have realized the design of full-digital transmitters and achieved impressive results [18], [19]. Furthermore, massive Machine Type of Communication (mMTC) [20], [21] requires more than one million links per square kilometer, and it is difficult for one single multiple access system to meet such a demand. Hence, combinations of multiple access technologies should be applied, where a typical solution is to combine multiple access methods, such as combining time division multiple access (TDMA) or frequency division multiple access (FDMA) with ADMA, to group users [11], [15], [16], [22]. Different groups of users are served with different time slots or frequencies, and users in the same group are served by ADMA at the same time and with the same frequency.
For the highly angular correlated channels in mmWave systems, the rational allocation of users can effectively improve the sum-rate of an ADMA system [15]. Previous work has focused on centimeter-wave MIMO and has used channel correlation, greedy algorithms, heuristic algorithms, and other algorithms [11], [22]- [25] to improve the sum-rate of SDMA or ADMA systems. Although these methods can provide good performance, their algorithmic complexity is often very high and expands rapidly with an increase in the number of users. The user grouping problem of mmWave MIMO is considered in some innovative and practical works [15]- [17]. However, most of the grouping algorithms for mmWave systems rely on hybrid precoding architectures and have performance approaching but not exceeding that of full-digital systems. In addition, except for a few algorithms such as the greedy algorithm in [15], most of these algorithms that rely on hybrid precoding architectures cannot be applied to full-digital systems.
Nonetheless, few studies have investigated the effect of angle distribution on precoding performance in mmWave systems. Therefore, in this paper, a more efficient grouping algorithm for the ADMA system is proposed according to the angle distribution. The major contributions are as follows.
• We first analyze the beamforming effect of linear precod-ing in a mmWave massive MIMO system under a uniform linear array (ULA). The results clarify that linear precoding can concentrate the electromagnetic wave signal in a small angular range to form a narrow beam that is directed to the intended user. In addition, the influence of the angle distribution on the system sum-rate is studied, and is later defined as the angular multiplexing gain. • Furthermore, we transform the problem of maximizing the sum-rate of the system through reasonable user groupings into a problem of equalizing the angular spacing of users in each group by the rule of angular multiplexing gain. Then, we prove and propose an angular spacing equalization grouping (ASEG) algorithm, which is optimal for the transformed problem. • Finally, the simulation results and the algorithm complexity analysis show that our algorithm can effectively improve the system sum-rate with an extremely low complexity while the number of users is large. The rest of the paper is organized as follows. Section II describes the mmWave massive MIMO system. Section III analyzes the beamforming effect of linear precoding and the effect of the angular distribution on the system rate. Section IV states the optimization problem and transforms it into an approximation problem. Section V proposes the grouping algorithm. Then, the angular multiplexing gain and proposed algorithm are simulated and analyzed in Section VI. Finally, Section VII concludes this paper.
Notations: (·) * , (·) T , (·) H and (·) † refer to the conjugate, transpose, Hermitian transpose, and Moore-Penrose pseudoinverse, respectively. · F denotes the 2-norm for vectors and the Frobenius norm for matrices. CN (0, σ 2 ) indicates a complex Gaussian distribution with a zero mean and a variance of σ 2 . I N is an N × N identity matrix.
II. SYSTEM MODEL AND CHANNEL CHARACTERISTICS
A. System Model
In this paper, we consider a mmWave full-digital time division duplex (TDD) massive MIMO system in a single cell, where K single-antenna users are served by a base station equipped with an N -antenna ULA. Considering the actual three-sector deployment of the antenna array, we focus on the users who are located in the range within the front 2π/3.
To satisfy the connection density requirement of mMTC, we need hybrid multiple access technologies. Therefore, in this article, we chose ADMA to cooperate with ideal TDMA or FDMA to serve massive numbers of users at the same time.
Remark 1: ADMA is one kind of SDMA. In the centimeter band, we usually refer to the multiple access method based on precoding as SDMA. However, in the mmWave band, we call it ADMA because of the high degree of correlation between precoding and the angle, which is reflected in [12] and the analysis that follows in this article.
B. Channel Characteristics
The downlink channel of user k to an N -antenna ULA can be written in the widely used Saleh-Valenzuela mmWave channel model [5], [12], [15] as
h k = β (0) k a (0) k + L l=1 β (l) k a (l) k ,(1)
where β (l) k is the complex gain of the l-path of user k, which consists of the path loss and shadow fading, and a (l) k ∈ C 1×N are steering vectors. l = 0 represents the line-of-sight (LOS) component of the user, while l ≥ 1 represents the nonline-ofsight (NLOS) components. The steering vectors a (l) k can be modeled as
a (l) k = e j·2·π·0·ξ (l) k e j·2·π·1·ξ (l) k . . . e j·2·π·(N −1)·ξ (l) k T ,(2)
where ξ
(l) k = d λ cos θ (l) k . θ (l)
k is the physical direction of arrival (DOA) of the l-path of user k, λ is the wavelength of the transmitted signal and d is the antenna distance, which is generally set as d = λ 2 in a large-scale multiantenna array. Due to the directionality of mmWaves, the NLOS components are usually much lower than the LOS elements [5], [26]. For simplicity, researchers generally only consider the LOS components [8], [15], [27]- [29] in their analysis, so we can ignore the NLOS components and superscript l. These NLOS components will be reconsidered in simulations.
C. Downlink Data Transmission
The downlink channel matrix of the cell can be expressed as
H = h T 1 , h T 2 , . . . , h T K ,(3)
For massive MIMO systems, precoding is the key to multiantenna signal processing. The precoding matrix is set as P = [p 1 , p 2 , ..., p K ] ∈ C N ×K ; then, the received signal vector of all users Y ∈ C K×1 is given by
Y = H √ pPx + n,(4)
where x ∈ C K×1 represents the data streams of K users, n ∈ C K×1 ∼ CN (0, σ 2 I K ) is the noise vector and p is the total power. In the multiplexing system, we divide users into G > 1 multiple access groups. We use the ideal TDMA or FDMA and ADMA in each group. Setting U as a universal set, we have a constraint that can be explicitly expressed as
U = {1, 2, ..., K} (5a) U = U 1 ∪ U 2 ∪ ... ∪ U G (5b) ∀i = k, U i ∩ U k = ∅,(5c)
In this multiplexing case, the rate of the k-th user in group U g can be written as
R k = log 2 1 + p h k p k 2 F i∈Ug i =k p h k p i 2 F + σ 2 .(6)
Then, the sum-rate of group U g is
R Ug = k∈Ug R k .(7)
Assume that time or frequency resources are allocated among groups by S g > 0, and G g=1 S g = 1; hence, the sum-rate of the whole system is given by
R = G g=1 S g R Ug .(8)
In general, to ensure fairness, we assume that resources are evenly distributed across groups with the same number of users in each group.
III. PHYSICAL BEAMFORMING EFFECT OF PRECODING AND ANGULAR MULTIPLEXING GAIN
Linear precoding can play the role of beamforming in massive MIMO. Due to the directionality and angle-related channel in a mmWave system, the physical beamforming effect of linear precoding is more obvious.
This section mainly analyzes the beamforming effect brought by linear precoding in mmWave MIMO systems. The beam formed by linear precoding has a narrow-angle, which is determined by the angles of users; thus, we correlate the system performance with the angular distribution of users.
Remark 2: In this article, a beam refers to electromagnetic waves whose energy is concentrated in a small angular range, beamforming refers to the process of generating beams, and beamforming effects are the effects of forming beams through beamforming processes.
A. Typical Linear Precoding Methods
There are three typical linear precoding methods and they are derived in different ways: maximum ratio transmission (MRT), zero forcing (ZF), and minimum mean-square error (MMSE) [30], [31].
MRT precoding aims to maximize the rate, and regardless of interference, we can obtain the MRT precoding matrix P ′ MRT and the normalized MRT precoding matrix P MRT , namely,
P ′ MRT = H H P MRT = P ′ MRT / P ′ MRT F .(9)
ZF aims to completely eliminate inter-user interference regardless of the rate; i.e.,
P ′ ZF = H H HH H −1 P ZF = P ′ ZF / P ′ ZF F .(10)
MMSE aims to maximize the signal-to-leakage-noise ratio (SLNR), which is a concession of the signal-to-interferencenoise ratio (SINR). Hence,
P ′ MMSE = H H HH H + σ 2 I K −1 P MMSE = P ′ MMSE / P ′ MMSE F .(11)
B. Physical Beamforming Effect of MRT
We consider a pair of users k and j, where k = j, which are served by MRT precoding. Their channels can be written as
h k = β k e j·π·0·cos θ k e j·π·1·cos θ k . . . e j·π·(N −1)·cos θ k T .(12)
Then, the MRT precoding vector can be expressed as
p k = β * k √ N |β k | e −j·π·0·cos θ k e −j·π·1·cos θ k . . . e −j·π·(N −1)·cos θ k .(13)
In general, the signal sent by precoding p k is the signal to user k but interferes with user j when j = k. Notably, the signal amplitude of user k is
SA k = h k p k = √ N |β k |,(14)
and the interference amplitude to user j is
IA k = h j p k = β j β * k √ N |β k | N −1 n=0 e j·π·n·(cos θj−cos θ k ) ,(15)
If only the relative amplitude of the signal is considered, we have
Φ(j, k) = √ N , θ j = θ k 1 √ N 1−e j·π·N ·(cos θ j −cos θ k ) 1−e j·π(cos θ j −cos θ k ) , otherwise ,(16)
which represents the signal that is sent to user k but received by user j. (16) is only related to θ j and θ k . We depict the relative amplitude of the signal in Fig. 1. Without loss of generality, we set θ k = 1 2 π and θ j to vary from 1 6 π to 5 6 π as the x-axis variable. As shown in Fig. 1, (16) reaches the maximum value when θ j = θ k and roughly decreases when |θ j −θ k | increases. Furthermore, Fig. 1 reveals the beamforming effect of MRT and the physical explanation
H z H H z −1 = |β j | 2 |β k | 2 N 2 − 1 − D N j,k 1 − D j,k 1 − D N k,j 1 − D k,j −1 |β k | 2 N −β j β * k 1−D N j,k 1−D j,k −β * j β k 1−D N k,j 1−D k,j |β j | 2 N .(20)P ZF = 1 2N · INV j,k |β j | 2 + |β k | 2 |β k | 2 β * j N a H j − 1 − D N k,j 1 − D k,j a H k , |β j | 2 β * k N a H k − 1 − D N j,k 1 − D j,k a H j .(23)
of MRT precoding: MRT precoding sends the signal to the user in a centralized manner and reduces the leakage in other directions. However, if other users are close to the target users, the MRT precoding beam will interfere with their signals, which must be avoided.
C. Physical Beamforming Effect of ZF 1) ZF of a pair of users: ZF precoding, which aims to zeroize interuser interference, is usually effective. However, when the angular spacing of users is too small, the cost of ZF will be expensive, and the signal strength will decay sharply. For ZF precoding, the signal pattern is more complex. Now, we introduce some notations to simplify the expression.
φ j = e jπ cos θj (17a) D j,k = φ j φ k (17b) t j,k = cos θ j − cos θ k .(17c)
We first calculate part of ZF; we set
H z = h j h k .(18)
Then,
H z H H z = β jβj N β jβk N −1 n=0 D n j,k β j β k N −1 n=0 D n k,j β kβk N .(19)
For a second-order matrix, the inverse matrix can be obtained directly, similar to (20), which is at the top of the next page.
Then, we simplify the coefficient of (20) as
INV j,k = |β j | 2 |β k | 2 N 2 − 1 − D N j,k 1 − D j,k 1 − D N k,j 1 − D k,j = |β j | 2 |β k | 2 N 2 − Ω(t j,k ) .(21)
Considering the normalized precoding power, we have
P ′ ZF F = Tr(P ′ H ZF P ′ ZF ) = √ N INV j,k |β j | 2 + |β k | 2 .(22)
Subsequently, ZF could be rewritten as (23), which is shown at the top of the next page. Finally, the precoding vector of user j is
p j = |β k | 2 β * j N a H j − 1−D N k,j 1−D k,j a H k 2N · INV j,k |β j | 2 + |β k | 2 ,(24)
and the received signal of user j under ZF precoding is
h j p j = β j a j |β k | 2 β * j N a H j − 1−D N k,j 1−D k,j a H k 2N · INV j,k |β j | 2 + |β k | 2 = |β j ||β k | N 2 − Ω(t j,k ) N (|β j | 2 + |β k | 2 ) .(25)
From the above derivations, we find that N a H j is the MRT precoding vector of direction a j , and
(1−D N k,j )/(1−D k,j )a H k
is the MRT precoding vector of direction a k , which means that the ZF precoding vector of a j is a combination of MRT precoding vectors.
Furthermore, the interference caused by beam N a H j in direction a k is a k N a H j = N (1 − D N k,j )/(1 − D k,j ). Moreover, the interference caused by beam (1 − D N k,j )/(1 − D k,j )a H k in direction a k is a k (1−D N k,j )/(1−D k,j )a H k = N (1−D N k,j )/(1− D k,j )
, which means the interference beam generated by the MRT beam of the served user is canceled by an MRT beam that is aligned with the interfered user at the same amplitude and antiphase as the interference.
In other words, when ZF sends a signal in the direction a j , it sends a cancellation signal in the direction a k simultaneously so that the signal leaked in the direction a k is canceled to zero by a H k . However, the cancellation of the interference comes at a price, and the cancellation beam
(1 − D N k,j )/(1 − D k,j )a H k
will also cause attenuation in the served user signal:
Ω(t j,k ) = a j (1 − D N k,j ) (1 − D k,j ) a H k = 1 − cos(N πt j,k ) 1 − cos(πt j,k ) .(26)
Now we can infer the following: When two directions a j , a k are too close to each other, the MRT beam of a j will greatly interfere with a k ; in turn, the cancellation signal will also cause great signal attenuation for a j . When the directions are far away, the attenuation is negligible. Then, we focus on the effect of the angular spacing of users on ZF beams. In Fig. 2, two users are close to each other, so we can see that the ZF beam combined with the signal beam and interference cancellation (IC) beam is much smaller than the signal beam. The IC beam not only weakens the peak value of the ZF beam but also changes the energy distribution and increases the energy leakage. In addition, the peak value of the ZF beam shifts from the target user position, which seriously affects the amplitude of the user's received signal. These are the costs of interference cancellation. However, in Fig. 3, the ZF beam is almost equal to the signal beam, which reveals that ZF performs well when there is sufficient angular spacing between users. In Fig. 4, the angle of one user is fixed at 1 2 π, and the x-axis represents the varying angle of another user. However, the amplitude now represents the signal amplitude of the target user. Different from MRT precoding, the interference in ZF is nearly zero, but the amplitude of the target user decays sharply as the other user moves close to the target user: if the angular spacing between other users and the target user is too small, the performance of the system will decline rapidly.
The sufficient angular spacing could be set as 2/N , which is the first zero point of Ω(t) according to IV-C and also half the width of the main lobe of Signal beam in Fig. 2 and 3.
2) ZF of multiuser systems: Similar to the situation of a pair of users, a multiuser ZF beam can still be decomposed into a linear combination of MRT beams. Set J = HH H ; then, the adjoint matrix of J is adj(J), and J i,j is the algebraic remainder of the element in row i and column j of J, which is also the element in row j and column i of adj(J), we have
p ′ j = 1 det(J) (J j,1β1 a H 1 +J j,2β2 a H 2 +· · ·+J j,KβK a H K ). (27)
The approximation and calculation of the determinant is a difficult problem in the field of mathematics, so we consider using the conclusion of III-C1 to approximate the ZF precoding vector. To facilitate the discussion, without loss of generality, the users are already sorted by angle
θ 1 < θ 2 < . . . < θ K .(28)
From III-C1 we know that with the increase of angle space, the cost of interference elimination decreases rapidly. At the same time, to avoid paying too much to interference elimination, the minimum angle space could not be too small for any user pair. This means, the angular spacing between adjacent users cannot be too small. This also makes the angular spacing between other users except adjacent users maintain a large level, so that the cost of interference elimination can be ignored. Therefore, ZF vector of user j, 2 ≤ j ≤ K − 1 could be approximate as
p ′ j ≈ 1 ζ j (J j,j−1βj−1 a H j−1 + J j,jβj a H j + J j,j+1βj+1 a H j+1 ),(29)
where ζ j is for power control. Since the orthogonality between vectors is independent of the module, we leave the module value problem for later. In fact, since the angular spacing between user j − 1 and j + 1 is large enough, the interaction between the two users is small enough to be ignored. Therefore, the conclusion of pair user ZF of (24) can be applied, then p ′ j could be further approximated as
p ′ j ≈ 1 ζ j (N a H j − 1 − D N j−1,j 1 − D j−1,j a H j−1 − 1 − D N j,j+1 1 − D j,j+1 a H j+1 ).(30)
After normalizing its power, it can be written as
ζ j = N a H j − 1 − D N j−1,j 1 − D j−1,j a H j−1 − 1 − D N j,j+1 1 − D j,j+1 a H j+1 F (31a) p j = 1 ζ j N a H j − 1 − D N j−1,j 1 − D j−1,j a H j−1 − 1 − D N j,j+1 1 − D j,j+1 a H j+1 (31b)
3) MMSE: The MMSE precoding method, which is a compromise of MRT and ZF, is similar to both MRT and ZF: when noise σ 2 is small, MMSE will act like ZF; when noise σ 2 is too large to be ignored, MMSE will act like MRT. In most cases, the above ZF rules are also applied to MMSE. However, MMSE precoding has its characteristics. When MMSE eliminates interference between users, it considers the impact of noise and balances the values of noise and interference. Compared with ZF, it increases a user's signal strength to combat noise instead of blindly seeking zero interference.
D. Free space multiuser signal strength
To show the beamforming effect more clearly, we set up an ideal transmission environment, which means that only the path loss and steering vector are considered. Fig. 5 simulates a rectangular cell: The base station is located at the origin, the ULA plane is parallel to the x-axis, the black circle in the figure represents the user's position, and the base station uses linear precoding to serve K = 8 users in the cell at the same time. The colors in the graph represent the intensity of all the received signals at each location and do not distinguish between signals and interference. Fig. 5 shows that multiple narrow areas with strong signals, which are beams, originating from the base station accurately point to multiple users in the cell. For a user located far away from other users, one beam can be sent by the base station. However, for adjacent users, the MRT cannot distinguish them well. For example, the beams of three adjacent users located in the upper left corner overlap each other, which provides the potential for interference. For ZF and MMSE, it can be seen that even if the angular spacing of users is small, different beams can be distinguished. Now, the principle of precoding in mmWave massive MIMO is revealed: Linear precoding concentrates signal energy and reduces interference by aligning the beams.
E. Angular Multiplexing Gain and Group Gain 1) Angular Multiplexing Gain:
In the previous part of this section, we discussed the beamforming effect of precoding and the impact of paired user angular spacing on precoding performance. It is not difficult to find that this effect will also have a great impact on the sum-rate of the system. Therefore, in the following, the improvement in the sum-rate of a system brought by precoding is called the angular multiplexing gain. The improvement in the sum-rate of a system brought by a reasonable angle distribution of users is called the angular multiplexing gain.
2) Grouping Gain: The above discussion shows that when the number of users is large, the performance loss of ZF and MMSE precoding is very serious. Therefore, to cope with a large number of users, that is, when the number of users in the cell is larger than the number of antennas, a mix of two or more multiple access systems to serve users must be considered. The sum-rate of the system that applies ZF precoding is almost reduced to zero when the number of users approaches the number of antennas. Thus, when we group users, the distance (or angular distance) between users will be naturally widened, which will alleviate the detriment caused by large numbers of users. Therefore, compared with the case of no grouping, user grouping will bring very large gains to the system when the number of users is large, especially when the users are located close together.
IV. PROBLEM FORMULATION
For multiple access systems implemented by precoding, especially MRT and MMSE, precoding itself can achieve any number of data streams, which is the key to multiple access systems. However, as the number of users grows, the sumrate of the system will reach a bottleneck. At this time, the time division or frequency division is combined and grouped according to certain rules based on the space or angle to effectively improve the sum-rate of the system.
In this section, we formulate the optimization problem and use the result of Section III to turn it into an approximation problem that is easier to solve.
A. Problem of Maximizing the sum-rate
The problem of maximizing the sum-rate can be stated as
P 0 : max U1,U2,...,UG 1 G G g=1 k∈Ug R k (32a) s.t. U = {1, 2, ..., K} (32b) U = U 1 ∪ U 2 ∪ ... ∪ U G (32c) ∀i = k, U i ∩ U k = ∅. (32d)
This grouping problem is a typical NP-hard problem, which is almost impossible to directly solve. Some approximate laws are needed to solve this problem.
B. The impact of angular spacing on precoding performance
This subsection takes advantage of the beamforming effect of linear precoding to simplify the optimization problem.
According to Section III, we know that in an mmWave Massive MIMO system, SINR k is highly correlated with the angular spacing. For example, in the MRT precoding method, the signal power is relatively constant, and the interference between user k and j is determined by |ψ k −ψ j | = d λ | cos θ k − cos θ j | = d λ |t k,j |. Especially when 1 6 π ≤ θ k , θ j ≤ 5 6 π, we have |t k,j | ∝ |θ k − θ j |. Moreover, interuser interference mainly occurs between adjacent users. Therefore, the angular space between adjacent users has a decisive influence on the performance of the system.
Due to the better performance and the smoother relationship between the signal and angle, we choose the ZF precoding method for research. When grouping and ZF precoding are adopted, the rate of user k could be rewritten as
R k = log 2 1 + p k h k p Ui k 2 F σ 2 ,(33)
where k ∈ U i , p k is the beam power to user k, and p Ui k is the ZF precoding vector of user k in group U i , which means p Ui k is calculated by using the channel matrix composed of users in U i . For the sake of brevity, we use p k to represent p Ui k .
p j h j p Ui j 2 F ≈ p j h j p j 2 F = p j ζ 2 j N a j a H j − 1 − D N j−1,j 1 − D j−1,j a j a H j−1 − 1 − D N j,j+1 1 − D j,j+1 a j a H j+1 2 . (34) 1 G G g=1 k∈Ug R k ≈ 1 G G g=1 k∈Ug R k = 1 G G g=1 k∈Ug log 2 (Γ k N 4 ) − 2 N 2 (Ω(t k−,k ) + Ω(t k,k+ )) (38a) = 1 G G g=1 k∈Ug log 2 (Γ k N 4 ) − 2 GN 2 G g=1 k∈Ug
(Ω(t k−,k ) + Ω(t k,k+ )).
(38b)
Then the received signal of user j under ZF precoding is shown as (34) at the top of the Page 7. Using Ω(t) in (26), (34) can be further reduced to
p k h k p k 2 F = p k ζ 2 k |β k | 2 N 2 − Ω(t k−,k ) − Ω(t k,k+ ) 2 ,(35)
where k + denotes the larger element that is closest to K in the same group, while k − denotes the smaller element that is closest to K in the same group. After that, (33) could be rewritten as
R k ≈ log 2 1 + Γ k N 2 − Ω(t k−,k ) − Ω(t k,k+ ) 2 ,(36)
where Γ k = (p k |β k | 2 )/(ζ 2 k σ 2 ). Generally speaking, the array gain of massive MIMO is high, and most users are in the high SNR region,
R k ≈ log 2 Γ k N 2 − Ω(t k−,k ) − Ω(t k,k+ ) 2 (37a) = log 2 (Γ k N 4 ) + 2 log 2 1 − 1 N 2 (Ω(t k−,k ) + Ω(t k,k+ )) (37b) ≈ log 2 (Γ k N 4 ) − 2 N 2 (Ω(t k−,k ) + Ω(t k,k+ )) (37c) = R k .(37d)
Therefore, the optimization objective can be approximated as (38b) at the top of the Page 7.
To facilitate the discussion, without loss of generality, the users are already sorted by angle
θ 1 < θ 2 < . . . < θ K .(39)
Set u i m ∈ {1, 2, ..., K} as the m-th user in group i, M is the number of elements in U i ,
U i = {u i 1 , u i 2 , . . . , u i M }.(40)
We set the elements in the group to be arranged in an increasing order, which means that u i 1 < u i 2 < . . . < u i M , and the corresponding angles of U i are also arranged
θ u i 1 < θ u i 2 < . . . < θ u i M .(41)
In addition, since the boundary elements, which are the elements with the smallest and largest angle in the group, lack one of the neighboring elements, we have
G g=1 k∈Ug −(Ω(t k−,k ) + Ω(t k,k+ )) = 2 G g=1 M−1 k=1 −Ω(t u g k ,u g k+1 ) (42) For equal power allocation, 1 G G g=1 k∈Ug log 2 (Γ k N 4 )(43)
will hardly change. Therefore, the problem can be approximated as
P 1 : max U1,U2,...,UG 1 G G g=1 M−1 k=1 −Ω(t u g k ,u g k+1 ) (44a) s.t. (32b), (32c), (32d),(44b)
C. Maximizing the Sum of a Certain Revenue Function for the Angular Spacing
The properties of Ω(t u g k ,u g k+1 ) are the key to the optimization problem. Next, we will get rid of the subscripts of t u g k ,u g k+1 for a moment and study the properties of this function alone:
Ω(t) = 1 − cos(N πt) 1 − cos(πt) (45a) dΩ(t) dt = πN sin (N πt)(1 − cos (πt)) (1 − cos (πt)) 2 − sin (πt)(1 − cos (1 − cos (N πt))) (1 − cos (πt)) 2 . (45b)
When 0 < t < 2 N , we have Ω(t) > 0, dΩ(t)/dt < 0. The analysis of the second derivative d 2 Ω(t)/dt 2 could be done by simulation. As shown in Fig. 6 at the top of Page. 8, we have d 2 Ω(t)/(dt) 2 > 0, 1/N < t < 2/N . According to the discussion in Section III, the larger the angular spacing t, the better performance of precoding we will get. As also shown in Fig. 6(a), when t is close to 2/N , the performance of precoding achieves the highest performance. At the same time, if t is too large, the group could not contain too much users, which will in turn cause performance reduction. Therefore, the t better be a little less than 2/N .
In this case, we define a function class g(t), where g ′ (t) > 0, g ′′ (t) < 0, g(t) = g(−t). In fact, −Ω(t) when 1/N < t < 2/N satisfies the definition of g(t), what is true for g(t) should also be true for −Ω(t).
Finally, the optimization problem can be approximated as
P 2 : max U1,U2,...,UG Obj = 1 G G g=1 M−1 k=1 g(θ u g k+1 − θ u g k ) (46a) s.t. (32b), (32c), (32d),(46b)
where θ u g k+1 − θ u g k ≈ t u g k+1 ,u g k , ∀x, g(x) = g(−x) and ∀x > 0, g ′ (x) > 0, g ′′ (x) < 0.
V. PROPOSED ANGULAR SPACING EQUALIZATION GROUPING ALGORITHM
In this section, we derive the optimal solution to Problem P 2 and the corresponding ASEG algorithm. Then, the comparison algorithms are introduced.
A. Angular Spacing Equalization Grouping
It can be seen from the above research that grouping methods are beneficial when the number of users is large, so we first make the large-scale user grouping assumption that each group has multiple users (at least greater than or equal to 3). This assumption is not hard to satisfy. We generally adopt the strategy of equal-member grouping.
In this case, it is difficult to design the optimal grouping algorithm directly, but we can analyze the best grouping results in some cases:
Proposition 1: Within a group (excluding the boundaries), there cannot be more than one element whose value is between a group of adjacent elements in any other group. Mathematically, for group U i = {u i 1 , u i 2 , . . . , u i M } and group U j = {u j 1 , u j 2 , . . . , u j M }, if there are u j k and u i l such that u i l < u j k < u j k+1 < u i l+1 , then there exists a better grouping than this grouping.
Proof: Exchange the elements before u j k and u j k in group U j and the elements before u i l and u i l in group U i ; namely,
U ′ i = {u j 1 , u j 2 , . . . , u j k , u i l+1 . . . , u i M } (47a) U ′ j = {u i 1 , u i 2 , . . . , u i l , u j k+1 . . . , u j M }.(47b)
Suppose the objective function before the exchange is Obj, and the objective function after the exchange is Obj ′ ; then, we have
Obj − Obj ′ = g(θ u j k+1 − θ u j k ) + g(θ u i l+1 − θ u i l ) − [g(θ u j k+1 − θ u i l ) + g(θ u i l+1 − θ u j k )].(48)
(48) can be rewritten as (49a) at the top of the Page 9. Notice that g ′′ (x) < 0, so we have (49b), which means (49c). Then, we have Obj < Obj ′ ; that is, U ′ i and U ′ j are better groupings Proposition 2: The other groups cannot have multiple elements larger than the boundary of one side of a certain group. Mathematically, if there is u i k such that u j M < u i k < u i k+1 , then there exists a better grouping than this grouping.
Obj − Obj ′ = θ u j k −θ u i l 0 [g ′ (x + θ u j k+1 − θ u j k + θ u i l+1 − θ u j k+1 ) − g ′ (x + θ u j k+1 − θ u j k )]dx (49a) [g ′ (x + θ u j k+1 − θ u j k + θ u i l+1 − θ u j k+1 ) − g ′ (x + θ u j k+1 − θ u j k )] < 0 (49b) θ u j k −θ u i l 0 [g ′ (x + θ u j k+1 − θ u j k + θ u i l+1 − θ u j k+1 ) − g ′ (x + θ u j k+1 − θ u j k )]dx < 0 (49c)
Proof: Transfer the element u i k+1 in group U i to group U j , and the objective function before and after the exchange can be expressed as
Obj − Obj ′ = g(θ u i k+1 − θ u i k ) − g(θ u i k+1 − θ u i M ).(50)Obviously, (θ u i k+1 − θ u i M ) > (θ u i k+1 − θ u i k )
, and g ′ (x) > 0. Then, we have Obj < Obj ′ , i.e., we have a better grouping.
Proposition 3: Suppose a complete set consists of any two groups; then, the adjacent elements in it must belong to different groups. Mathematically, for
U ′ = {U i , U j }, if u i k is not the largest element of U ′ , then u i k + 1 ∈ U j ; if u i k is not the smallest element of U ′ , then u i k − 1 ∈ U j .
Proof: Now consider any two groups, for example, U i , U j . Due to Proposition 1 and Proposition 2, we know that the elements of the two groups must be staggered elements in a complete set in increasing order. That is, for any nonboundary element in U i , its adjacent elements in the complete set composed of U i , U j must be elements of U j . Otherwise, it will violate Proposition 1. The elements in U j also follow this rule. The boundary elements also comply with the above rules.The inner adjacent element of boundary elements in the complete set must be one of the elements in another set, and there are no elements on the outside of the boundary. Thus, the numbers of elements in any two groups differ by no more than 1, and they are staggered in a complete set.
Proposition 4: For any adjacent elements in any group, there must be G − 1 elements between them in a universal set. Mathematically,
∀u i k , 1 ≤ k ≤ M − 1, there exists only G − 1 elements ∈ {u i k + 1, u i k + 2, ..., u i k + G − 1} between [u i k , u i k+1 ]
. Proof: Due to Proposition 2 and Proposition 3, we might as well set u 1 1 = 1, that is, the first element of the first set of U 1 . Guaranteed by Proposition 3, at this time, there must be a unique element between u 1 1 and u 1 2 in each other group, and they must be the G − 1 elements between [2, G]. If there are t 1 < t 2 < · · · < t T ∈ [u 1 1 , u 1 2 ], T ≥ 1 and they do not belong to [2, G], then in the remaining G − 1 groups, there must be G + T − 1 elements between [u 1 1 , u 1 2 ] and there must be at least one group that contains multiple elements between [u 1 1 , u 1 2 ], which violates Proposition 1. Now, we consider any non boundary element u i k of any group; there exist and only exist G − 1 elements between [u i k−1 , u i k ], and the same between [u i k , u i k+1 ]. The proof of this conclusion is the same as the proof above, so it will not be repeated here.
From Proposition 4, we know that the difference in the global sequence number for a certain group of adjacent elements is the number of groups G. According to Proposition 2, the boundary elements of G groups must be {1, 2, . . . , G}. These G elements, without loss of generality, are placed into the U 1 , U 2 , . . . , U G group; namely,
1 ∈ U 1 , 2 ∈ U 2 , . . . , G ∈ U G(51)
Then, according to the above grouping rules, we can determine a unique grouping method that satisfies all the above rules:
U 1 = {1, G + 1, . . . , (M − 1)G + 1} (52a) U 2 = {2, G + 2, . . . , (M − 1)G + 2} (52b) . . . (52c) U G = {G, 2G, . . . , M G}.(52d)
All the rules jointly determine a unique grouping fraction. Then, this grouping method is the optimal grouping method. Proposition 5: The optimal solution of Problem P 2 is given by (52).
Then, the ASEG algorithm can be constructed as Algorithm 1.
Algorithm 1 Angular Spacing Equalization Grouping (ASEG) 1: Input: angle of all users θ = [θ 1 , θ 2 , ..., θ K ] 2: apply a sorting algorithm, such as quicksort or mergesort, to θ 3: group users according to (52) into U 1 , U 2 , ..., U G 4: Output: U 1 , U 2 , ..., U G
B. Self-Evolution Genetic Algorithm for Comparison
Genetic algorithms are widely used in grouping and selection problems [32]- [34]. Given the particularity of the grouping problem, we propose a self-evolution genetic algorithm (SEGA) based on the idea of a genetic algorithm. SEGA is an improved genetic algorithm that fits the particularity of the grouping problem.
First, we need to return to Problem P 0 .
where U g is naturally coded. Now, we redefine U g,e as the g-th group of elite e elements, which is a special group with the best gene. Due to the uniqueness of users described by(32b),(32c) and (32d), the crossover and mutation operations of the standard genetic algorithm cannot be carried out directly. However, different groups with the same elite elements can cross each other, which is equivalent to exchanging users in different groups. Therefore, we use the crossover operation to create new individuals from the elites as offspring. To prevent better parent individuals from being replaced by child individuals, we engage all parents and child individuals in natural selection and choose the individuals with the highest fitness levels among them as elites.
C. SUS Algorithm and Greedy Algorithm for Comparison
Semiorthogonal user selection (SUS) [11] is a traditional efficient user scheduling algorithm in a linear beamforming system for MU-MIMO downlink transmission by ZF. The algorithm aims to select semiorthogonal users and place them into a group to be served by ZF. simultaneously. This algorithm has achieved significant results in MIMO scenarios, so it is widely used to compare the performance of grouping algorithms [25].
The main idea of the greedy algorithm in [15] is to traverse all users each time, simulate joining the group, select the user with the highest rate after joining the group as the user that joined the group, and repeat until all users are selected. Different from [15], we do not have the concept of a spatial support index interval in the FD system, and to keep the number of groups fixed, we lift this restriction.
VI. SIMULATION AND ANALYSIS
In this section, the numerical results are presented to verify our discussion and algorithms first. A TDD massive MIMO system is considered in this section, where the base station is equipped with N = 128 antenna ULAs. K single-antenna users are uniformly located inside a fan area cell with a radius of 300m, in which the central angle is usually set as 2π/3. The path loss is given according to
where PL L for LOS paths, PL N is for NLOS paths,X L ∼ CN (0, 10 0.72 ) and X L ∼ CN (0, 10 1.94 ) are shadow fadings, d is the distance between a user and the base station in meters and f c = 28000MHz [35]- [37]. We set the number of NLOS paths as L = 2. ρ DL = 50dBm is the downlink power constraint on the base station side, and σ 2 = −104dBm is the noise variance on the user side.
A. Angular Spacing Equalization Grouping
The ASEG algorithm can improve the spectral efficiency of the system when the number of users is high and increase the maximum spectral efficiency of the system at the same time. The case of G = 1 in Fig. 7 is equivalent to no grouping. The peak value of G = 1 in Fig. 7 can be used as a reference standard to observe the improvement in the system rate obtained by a reasonable grouping. Unlike random grouping, the ASEG grouping method sorts users according to their angles and then divides them into ordered groups. This method divides users with larger angular spacing into the same group to make better use of the angular multiplexing gain. The simulation results in Fig. 7 show that this method brings obvious spectral efficiency gains, indicating that an increase in angular spacing can increase the angular multiplexing gain.
Although this algorithm is proposed based on the analysis of ZF, it is still suitable for MRT and MMSE precoding and achieves good results, as shown in Fig. 7, which shows that the algorithm has a certain universality.
B. Comparison and algorithm analysis
In this subsection, five different grouping algorithms are presented and their performance when ZF is used is compared when G = 4. Note that SUS-G1 is the first group of SUS in [11], and GREED represents the greedy algorithm in [15].
For ZF, as shown in Fig. 8(a), ASEG has a clear advantage over other algorithms before reaching the peak. However, it dropped more in the second half and was overtaken by the other algorithms. This illustrates the correctness of our previous approximation behavior around its peak and that ASEG also suffers severe performance degradation when the 2/N assumption gradually breaks down as the number of users increases, which is to be expected. It is worth noting that, due to the conclusion of the previous section, we can see that for the grouping problem, the most important performance indicator is the peak value because the performance drop after the peak value can be addressed by adding more groups. The SUS algorithm still does not achieve meaningful results, and we can see that the first group of SUS does increase the user rate in Fig. 8(b). Since SUS-G1 reached the highest rate in [11], [25], here, we research other algorithms that are reasonably overlooked.
It can be seen that through the study of the actual physical performance of a communication system and the algorithm designed in a physical sense, it is possible to obtain a great comprehensive performance advantage compared to the greedy algorithm or the intelligent algorithm, which are only designed from a mathematical point of view.
Through the comparison of the above three algorithms, it can also be found that none of the three algorithms outperforms the other algorithms in all scenarios, indicating that the three algorithms only partially utilize the angular multiplexing gain under specific circumstances, which means the current research on how to exhaust the angular multiplexing gain by grouping is far from complete . We need to conduct more research on the angular multiplexing gain.
C. Analysis of the algorithm complexity
The algorithm complexity is an important indicator used to evaluate the algorithm, and it is also the key to whether the algorithm can be applied in practical scenarios. For example, for DPC precoding, although its performance is very good, (c) Time used during the simulation when ZF is used Fig. 8. Comparison of the rates and times of different grouping algorithms when ZF is used its high complexity makes it more applicable in theoretical analyses rather than in practical applications. The most complex part of the ASEG algorithm is sorting, so its algorithm complexity is affected by the sorting algorithm used. The research on sorting algorithms is very mature. Generally, the algorithm complexity of traditional sorting algorithms, such as merge sort and quick sort, can easily reach O (K log 2 (K)).
The algorithmic complexity of GREED is mainly composed of a greedy process, and the rate calculation O K 4 N far exceeds that of the ASED algorithm. The algorithmic complexity of SEGA is iterative, the elite and rate calculations require O IE(K 2 N ) , where the values of I and E are usually on the same order of magnitude as K.
As a result, the ASED algorithm has an extremely low algorithm complexity, which is even lower than the complexity of multiplying two matrices, and the increase in algorithm complexity caused by user growth is not obvious. Furthermore, the complexity of the greedy algorithm increases by the fourth power with the number of users. Although in theory, SEGA only doubles in complexity, to improve the performance of SEGA, the number of iterations and the amount of retention are often increased simultaneously. The number of elites eventually leads to an increase in complexity that is close to the increase experienced by GREED as the number of users increases. Fig. 8(c) shows the average simulation time for a single simulation. To cope with a large number of simulation tasks, we use a multicore server to perform 60 sets of Monte-Carlo simulations simultaneously with parallel computing. Fig. 8(c) shows that SEGA has the highest complexity, followed by GREED and SUS. The complexity of ASEG is not significantly different from RAND and is much lower than that of the other algorithms. The complexity of these algorithms increases rapidly with the increase in users and is three to four orders of magnitude higher than the complexity of ASEG when the number of users is large. Fig. 8(c) also shows the extremely low algorithmic complexity of the ASEG algorithm in comparison to that of the other algorithms. The algorithmic complexity of ASEG is not meaningfully different from that of RAND, which means that the increased complexity of the ASEG algorithm is even partially masked by other simulation contents, such as the precoding calculation and rate calculation.
In general, ASEG has extremely low and even negligible algorithm complexity while maintaining excellent performance; hence, the ASEG algorithm has potential practical application value.
VII. CONCLUSION
In this paper, we investigated the angular multiplexing gain and proposed a grouping scheme for ADMA mixed with a TDMA system. Through the study of the physical beamforming effect of precoding in mmWave massive MIMO systems, we introduced the relationship between the angular multiplexing gain and the angle range distribution of users.
By the above theory of the angular multiplexing gain, we transformed the sum-rate maximization problem for ADMA systems into the problem of equalizing user angular spacing, proposed the ASEG algorithm, and proved that it is optimal for the above approximation problems. The simulation results show that this algorithm has very low complexity and high performance.
Fig. 1 .
1The relationship between the signal amplitude and direction.
Fig. 4 .
4Amplitude of the ZF beam of a user at a fixed angle while the angle of another user varies along the x-axis.
Fig. 5 .
5Distribution of radiation intensity for multiuser linear precoding.
Fig. 6 .
6Function Ω(t) and its first-order and second-order derivatives when N = 16.
P 0
0: max U1,U2,...,UG 1 G G g=1 k∈Ug R k (53a) s.t. (32b), (32c), (32d).
PL L (dB) = −30.18 + 21 log 10 (d) + 20 log 10 (f c ) + X L (54a) PL N (dB) = −34.53 + 34 log 10 (d) + 20 log 10 (f c ) + X N ,
Fig. 7 .
7sum-rate versus the number of groups when using ASEG.
Fig. 2. Combination of ZF beams when the angular spacing of users is small.Fig. 3. Combination of ZF beams when the angular spacing of users is large.1
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A study on millimeter-wave multiuser directional beamforming based on measurements and ray tracing simulations. F Fuschini, M Zoli, E M Vitucci, M Barbiroli, V Degli-Esposti, IEEE Transactions on Antennas and Propagation. 674F. Fuschini, M. Zoli, E. M. Vitucci, M. Barbiroli, and V. Degli-Esposti, "A study on millimeter-wave multiuser directional beamforming based on measurements and ray tracing simulations," IEEE Transactions on Antennas and Propagation, vol. 67, no. 4, pp. 2633-2644, 2019.
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On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming. T Yoo, A Goldsmith, IEEE journal on selected areas in communications. 243journal ArticleT. Yoo and A. Goldsmith, "On the optimality of multiantenna broadcast scheduling using zero-forcing beamforming," IEEE journal on selected areas in communications, vol. 24, no. 3, pp. 528-541, 2006, journal Article.
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Efficient channel estimator with angle-division multiple access. X Liu, J Sha, H Xie, F Gao, S Jin, Z Zhang, X You, C Zhang, IEEE Transactions on Circuits and Systems. 662X. Liu, J. Sha, H. Xie, F. Gao, S. Jin, Z. Zhang, X. You, and C. Zhang, "Efficient channel estimator with angle-division multiple access," IEEE Transactions on Circuits and Systems, vol. 66, no. 2, pp. 708-718, 2019.
An overview of enhanced massive MIMO with array signal processing techniques. M Wang, F Gao, S Jin, H Lin, IEEE Journal of Selected Topics in Signal Processing. 135M. Wang, F. Gao, S. Jin, and H. Lin, "An overview of enhanced massive MIMO with array signal processing techniques," IEEE Journal of Selected Topics in Signal Processing, vol. 13, no. 5, pp. 886-901, 2019.
Angle domain hybrid precoding and channel tracking for millimeter wave massive MIMO systems. J Zhao, F Gao, W Jia, S Zhang, S Jin, H Lin, IEEE Transactions on Wireless Communications. 1610J. Zhao, F. Gao, W. Jia, S. Zhang, S. Jin, and H. Lin, "Angle domain hybrid precoding and channel tracking for millimeter wave massive MIMO systems," IEEE Transactions on Wireless Communications, vol. 16, no. 10, pp. 6868-6880, 2017.
Boost: A user association and scheduling framework for beamforming mmWave networks. P Paul, H Wu, C Xin, IEEE Transactions on Mobile Computing. 2010P. Paul, H. Wu, and C. Xin, "Boost: A user association and scheduling framework for beamforming mmWave networks," IEEE Transactions on Mobile Computing, vol. 20, no. 10, pp. 2924-2935, 2021.
3D angular-based hybrid precoding and user grouping for uniform rectangular arrays in massive MU-MIMO systems. A Koc, A Masmoudi, T Le-Ngoc, IEEE Access. 8A. Koc, A. Masmoudi, and T. Le-Ngoc, "3D angular-based hybrid precoding and user grouping for uniform rectangular arrays in massive MU-MIMO systems," IEEE Access, vol. 8, pp. 84 689-84 712, 2020.
Digital beamforming-based massive MIMO transceiver for 5G millimeter-wave communications. B Yang, Z Yu, J Lan, R Zhang, J Zhou, W Hong, IEEE Transactions on Microwave Theory and Techniques. 667B. Yang, Z. Yu, J. Lan, R. Zhang, J. Zhou, and W. Hong, "Digital beamforming-based massive MIMO transceiver for 5G millimeter-wave communications," IEEE Transactions on Microwave Theory and Tech- niques, vol. 66, no. 7, pp. 3403-3418, 2018.
Design and implementation of a fulldigital beamforming array with nonreciprocal Tx/Rx beam patterns. C Guo, W Hong, L Tian, IEEE Antennas and Wireless Propagation Letters. 1911C. Guo, W. Hong, L. Tian et al., "Design and implementation of a full- digital beamforming array with nonreciprocal Tx/Rx beam patterns," IEEE Antennas and Wireless Propagation Letters, vol. 19, no. 11, pp. 1978-1982, 2020.
Toward massive machine type communications in ultra-dense cellular IoT networks: Current issues and machine learning-assisted solutions. S K Sharma, X Wang, IEEE Communications Surveys & Tutorials. 221S. K. Sharma and X. Wang, "Toward massive machine type communi- cations in ultra-dense cellular IoT networks: Current issues and machine learning-assisted solutions," IEEE Communications Surveys & Tutorials, vol. 22, no. 1, pp. 426-471, 2020.
Massive access for 5G and beyond. X Chen, D W K Ng, W Yu, E G Larsson, N Al-Dhahir, R Schober, IEEE Journal on Selected Areas in Communications. 393X. Chen, D. W. K. Ng, W. Yu, E. G. Larsson, N. Al-Dhahir, and R. Schober, "Massive access for 5G and beyond," IEEE Journal on Selected Areas in Communications, vol. 39, no. 3, pp. 615-637, 2021.
Differentiated service support in wireless networks with multibeam antennas. L R Lakshmi, B Sikdar, IEEE Transactions on Communications. 695L. R. Lakshmi and B. Sikdar, "Differentiated service support in wireless networks with multibeam antennas," IEEE Transactions on Communi- cations, vol. 69, no. 5, pp. 3162-3175, 2021.
User scheduling and antenna topology in dense massive MIMO networks: An experimental study. C.-M Chen, Q Wang, A Gaber, A P Guevara, S Pollin, IEEE Transactions on Wireless Communications. 199C.-M. Chen, Q. Wang, A. Gaber, A. P. Guevara, and S. Pollin, "User scheduling and antenna topology in dense massive MIMO networks: An experimental study," IEEE Transactions on Wireless Communications, vol. 19, no. 9, pp. 6210-6223, 2020.
A framework for radio resource allocation and SDMA grouping in massive MIMO systems. W V F Mauricio, D C Araujo, T F Maciel, F R M Lima, IEEE Access. 9W. V. F. Mauricio, D. C. Araujo, T. F. Maciel, and F. R. M. Lima, "A framework for radio resource allocation and SDMA grouping in massive MIMO systems," IEEE Access, vol. 9, pp. 61 680-61 696, 2021.
A new approach to user scheduling in massive multi-user MIMO broadcast channels. G Lee, Y Sung, IEEE Transactions on Communications. 664G. Lee and Y. Sung, "A new approach to user scheduling in massive multi-user MIMO broadcast channels," IEEE Transactions on Commu- nications, vol. 66, no. 4, pp. 1481-1495, 2018.
Inbuilding coverage of millimeter-wave wireless networks from channel measurement and modeling perspectives. P Zhang, C Yi, B Yang, C.-X Wang, H Wang, X You, Science China Information Sciences. 638P. Zhang, C. Yi, B. Yang, C.-X. Wang, H. Wang, and X. You, "In- building coverage of millimeter-wave wireless networks from channel measurement and modeling perspectives," Science China Information Sciences, vol. 63, no. 8, 2020.
Capacity analysis of NOMA with mmWave massive MIMO systems. D Zhang, Z Zhou, C Xu, Y Zhang, J Rodriguez, T Sato, IEEE Journal on Selected Areas in Communications. 357D. Zhang, Z. Zhou, C. Xu, Y. Zhang, J. Rodriguez, and T. Sato, "Capacity analysis of NOMA with mmWave massive MIMO systems," IEEE Journal on Selected Areas in Communications, vol. 35, no. 7, pp. 1606-1618, 2017.
Angle feedback for NOMA transmission in mmWave drone networks. N Rupasinghe, Y Yapici, I Guvenc, M Ghosh, Y Kakishima, IEEE Journal of Selected Topics in Signal Processing. 133N. Rupasinghe, Y. Yapici, I. Guvenc, M. Ghosh, and Y. Kakishima, "Angle feedback for NOMA transmission in mmWave drone networks," IEEE Journal of Selected Topics in Signal Processing, vol. 13, no. 3, pp. 628-643, 2019.
Energy-efficient design for mmWave-enabled NOMA-UAV networks. X Pang, J Tang, N Zhao, X Zhang, Y Qian, Science China Information Sciences. 644X. Pang, J. Tang, N. Zhao, X. Zhang, and Y. Qian, "Energy-efficient design for mmWave-enabled NOMA-UAV networks," Science China Information Sciences, vol. 64, no. 4, 2021.
Precoding and beamforming techniques in mmWave-massive MIMO: Performance assessment. T Kebede, Y Wondie, J Steinbrunn, H B Kassa, K T Kornegay, IEEE access. 10T. Kebede, Y. Wondie, J. Steinbrunn, H. B. Kassa, and K. T. Kornegay, "Precoding and beamforming techniques in mmWave-massive MIMO: Performance assessment," IEEE access, vol. 10, pp. 16 365-16 387, 2022.
Massive MIMO linear precoding: A survey. N Fatema, G Hua, Y Xiang, D Peng, I Natgunanathan, IEEE systems journal. 124N. Fatema, G. Hua, Y. Xiang, D. Peng, and I. Natgunanathan, "Massive MIMO linear precoding: A survey," IEEE systems journal, vol. 12, no. 4, pp. 3920-3931, 2018.
On the throughput of largebut-finite MIMO networks using schedulers. B Makki, T Svensson, M.-S Alouini, IEEE Transactions on Wireless Communications. 181B. Makki, T. Svensson, and M.-S. Alouini, "On the throughput of large- but-finite MIMO networks using schedulers," IEEE Transactions on Wireless Communications, vol. 18, no. 1, pp. 152-166, 2019.
Performance of network-assisted full-duplex for cell-free massive MIMO. D Wang, M Wang, P Zhu, J Li, J Wang, X You, IEEE Transactions on Communications. 683D. Wang, M. Wang, P. Zhu, J. Li, J. Wang, and X. You, "Performance of network-assisted full-duplex for cell-free massive MIMO," IEEE Transactions on Communications, vol. 68, no. 3, pp. 1464-1478, 2020.
Antenna selection for full-duplex distributed massive MIMO via the elite preservation genetic algorithm. P Zhu, Z Sheng, J Bao, J Li, IEEE Communications Letters. P. Zhu, Z. Sheng, J. Bao, and J. Li, "Antenna selection for full-duplex distributed massive MIMO via the elite preservation genetic algorithm," IEEE Communications Letters, pp. 1-1, 2022.
Heuristic search inspired beam selection algorithms for mmWave MU-MIMO system with discrete lens array. D Wang, W Zhang, Q Zhu, IEEE Access. 9D. Wang, W. Zhang, and Q. Zhu, "Heuristic search inspired beam selection algorithms for mmWave MU-MIMO system with discrete lens array," IEEE Access, vol. 9, pp. 61 324-61 333, 2021.
Angle domain channel estimation in hybrid millimeter wave massive MIMO systems. D Fan, F Gao, Y Liu, Y Deng, G Wang, Z Zhong, A Nallanathan, IEEE transactions on wireless communications. 1712D. Fan, F. Gao, Y. Liu, Y. Deng, G. Wang, Z. Zhong, and A. Nallanathan, "Angle domain channel estimation in hybrid millimeter wave mas- sive MIMO systems," IEEE transactions on wireless communications, vol. 17, no. 12, pp. 8165-8179, 2018.
Millimeter-wave omnidirectional path loss data for small cell 5G channel modeling. G R Maccartney, T S Rappaport, M K Samimi, S Sun, IEEE access. 3G. R. Maccartney, T. S. Rappaport, M. K. Samimi, and S. Sun, "Millimeter-wave omnidirectional path loss data for small cell 5G channel modeling," IEEE access, vol. 3, pp. 1573-1580, 2015.
| {'fraction_non_alphanumeric': 0.06468149386128431, 'fraction_numerical': 0.025390499453622165, 'mean_word_length': 3.7304446978335233, 'pattern_counts': {'":': 0, '<': 34, '<?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': 76, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}} | {'abstract': 'A future millimeter-wave (mmWave) massive multiple-input and multiple-output (MIMO) system may serve hundreds or thousands of users at the same time; thus, research on multiple access technology is particularly important. Moreover, due to the short-wavelength nature of a mmWave, largescale arrays are easier to implement than microwaves, while their directivity and sparseness make the physical beamforming effect of precoding more prominent. In consideration of the mmWave angle division multiple access (ADMA) system based on precoding, this paper investigates the influence of the angle distribution on system performance, which is denoted as the angular multiplexing gain. Furthermore, inspired by the above research, we transform the ADMA user grouping problem to maximize the system sum-rate into the inter-user angular spacing equalization problem. Then, the form of the optimal solution for the approximate problem is derived, and the corresponding grouping algorithm is proposed. The simulation results demonstrate that the proposed algorithm performs better than the comparison methods. Finally, a complexity analysis also shows that the proposed algorithm has extremely low complexity.Index Terms-Massive MIMO, user scheduling, ADMA grouping, millimeter-wave communication, digital beamforming.', 'arxivid': '2305.15935', 'author': ['Peng Jiang ', 'Pengcheng Zhu ', 'Jiamin Li ', 'Dongming Wang '], 'authoraffiliation': [], 'corpusid': 258887487, 'doi': '10.48550/arxiv.2305.15935', 'github_urls': [], 'n_tokens_mistral': 19713, 'n_tokens_neox': 17490, 'n_words': 11233, 'pdfsha': 'f135e80193db938ef0a72418ed6b896d09258890', 'pdfurls': ['https://export.arxiv.org/pdf/2305.15935v1.pdf'], 'title': ['Grouping Method for mmWave Massive MIMO System: Exploitation of Angular Multiplexing Gain', 'Grouping Method for mmWave Massive MIMO System: Exploitation of Angular Multiplexing Gain'], 'venue': []} |
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